Ch5 Keys Study Guide Intervention 6e6y4f

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

5

DATE

PERIOD

NAME

DATE

5-1

Anticipation Guide

Study Guide and Intervention Bisectors of Triangles

Before you begin Chapter 5

Read each statement.

•

Decide whether you Agree (A) or Disagree (D) with the statement.

•

Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). STEP 2 A or D

Statement 1. Any point that is on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.

A

2. The circumcenter of a triangle is equidistant from the midpoints of each side of the triangle.

D

4. Three altitudes can be drawn for any one triangle.

A A

5. A median of a triangle is any segment that contains the midpoint of a side of the triangle.

D

6. The measure of an exterior angle of a triangle is always greater than the measures of either of its corresponding remote interior angles.

A

7. The longest side in a triangle is opposite the smallest angle in that triangle.

D

8. To write an indirect proof that two lines are perpendicular, begin by assuming the two lines are not perpendicular.

A


A1

9. The length of the longest side of a triangle is always greater than the sum of the lengths of the other two sides.

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Circumcenter Theorem

The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.

(

A

Glencoe Geometry

Reread each statement and complete the last column by entering an A or a D.

•

Did any of your opinions about the statements change from the first column?

•

For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

5x - 6

Glencoe Geometry

B

D 3x + 8 A

.

,

AD = DC 3x + 8 = 5x - 6 14 = 2x 7=x

−−− −− FK is the perpendicular bisector of GM. FG = FM 2.8 = FM

Exercises Find each measure. 1. XW

2. BF : 7.5

" 4.2 # '

5

;

5

19

8

19

9 & &

7.5

4.2

Point P is the circumcenter of EMK. List any segment(s) congruent to each segment below. −−− −− 3. MY YE

−− −− −− 4. KP MP, EP

−−− −− 5. MN NK

−−− −− 6. ER RK

Chapter 5

6/6/08 12:51:46 001_024_GEOCRMC05_890514.indd PM 5

Answers

−− Example 2 BD is the perpendicular −− bisector of AC. Find x.

Find the measure of FM.

C

•

001_024_GEOCRMC05_890514.indd 3

Converse of Perpendicular Bisector Theorem

2.8

After you complete Chapter 5

3

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Example 1

D

10. In two triangles, if two pairs of sides are congruent, then the measure of the included angles determines which triangle has the longer third side.

Chapter 5

Perpendicular Bisector Theorem

'

3. The altitudes of a triangle meet at the orthocenter.

Step 2

A perpendicular bisector is a line, segment, or ray that is perpendicular to the given segment and es through its midpoint. Some theorems deal with perpendicular bisectors.


STEP 1 A, D, or NS

Perpendicular Bisector

5

:

3 1

.

/

,

Glencoe Geometry

4/11/08 8:13:29 AM

Answers (Anticipation Guide and Lesson 5-1)

•

Chapter Resources

Relationships in Triangles Step 1

PERIOD

Lesson 5-1

Chapter 5

NAME

DATE

5-1

PERIOD

Study Guide and Intervention

NAME

DATE

5-1

(continued)

Skills Practice

Bisectors of Triangles

Bisectors of Triangles Find each measure.

Angle Bisectors

Another special segment, ray, or line is an angle bisector, which divides an angle into two congruent angles. Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

Converse of Angle Bisector Theorem

If a point in the interior of an angle if equidistant from the sides of the angle, then it is on the bisector of the angle.

Incenter Theorem

The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle.

1. FG

2. KL 3

'

13

5x - 17

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&

-

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,

28

4. ∠LYF 5 -

4

2

60


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8" 8

47°

#

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43

47 4. ∠EWL 8

7 .

(7x + 5)° (3x + 21)°

+

&

5

2x + 1 3x - 8

-

,

19

33

)

5. MU 5 7. ∠PHU 21 Chapter 5

.

6. ∠UGM 28 8. HU 13

6

12

21°

5

28°

#

3

(2x + 5)°

6 1

7x

* .

7

8

11 "

Point P is the circumcenter of ABC. List any segment(s) congruent to each segment below.

3

−−− −− 7. BR AR −− 8. CS −− 9. BP

4 1

−− AS −− −− AP,

# (3x + 2)°

5 1

$ (4x - 9)°

Point A is the incenter of PQR. Find each measure below. 6

5

10. ∠ARU 40

20° "

12. ∠QPK 35

V (

19° 19°

(4x - 1)° :

2x + 5

11. AU 20

1

2

40°

,

3

:

Glencoe Geometry

001_024_GEOCRMC05_890514.indd 6

Chapter 5

4/11/08001_024_GEOCRMC05_890514.indd 8:13:37 AM 7


Glencoe Geometry

Point U is the incenter of GHY. Find each measure below.


A2

2. ∠YBA "

3. MK

6. ∠MYW 5

Find each measure.

:

58

5. IU

Exercises 1. ∠ABE

58°

3

is the angle bisector of ∠NMP, so m∠1 = m∠2. MR 5x + 8 = 8x - 16 24 = 3x 8=x

43°

'

5x - 30

P

#

+

6

7

Glencoe Geometry

6/6/08 12:52:24 PM

Answers (Lesson 5-1)

2x + 24

R 1

M

4.2

3. TU

⎯⎯ is the angle bisector of ∠NMP. Find x if m∠1 = 5x + 8 MR and m∠2 = 8x - 16.

.

4.2

3x + 1

13

Example

N

PERIOD

Lesson 5-1

Chapter 5

NAME

Chapter 5

NAME

5-1

DATE

PERIOD

NAME

DATE

5-2

Enrichment

PERIOD

Study Guide and Intervention Medians and Altitudes of Triangles

Medians

Inscribed and Circumscribed Circles

A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle intersect at the centroid of the triangle. The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

The three angle bisectors of a triangle intersect in a single point called the incenter. This point is the center of a circle that just touches the three sides of the triangle. Except for the three points where the circle touches the sides, the circle is inside the triangle. The circle is said to be inscribed in the triangle.

Example

"

2 BK BU = − 3

3

,

2 16 = − BK

6

3

A R

24 = BK

#

$

4

Q

B

Lesson 5-2

BU + UK = BK 16 + UK = 24 UK = 8

Construct the inscribed circle in each triangle. 2.

In ABC, U is the centroid and

BU = 16. Find UK and BK.

P

3.

Exercises

G

4. Follow the steps below to construct the circumscribed circle for FGH. −−− −−− Step 1 Construct the perpendicular bisectors of FG and FH. Use the letter A to label the point where the perpendicular bisectors meet. −− Step 2 Draw the circle that has center A and radius AF.

A F

H

Construct the circumscribed circle for each triangle. 5.

10

Glencoe Geometry

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1. UD 8

$

2. EU 6

&

% #

12

6 16

'

3. CU 12

4. AD 24

5. UF 6

6. BE 18

"

&

In CDE, U is the centroid, UK = 12, EM = 21, and UD = 9. Find each measure. 7. CU 24

8. MU 7

+ 6

9. CK 36

10. JU 4.5

11. EU 14

12. JD 13.5

Chapter 5

11

12

,

9

$ .

4/11/08001_024_GEOCRMC05_890514.indd 8:14:07 AM 11


Glencoe Geometry

Chapter 5

6.


The three perpendicular bisectors of the sides of a triangle also meet in a single point. This point is the center of the circumscribed circle, which es through each vertex of the triangle. Except for the three points where the circle touches the triangle, the circle is outside the triangle.


A4

In ABC, AU = 16, BU = 12, and CF = 18. Find each measure.

%

Glencoe Geometry

4/11/08 8:14:15 AM

Answers (Lesson 5-1 and Lesson 5-2)

1. With a com and a straightedge, construct the inscribed circle for PQR by following the steps below. Step 1 Construct the bisectors of ∠R and ∠Q. Label the point where the bisectors meet, A. −−− Step 2 Construct a perpendicular segment from A to RQ. Use the letter B to label the point where the perpendicular −−− segment intersects RQ. Step 3 Use a com to draw the circle with center at A and −− radius AB .


DATE

5-2


PERIOD

NAME

5-2

(continued)

Medians and Altitudes of Triangles

1. KM

# (7, 7)

3. LK

(9, 3) $

(1, 3)

2

A5

Distributive Property Simplify.

2

Distributive Property

2

2

5 33 3 1 − x+− =-− x+− 2

2

2

1 Subtract − x from each side.

2

2

5 33 − = −2x + − 2

33 Subtract − from each side.

2

2

−14 = −2x 7=x 5 5 7 5 1 1 y=− x+− =− (7) + − =− +− =6 2

2

2

2

2

Divide both sides by -2.

2

The coordinates of the orthocenter of ABC is (6, 7).

Exercises COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle.

Glencoe Geometry

1. J(1, 0), H(6, 0), I(3, 6)

6 4 %

8. HM

&

3

)

8

9. TH

8

5

10. HR

12

11. TD

12. ER

12

18

COORDINATE GEOMETRY Find the coordinates of the centroid of each triangle. 13. X(−3, 15) Y(1, 5), Z(5, 10)

14. S(2, 5), T(6, 5), R(10, 0)

(6, 3 −13 )

(1, 10)

COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle. 15. L(8, 0), M(10, 8), N(14, 0)

16. D(−9, 9), E(−6, 6), F(0, 6)

(10, 1)

2. S(1, 0), T(4, 7), U(8, −3)

(-9, -3)

5 3 − ,− 2 2

(3, 1)

Chapter 5

6. PM

3

.

33 3 y = -− x+−

2

5. NK

16

Simplify. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

2

4. LR

4.5

7. SH

3 m = -− , (x1, y1) = C(9, 3)

Solve the system of equations and find where the altitudes meet. 5 1 y=− x+−

3

In STR, H is the centroid, EH = 6, DH = 4, and SM = 24. Find each length.

Point-slope form


1 m=− , (x1, y1) = A(1, 3)

3 y - 3 = -− (x - 9) 2 3 27 y - 3 = -−x + − 2 2 33 3 y = - −x + − 2 2

.

4

1.5

2

2

1 y-3=− (x – 1) 2 1 1 y - 3 = −x – − 2 2 5 1 y = −x + − 2 2

4 ,

12

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Glencoe Geometry

Chapter 5

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Answers

13

Glencoe Geometry

6/6/08 12:53:03 PM


−− C to AB. −− 2 , then the altitude has a If AB has a slope of − 3 3 slope of - − . y - y1 = m(x - x1)

Point-slope form

-

2. KQ

/

"

0

y - y1 = m(x – x1)

2

1

2

x

1 . has a slope of − 2

Skills Practice Medians and Altitudes of Triangles

y

Find the equation of the altitude from −−− A to BC. −−− If BC has a slope of −2, then the altitude

PERIOD

In PQR, NQ = 6, RK = 3, and PK = 4. Find each length.

Altitudes An altitude of a triangle is a segment from a vertex to the line containing the opposite side meeting at a right angle. Every triangle has three altitudes which meet at a point called the orthocenter. Example The vertices of ABC are A(1, 3), B(7, 7) and C(9, 3). Find the coordinates of the orthocenter of ABC. Find the point where two of the three altitudes intersect.

DATE

Lesson 5-2

Chapter 5

NAME

5-2

DATE

PERIOD

NAME

5-2

Practice Medians and Altitudes of Triangles 2. FP

15

13

Word Problem Practice

E B

1. BALANCING Johanna balanced a triangle flat on her finger tip. What point of the triangle must Johanna be touching?

C 18

30

F

P

4. MEDIANS Look at the right triangle below. What do you notice about the orthocenter and the vertices of the triangle?

D

3. BP

4. CD

26

45

5. PA

6. EA

36

54

6

.

/

9. MR

10. ZV

:

5. PLAZAS An architect is deg a triangular plaza. For aesthetic purposes, the architect pays special attention to the location of the centroid C and the circumcenter O.

24

A6

11. NV

12. IZ

36

12

*

3

7

13. I(3, 1), J(6, 3), K(3, 5)

14. H(0, 1), U(4, 3), P(2, 5)

(4, 3)

(2, 3)

COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle. 15. P(-1, 2), Q(5, 2), R(2, 1)

16. S(0, 0), T(3, 3), U(3, 6)

(2, -1)

(0, 9)

17. MOBILES Nabuko wants to construct a mobile out of flat triangles so that the surfaces of the triangles hang parallel to the floor when the mobile is suspended. How can Nabuko be certain that she hangs the triangles to achieve this effect?

She needs to hang each triangle from its center of gravity or centroid, which is the point at which the three medians of the triangle intersect.

Glencoe Geometry

001_024_GEOCRMC05_890514.indd 14

a. Give an example of a triangular plaza where C = O. If no such example exists, state that this is impossible.

3. DISTANCES For what kind of triangle is there a point where the distance to each side is half the distance to each vertex? Explain.

an equilateral triangle

equilateral: incenter = centroid = circumcenter

b. Give an example of a triangular plaza where C is inside the plaza and O is outside the plaza. If no such example exists, state that this is impossible.

an obtuse triangle c. Give an example of a triangular plaza where C is outside the plaza and O is inside the plaza. If no such example exists, state that this is impossible.

impossible

Chapter 5

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Glencoe Geometry

14


COORDINATE GEOMETRY Find the coordinates of the centroid of each triangle.

Chapter 5

The orthocenter coincides with one of the vertices.

orthocenter

;


9

2. REFLECTIONS Part of the working space in Paulette’s loft is partitioned in the shape of a nearly equilateral triangle with mirrors hanging on all three partitions. From which point could someone see the opposite corner behind his or her reflection in any of the three mirrors?

15

Glencoe Geometry

4/11/08 8:14:33 AM


8. YZ

3

centroid

A

In MIV, Z is the centroid, MZ = 6, YI = 18, and NZ = 12. Find each measure. 7. ZR

PERIOD

Medians and Altitudes of Triangles

In ABC, = 30, EP = 18, and BF = 39. Find each length. 1. PD

DATE

Lesson 5-2

Chapter 5

NAME


Chapter 5

NAME

5-2

DATE

PERIOD

NAME

5-3

Enrichment

DATE

PERIOD

Study Guide and Intervention Inequalities in One Triangle

Angle Inequalities

Constructing Centroids and Orthocenters

Properties of inequalities, including the Transitive, Addition, and Subtraction Properties of Inequality, can be used with measures of angles and segments. There is also a Comparison Property of Inequality. For any real numbers a and b, either a < b, a = b, or a > b.

The three medians of a triangle intersect at a single point called the centroid. You can use a straightedge and com to find the centroid of a triangle. 6

1. With a straightedge and com, construct the centroid for STU by following the steps below.

The Exterior Angle Inequality Theorem can be used to prove this inequality involving an exterior angle. B 6 #

Construct the centroid of each triangle. 2.

5

4

Exterior Angle Inequality Theorem

"

)

The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles.

5

4

1

A

C

D

m∠1 > m∠A, m∠1 > m∠B

3. Example List all angles of EFG whose measures are less than m∠1.

) )

G 4 1 2

The measure of an exterior angle is greater than the measure of either remote interior angle. So m∠3 < m∠1 and m∠4 < m∠1.

3

E

H

F

Step 1 Extend segments CD and DE past point D long enough to meet perpendiculars from E and C as shown. Step 2 Construct the perpendicular from point C to the line DE and label the point of intersection X. Likewise, label the point of intersection of line CD with the perpendicular from E as point Z. In this case both X and Z lie outside CDE. Step 3 Label O the point where perpendiculars and EZ intersect. This is the CX orthocenter of CDE.

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&

$ 9 % 0

&

;

Construct the orthocenter of each triangle. 6.

5.

Glencoe Geometry

0

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

16

2. measures are greater than m∠3 ∠1, ∠5 3. measures are less than m∠1 ∠5, ∠6

U 3 5

4. measures are greater than m∠1 ∠7

7

X

5. measures are less than m∠7 ∠1, ∠3, ∠5, ∠6, ∠TUV

001_024_GEOCRMC05_890514.indd 16

7. measures are greater than m∠5 ∠1, ∠7, ∠TUV

V

S 8

8. measures are less than m∠4 ∠2, ∠3

N 7

Q

Chapter 5

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Answers

6

1 4

2

T W Exercises 3–8


1

R

Glencoe Geometry

5

4

J K Exercises 1–2

M


9. measures are less than m∠1 ∠4, ∠5, ∠7, ∠NPR

0

3

1 2

10. measures are greater than m∠4 ∠1, ∠8, ∠OPN, ∠ROQ

Chapter 5

L

Lesson 5-3

$


4. Follow the steps below to construct the orthocenter of CDE using a straightedge and com.


A7

Exercises The three altitudes of a triangle meet in a single point called the orthocenter of the triangle.

17

2

3

6

5 4

O Exercises 9–10

P

Glencoe Geometry

4/11/08 8:14:46 AM


Step 1 Locate the midpoints of sides TU and SU. Label the midpoints A and B respectively. Step 2 Draw the segments SA and TB. Use the letter H to label their point of intersection, which is the centroid of STU.


Chapter 5

NAME

5-2

DATE

PERIOD

NAME

5-3

Enrichment

DATE

PERIOD

Study Guide and Intervention Inequalities in One Triangle

Angle Inequalities

Constructing Centroids and Orthocenters

Properties of inequalities, including the Transitive, Addition, and Subtraction Properties of Inequality, can be used with measures of angles and segments. There is also a Comparison Property of Inequality. For any real numbers a and b, either a < b, a = b, or a > b.

The three medians of a triangle intersect at a single point called the centroid. You can use a straightedge and com to find the centroid of a triangle. 6

1. With a straightedge and com, construct the centroid for STU by following the steps below.

The Exterior Angle Inequality Theorem can be used to prove this inequality involving an exterior angle. B 6 #

Construct the centroid of each triangle. 2.

5

4

Exterior Angle Inequality Theorem

"

)

The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles.

5

4

1

A

C

D

m∠1 > m∠A, m∠1 > m∠B

3. Example List all angles of EFG whose measures are less than m∠1.

) )

G 4 1 2

The measure of an exterior angle is greater than the measure of either remote interior angle. So m∠3 < m∠1 and m∠4 < m∠1.

3

E

H

F

Step 1 Extend segments CD and DE past point D long enough to meet perpendiculars from E and C as shown. Step 2 Construct the perpendicular from point C to the line DE and label the point of intersection X. Likewise, label the point of intersection of line CD with the perpendicular from E as point Z. In this case both X and Z lie outside CDE. Step 3 Label O the point where perpendiculars and EZ intersect. This is the CX orthocenter of CDE.

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&

$ 9 % 0

&

;

Construct the orthocenter of each triangle. 6.

5.

Glencoe Geometry

0


16

2. measures are greater than m∠3 ∠1, ∠5 3. measures are less than m∠1 ∠5, ∠6

U 3 5


7

X

5. measures are less than m∠7 ∠1, ∠3, ∠5, ∠6, ∠TUV

001_024_GEOCRMC05_890514.indd 16

7. measures are greater than m∠5 ∠1, ∠7, ∠TUV

V

S 8


N 7

Q

Chapter 5

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Answers

6

1 4

2

T W Exercises 3–8


1

R

Glencoe Geometry

5

4

J K Exercises 1–2

M


9. measures are less than m∠1 ∠4, ∠5, ∠7, ∠NPR

0

3

1 2

10. measures are greater than m∠4 ∠1, ∠8, ∠OPN, ∠ROQ

Chapter 5

L

Lesson 5-3

$


4. Follow the steps below to construct the orthocenter of CDE using a straightedge and com.


A7

Exercises The three altitudes of a triangle meet in a single point called the orthocenter of the triangle.

17

2

3

6

5 4

O Exercises 9–10

P

Glencoe Geometry

4/11/08 8:14:46 AM


Step 1 Locate the midpoints of sides TU and SU. Label the midpoints A and B respectively. Step 2 Draw the segments SA and TB. Use the letter H to label their point of intersection, which is the centroid of STU.

DATE

5-3

PERIOD


NAME

DATE

5-3

(continued)

Skills Practice

Inequalities in One Triangle


Angle-Side Relationships

When the sides of triangles are not congruent, there is a relationship between the sides and angles of the triangles.


A

• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.

B

C

1. measures less than m∠1

If AC > AB, then m∠B > m∠C.

20°

A

−−− −− −− CB, AB, AC

∠T, ∠R, ∠S

∠1, ∠3 ∠1, ∠3, ∠5

125°

B

List the angles and sides of each triangle in order from smallest to largest. 5. 3

2.

35 cm

80°

23.7 cm

R

S

∠T, ∠R, ∠S −− −− −− RS, ST, RT 6 14

4

40°

60°

5.

A

T

#

5

11

∠S, ∠U, ∠T, −− −− −− UT, ST, SU

6. 12

$

8

8.

3

4

1

∠B, ∠C, ∠A, −− −− −− AC, BA, CB

7. $

C

4.0

∠C, ∠B, ∠A −− −− −− AB, AC, BC

" 5

5

4.3

3.8

∠T, ∠R, ∠S −− −− −− RS, ST, RT

4.

1

B

2

20

∠Q, ∠P, ∠R, −− −− −− PR, RQ, QP 9.

:

3

35° 120°

25°

& 9

∠E, ∠C, ∠D, −− −− −− CD, DE, CE Chapter 5

56°

58°

∠X, ∠Z, ∠Y, −− −− −− YZ, XY, XZ 18

;

4

60°

54°

98°

2

5

-

−− −− −− ∠Q, ∠R, ∠S, RP, PQ, RQ 7.

−− −− −− ∠K, ∠M, ∠L, ML, KL, KM 8.

)

9 39

16

9

: '

38

(

15

34

;

−− −− −− ∠F, ∠H, ∠G, HG, FG, FH 9. #

−− −− −− ∠X, ∠Y, ∠Z, YZ, XZ, XY 10.

5 98°

42°

4

"

5

∠T, ∠S, ∠R, −− −− −− RS, RT, ST Glencoe Geometry

001_024_GEOCRMC05_890514.indd 18

43°

$

6

−− −− −− ∠A, ∠B, ∠C, BC, AC, AB Chapter 5

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Glencoe Geometry

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48 cm

3.


A8

1.

.

24°

2

List the angles and sides in order from smallest to largest.

T

6. , 6

S

9

4. measures greater than m∠8

Exercises R

8

∠2, ∠4, ∠6, ∠7

35°

T

9 cm

5 6

−− −− −− ∠S, ∠U, ∠T, UT, ST, SU 19

Glencoe Geometry

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R

3

3. measures greater than m∠5

C

7 cm

1

2. measures less than m∠9

Example 2 List the sides in order from shortest to longest.

S 6 cm

2 4 7

∠2, ∠3, ∠4, ∠5, ∠7, ∠8

If m∠A > m∠C, then BC > AB.

• If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. Example 1 List the angles in order from smallest to largest measure.

PERIOD

Lesson 5-3

Chapter 5

NAME


5-3

DATE

PERIOD

NAME

5-3

Practice Inequalities in One Triangle 2. ∠4, ∠8, ∠9

∠1

3

∠4

3. ∠2, ∠3, ∠7

8 4

4. ∠7, ∠8, ∠10

∠7

1

∠10

7

2

10 9

6 5

2. OBTUSE TRIANGLES Don notices that the side opposite the right angle in a right triangle is always the longest of the three sides. Is this also true of the side opposite the obtuse angle in an obtuse triangle? Explain.

1

2

6. measures are less than m∠3

3 5 6

4

7

∠5, ∠7, ∠8

8

∠1, ∠3, ∠5, ∠9

m∠RST > m∠TRS

10. m∠RTW, m∠TWR

−− −− DH > GH

−−− −−− 15. EG, FG

Glencoe Geometry

−− −− EG < FG

44 35

34 45

W

14

S T

22

12. m∠WQR, m∠QRW

m∠WQR < m∠QRW D

E 48°

113°

F

−−− −−− 14. DE, DG

−− −− DE < DG

H

120°

17°

32°

G

−−− −−− 16. DE, EG

−− −− DE > EG

17. SPORTS The figure shows the position of three trees on one part of a Frisbee™ course. At which tree position is the angle between the trees the greatest? 2

Chapter 5

Q

m∠RTW < m∠TWR

Use the figure at the right to determine the relationship between the lengths of the given sides. −−− −−− 13. DH, GH

R 47

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2 40 ft 3

37.5 ft 53 ft

1

Glencoe Geometry


∠6, ∠9


A9

8. measures are greater than m∠2

2, 1, 3 Dallas 5. CITIES Stella is going to Texas to visit a friend. 64˚ As she was looking at 59˚ a map to see where Abilene she might want to go, she noticed the cities Austin Austin, Dallas, and Abilene formed a triangle. She wanted to determine how the distances between the cities were related, so she used a protractor to measure two angles.

M string

a. Based on the information in the figure, which of the two cities are nearest to each other?

Sample answer: The string divides the triangle in two; one of these triangles is right or obtuse because one side of the string must make a right or obtuse angle with the stick. In this triangle, the side opposite the right or obtuse angle is longer than the string and that side is also a side of the triangle.

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Answers

54˚

3

3. STRING Jake built a triangular structure with three black sticks. He tied one end of a string to vertex M and the other end to a point on the stick opposite M, pulling the string taut. Prove that the length of the string cannot exceed the longer of the two sides of the structure.

Chapter 5

2

47˚

Yes. Since an obtuse triangle only has 1 obtuse angle and 2 acute angles, the side opposite the obtuse angle is the longest side.

9

7. measures are greater than m∠7

Use the figure at the right to determine the relationship between the measures of the given angles.

1

Dallas and Abilene b. Based on the information in the figure, which of the two cities are farthest apart from each other?

Abilene and Austin

21

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∠3, ∠4, ∠5, ∠7, ∠8

11. m∠RST, m∠TRS

4. SQUARES Matthew has three different squares. He arranges the squares to form a triangle as shown. Based on the information, list the squares in order from the one with the smallest perimeter to the one with the largest perimeter.

They are equal.

5. measures are less than m∠1

m∠QRW < m∠RWQ


1. DISTANCE Carl and Rose live on the same straight road. From their balconies they can see a flagpole in the distance. The angle that each person’s line of sight to the flagpole makes with the road is the same. How do their distances from the flagpole compare?

Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition.

9. m∠QRW, m∠RWQ

PERIOD


Use the figure at the right to determine which angle has the greatest measure. 1. ∠1, ∠3, ∠4

DATE

Lesson 5-3

Chapter 5

NAME

Chapter 5

NAME

DATE

5-3

PERIOD

NAME

5-3

Enrichment

DATE

PERIOD

Graphing Calculator Activity Cabri Junior: Inequalities in One Triangle

The diagram below shows segment AB adjacent to a closed region. The problem requires that you construct another segment XY to the right of the closed region such that points A, B, X, and Y are collinear. You are not allowed to touch or cross the closed region with your com or straightedge.

Q D

n

R

m

E

Existing Road

B

Closed Region (Lake)

T X Y

A10

−− 1. Construct the perpendicular bisector of AB. Label the midpoint as point C, and the line as m. 2. Mark two points P and Q on line m that lie well above the closed region. Construct the −−− perpendicular bisector, n, of PQ. Label the intersection of lines m and n as point D. 3. Mark points R and S on line n that lie well to the right of the closed region. Construct −− the perpendicular bisector, k , of RS. Label the intersection of lines n and k as point E. 4. Mark point X on line

k so that X is below line n

−−− −− and so that EX is congruent to DC.

−− −− 5. Mark points T and V on line k and on opposite sides of X, so that XT and XV are −− congruent. Construct the perpendicular bisector, , of TV. Call the point where the −− line hits the boundary of the closed region point Y. XY corresponds to the new road.


Follow these instructions to construct a segment XY so that it is collinear with segment AB.


V

Exercises Analyze your drawing. 1. What is the relationship between m∠ACD and m∠ ABC? m∠ ACD and m∠BAC?

Sample answer: m∠ACD > m∠ABC; m∠ACD > m∠BAC 2. Make a conjecture about the relationship between the measures of an exterior angle (∠ ACD) and its two remote interior angles (∠ ABC and ∠ BAC).

The measure of an exterior angle is equal to the sum of the measure of the two remote interior angles. 3. Change the dimensions of the triangle by moving point A. (Press CLEAR so the pointer becomes a black arrow. Move the pointer close to point A until the arrow becomes transparent and point A is blinking. Press ALPHA to change the arrow to a hand. Then move the point.) Is your conjecture still true? yes 4. Which side of the triangle is the longest? the shortest? See students’ work.

See students’ work. 6. Make a conjecture about where the longest side is in relationship to the greatest angle and where the shortest side is in relationship to the least angle.

The longest side is opposite the greatest angle. The shortest side is opposite the least angle. Chapter 5

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5. Which angle measure (not including the exterior angle) is the greatest? the least?

23

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C A

S

k

P

Cabri Junior can be used to investigate the relationships between angles and sides of a triangle. Step 1 Use Cabri Junior. to draw and label a triangle. • Select F2 Triangle to draw a triangle. • Move the cursor to where you want the first vertex. Press ENTER . • Repeat this procedure to determine the next two vertices of the triangle. • Select F5 Alph-num to label each vertex. • Move the cursor to a vertex, press ENTER , enter A, and press ENTER again. • Repeat this procedure to label vertex B and vertex C. Step 2 Draw an exterior angle of ABC. −−− • Select F2 Line to draw a line through BC. so that C is between B and • Select F2 Point, Point on to draw a point on BC the new point. • Select F5 Alph-num to label the point D. Step 3 Find the measures of the three interior angles and the exterior angle, ∠ ACD. • Select F5 Measure, Angle. • To find the measure of ∠ ABC, select points A, B, and C (with the vertex B as the second point selected). • Repeat to find the remaining angle measures. Step 4 Find the measure of each side of ABC. • Select F5 Measure, D. & Length. −− • To find the length of AB, select point A and then select point B. −−− −− • Repeat this procedure to find the lengths of BC and CA.

Lesson 5-3

Construction Problem


DATE

5-3

PERIOD

NAME

DATE

5-4

Geometer’s Sketchpad Activity Inequalities in One Triangle

Analyze your drawing. 1. What is the relationship between m∠BCD and m∠ ABC? m∠BCD and m∠ BAC?

Sample answer: m∠BCD > m∠ABC; m∠BCD > m∠BAC 2. Make a conjecture about the relationship between the measures of an exterior angle (∠BCD) and its two remote interior angles (∠ABC and ∠ BAC).

The measure of an exterior angle is equal to the sum of the measure of the two remote interior angles. 3. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true? yes 4. Which side of the triangle is the longest? the shortest? See students’ work.

Glencoe Geometry


See students’ work.

1. Assume that the conclusion is false by assuming the oppposite is true. 2. Show that this assumption leads to a contradiction of the hypothesis or some other fact. 3. Point out that the assumption must be false, and therefore, the conclusion must be true.

Example

Given: 3x + 5 > 8 Prove: x > 1

Step 1 Assume that x is not greater than 1. That is, x = 1 or x < 1. Step 2 Make a table for several possibilities for x = 1 or x < 1. When x = 1 or x < 1, then 3x + 5 is not greater than 8. Step 3 This contradicts the given information that 3x + 5 > 8. The assumption that x is not greater than 1 must be false, which means that the statement “x > 1” must be true.

x 1

3x + 5 8

0

5

-1

2

-2

-1

-3

-4

Exercises Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Exercises

Steps for Writing an Indirect Proof


A11 Step 3

Use The Geometer’s Sketchpad to draw a triangle and one exterior angle. • Construct a ray by selecting the Ray tool from the toolbar. First, click where you want the first point. Then click a second point to m⬔ABC 69.29˚ m⬔BCA 55.92˚ draw the ray. m⬔BAC 54.78˚ • Next, select the Segment tool from B m⬔BCD 124.08˚ the toolbar. Use the endpoint of AB 2.20 cm the ray as the first point for the BC 2.17 cm segment and click on a second AC 2.49 cm A C D point to construct the segment. • Construct another segment ing the second point of the previous segment to a point on the ray. • Display the labels for each point. Use the Selection Arrow tool to select all four points. Display the labels by selecting Show Label from the Display menu. Find the measures of each angle. • To find the measure of ∠ABC, use the Selection Arrow tool to select points A, B, and C (with the vertex B as the second point selected). Then, under the Measure menu, select Angle. Use this method to find the remaining angle measures, including the exterior angle, ∠BCD. Find the measures of each side of the triangle. • To find the measure of side AB, select A and then B. Next, under the Measure menu, select Distance. Use this method to find the length of the other two sides.

Indirect Algebraic Proof One way to prove that a statement is true is to temporarily assume that what you are trying to prove is false. By showing this assumption to be logically impossible, you prove your assumption false and the original conclusion true. This is known as an indirect proof.

State the assumption you would make to start an indirect proof of each statement. 1. If 2x > 14, then x > 7. x ≤ 7 2. For all real numbers, if a + b > c, then a > c - b. a ≤ c - b Complete the indirect proof. Given: n is an integer and n2 is even. Prove: n is even. 3. Assume that n is not even. That is, assume n is odd. 4. Then n can be expressed as 2a + 1 by the meaning of odd number. 2 5. n2 = (2a + 1)

Substitution

6.

= (2a + 1)(2a + 1)

Multiply.

7.

2 = 4a + 4a + 1

Simplify.

8.

= 2(2a2 + 2a) + 1 Distributive Property

9. 2(2a2 + 2a)+ 1 is an odd number. This contradicts the given that n2 is even, so the assumption must be false.

6. Make a conjecture about where the longest side is in relationship to the greatest angle and where the shortest side is in relationship to the least angle.

10. Therefore, n is even.


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Step 2

Study Guide and Intervention Indirect Proof

The Geometer’s Sketchpad can be used to investigate the relationships between angles and sides of a triangle. Step 1

PERIOD

Lesson 5-4

Chapter 5

NAME


DATE

5-3

PERIOD

NAME

DATE

5-4

Geometer’s Sketchpad Activity Inequalities in One Triangle

Analyze your drawing. 1. What is the relationship between m∠BCD and m∠ ABC? m∠BCD and m∠ BAC?

Sample answer: m∠BCD > m∠ABC; m∠BCD > m∠BAC 2. Make a conjecture about the relationship between the measures of an exterior angle (∠BCD) and its two remote interior angles (∠ABC and ∠ BAC).

The measure of an exterior angle is equal to the sum of the measure of the two remote interior angles. 3. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true? yes 4. Which side of the triangle is the longest? the shortest? See students’ work.

Glencoe Geometry


See students’ work.

1. Assume that the conclusion is false by assuming the oppposite is true. 2. Show that this assumption leads to a contradiction of the hypothesis or some other fact. 3. Point out that the assumption must be false, and therefore, the conclusion must be true.

Example

Given: 3x + 5 > 8 Prove: x > 1

Step 1 Assume that x is not greater than 1. That is, x = 1 or x < 1. Step 2 Make a table for several possibilities for x = 1 or x < 1. When x = 1 or x < 1, then 3x + 5 is not greater than 8. Step 3 This contradicts the given information that 3x + 5 > 8. The assumption that x is not greater than 1 must be false, which means that the statement “x > 1” must be true.

x 1

3x + 5 8

0

5

-1

2

-2

-1

-3

-4

Exercises Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Exercises

Steps for Writing an Indirect Proof


A11 Step 3

Use The Geometer’s Sketchpad to draw a triangle and one exterior angle. • Construct a ray by selecting the Ray tool from the toolbar. First, click where you want the first point. Then click a second point to m⬔ABC 69.29˚ m⬔BCA 55.92˚ draw the ray. m⬔BAC 54.78˚ • Next, select the Segment tool from B m⬔BCD 124.08˚ the toolbar. Use the endpoint of AB 2.20 cm the ray as the first point for the BC 2.17 cm segment and click on a second AC 2.49 cm A C D point to construct the segment. • Construct another segment ing the second point of the previous segment to a point on the ray. • Display the labels for each point. Use the Selection Arrow tool to select all four points. Display the labels by selecting Show Label from the Display menu. Find the measures of each angle. • To find the measure of ∠ABC, use the Selection Arrow tool to select points A, B, and C (with the vertex B as the second point selected). Then, under the Measure menu, select Angle. Use this method to find the remaining angle measures, including the exterior angle, ∠BCD. Find the measures of each side of the triangle. • To find the measure of side AB, select A and then B. Next, under the Measure menu, select Distance. Use this method to find the length of the other two sides.

Indirect Algebraic Proof One way to prove that a statement is true is to temporarily assume that what you are trying to prove is false. By showing this assumption to be logically impossible, you prove your assumption false and the original conclusion true. This is known as an indirect proof.

State the assumption you would make to start an indirect proof of each statement. 1. If 2x > 14, then x > 7. x ≤ 7 2. For all real numbers, if a + b > c, then a > c - b. a ≤ c - b Complete the indirect proof. Given: n is an integer and n2 is even. Prove: n is even. 3. Assume that n is not even. That is, assume n is odd. 4. Then n can be expressed as 2a + 1 by the meaning of odd number. 2 5. n2 = (2a + 1)

Substitution

6.

= (2a + 1)(2a + 1)

Multiply.

7.

2 = 4a + 4a + 1

Simplify.

8.

= 2(2a2 + 2a) + 1 Distributive Property

9. 2(2a2 + 2a)+ 1 is an odd number. This contradicts the given that n2 is even, so the assumption must be false.

6. Make a conjecture about where the longest side is in relationship to the greatest angle and where the shortest side is in relationship to the least angle.

10. Therefore, n is even.


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Glencoe Geometry

4/11/08 8:15:35 AM


Step 2

Study Guide and Intervention Indirect Proof

The Geometer’s Sketchpad can be used to investigate the relationships between angles and sides of a triangle. Step 1

PERIOD

Lesson 5-4

Chapter 5

NAME

Chapter 5

NAME

5-4

DATE


PERIOD

NAME

5-4

(continued)

Indirect Proof

DATE

PERIOD

Skills Practice Indirect Proof

State the assumption you would make to start an indirect proof of each statement.

Indirect Proof with Geometry

To write an indirect proof in geometry, you assume that the conclusion is false. Then you show that the assumption leads to a contradiction. The contradiction shows that the conclusion cannot be false, so it must be true.

1. m∠ABC < m∠CBA

m∠ABC ≥ m∠CBA

Example A

Given: m∠C = 100 Prove: ∠A is not a right angle.

2. DEF RST

B

DEF RST C

3. Line a is perpendicular to line b.

Step 1 Assume that ∠A is a right angle.

Line a is not perpendicular to line b.

Step 2 Show that this leads to a contradiction. If ∠A is a right angle, then m∠A = 90 and m∠C + m∠A = 100 + 90 = 190. Thus the sum of the measures of the angles of ABC is greater than 180.

4. ∠5 is supplementary to ∠6.

Write an indirect proof of each statement. 5. Given: x2 + 8 ≤ 12 Prove: x ≤ 2

m∠B ≠ 45 −− −− 2. If AV is not congruent to VE, then AVE is not isosceles.

AVE is isosceles.

−−

−−

3. Assume that DE FE. −−− −−− 4. EG EG

Reflexive Property

5. EDG EFG

SAS

−− −− 6. DG FG

D

G 1

E

2

F

Assume the conclusion is false.

CTC

7. This contradicts the given information, so the assumption must

−−

E

6. Given: ∠D ∠F Prove: DE ≠ EF D

F

Proof: Step 1: Assume DE = EF. −− −− Step 2: If DE = EF, then DE −− EF −−by the definition of congruent segments. But if DE EF, then ∠D ∠F by the Isosceles Triangle Theorem. This contradicts the given information that ∠D ∠F. Step 3: Since the assumption that DE = EF leads to a contradiction, it must be false. Therefore, it must be true that DE ≠ EF.

−−

8. Therefore, DE is not congruent to FE.

Chapter 5

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Glencoe Geometry

be false.


1. If m∠A = 90, then m∠B = 45.

Proof: Step 1: Assume x > 2. Step 2: If x > 2, then x2 > 4. But if x2 > 4, it follows that x2 + 8 > 12. This contradicts the given fact that x2 + 8 ≤ 12. Step 3: Since the assumption of x > 2 leads to a contradiction, it must be false. Therefore, x ≤ 2 must be true.

Lesson 5-4

State the assumption you would make to start an indirect proof of each statement. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

A12

Exercises

Complete the indirect proof. −−− −−− Given: ∠1 ∠2 and DG is not congruent to FG. −−− −− Prove: DE is not congruent to FE.


∠5 is not supplementary to ∠6.

Step 3 The conclusion that the sum of the measures of the angles of ABC is greater than 180 is a contradiction of a known property. The assumption that ∠A is a right angle must be false, which means that the statement “∠A is not a right angle” must be true.

27

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5-4

DATE

PERIOD

NAME

5-4

Practice Indirect Proof

−− BD does not bisect ∠ABC.

RT ≠ TS Write an indirect proof of each statement.

Step 2 If a || b, then the consecutive interior angles ∠2 and ∠3 are supplementary. Thus m∠2 + m∠3 = 180. This contradicts the given statement that m∠2 + m∠3 ≠ 180. Step 3 Since the assumption leads to a contradiction, the statement a || b must be false. Therefore, a ∦ b must be true. 5. PHYSICS Sound travels through air at about 344 meters per second when the temperature is 20°C. If Enrique lives 2 kilometers from the fire station and it takes 5 seconds for the sound of the fire station siren to reach him, how can you prove indirectly that it is not 20°C when Enrique hears the siren?

Glencoe Geometry

Assume that it is 20°C when Enrique hears the siren, then show that at this temperature it will take more than 5 seconds for the sound of the siren to reach him. Since the assumption is false, you will have proved that it is not 20°C when Enrique hears the siren.

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b


A13

Step 2 If x ≤ 3, then -4x ≥ -12. But -4x ≥ -12 implies that -4x + 2 ≥ -10, which contradicts the given inequality. Step 3 Since the assumption that x ≤ 3 leads to a contradiction, it must be true that x > 3. a

Sample answer: Suppose no state has area > 120,000 mi2. Then the total area could not exceed 120,000 × 49 + 11,000 = 5,891,000, a contradiction.

y

B 5

A

O

3. CONSECUTIVE NUMBERS David was trying to find a common factor other than 1 between various pairs of consecutive integers. Write an indirect proof to show David that two consecutive integers do not share a common factor other than 1.

y

2a 2a+1 x − =− is an integer and − =− n n n n 2a+1 2a =− is also an integer. But − n n

1 1 and − is not an integer unless +− n n n = 1, a contradiction. Chapter 5

5

C x

a. Suppose ABC has a lattice point in its interior. Show that the lattice triangle can be partitioned into three smaller lattice triangles.

Sample answer: Assume x and y are integers with a common factor greater than 1. For consecutive integers one is even and the other is odd, so x = 2a and y = 2a + 1, for an integer a. Let n be the common factor greater that 1. Therefore

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Answers

5. LATTICE TRIANGLES A lattice point is a point whose coordinates are both integers. A lattice triangle is a triangle whose vertices are lattice points. It is a fact that a lattice triangle has an area of at least 0.5 square units.

2. AREA The area of the United States is about 6,000,000 square miles. The area of Hawaii is about 11,000 square miles. Use an indirect proof to show that at least one of the fifty states has an area greater than 120,000 square miles.

Proof: Step 1 Assume x ≤ 3.

3

Suppose the letters are distinct and nonconsecutive. Then the alphabet must have at least 14 + 13 or 27 letters, a contradiction.


3. Given: -4x + 2 < -10 Prove: x > 3

1 2

4. WORDS The words accomplishment, counterexample, and extemporaneous all have 14 letters. Use an indirect proof to show that any word with 14 letters must use a repeated letter or have two letters that are consecutive in the alphabet.

Sample answer: Suppose all canoes had ≤ 2 students, then the total would be less than or equal to 17 × 2 = 34, a contradiction.

2. RT = TS

Chapter 5


1. CANOES Thirty-five students went on a canoeing expedition. They rented 17 canoes for the trip. Use an indirect proof to show that at least one canoe had more than two students in it.

−−− 1. BD bisects ∠ABC.

Proof: Step 1 Assume a || b.

PERIOD

Indirect Proof

State the assumption you would make to start an indirect proof of each statement.

4. Given: m∠2 + m∠3 ≠ 180 Prove: a ∦ b

DATE

29

Sample answer in diagram above. b. Prove indirectly that a lattice triangle with area 0.5 square units contains no lattice point. (Being on the boundary does not count as inside.)

Sample answer: From Exercise 5a, the lattice triangle contains 3 smaller lattice triangles, each of which has area at least 0.5 square units. The original would then have area at least 1.5 square units, a contradiction.

Lesson 5-4

Chapter 5

NAME

Glencoe Geometry

4/11/08 8:15:51 AM

Chapter 5

NAME

DATE

5-4

PERIOD

NAME

5-5

Enrichment

DATE

PERIOD

Study Guide and Intervention The Triangle Inequality

More Counterexamples

The Triangle Inequality

If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. This illustrates the Triangle Inequality Theorem.

Some statements in mathematics can be proven false by counterexamples. Consider the following statement.

A

For any numbers a and b, a - b = b - a. Triangle Inequality Theorem

You can prove that this statement is false in general if you can ﬁnd one example for which the statement is false.

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

B

Example The measures of two sides of a triangle are 5 and 8. Find a range for the length of the third side.

In general, for any numbers a and b, the statement a - b = b - a is false. You can make the equivalent verbal statement: subtraction is not a commutative operation.

By the Triangle Inequality Theorem, all three of the following inequalities must be true. 5+x>8 8+x>5 5+8>x x>3 x > -3 13 > x Therefore x must be between 3 and 13.

In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample.

Sample answers are given. 2. a ÷ (b ÷ c) (a ÷ b) ÷ c

6 ÷ (4 ÷ 2) (6 ÷ 4) ÷ 2

Exercises

6 1.5 − − 2

Is it possible to form a triangle with the given side lengths? If not, explain why not.

2

3 ≠ 0.75 4. a ÷ (b + c) (a ÷ b) + (a ÷ c)

6÷44÷6

6 ÷ (4 + 2) (6 ÷ 4) +(6 ÷ 2)

3 2 − ≠− 2 3

6 ÷ 6 1.5 + 3 1 ≠ 4.5 6. a2 + a2 a4

62 + 62 64

6 + (4 . 2) (6 + 4) (6 + 2) 6 + 8 (10) (8)

36 + 36 1296

14 80

72 ≠ 1296

7. Write the verbal equivalents for Exercises 1, 2, and 3.

1. Subtraction is not an associative operation. 2. Division is not an associative operation. 3. Division is not a commutative operation.

1. 3, 4, 6 yes

2. 6, 9, 15 no; 6 + 9 = 15

3. 8, 8, 8 yes

4. 2, 4, 5

5. 4, 8, 16 no; 4 + 8 < 16

6. 1.5, 2.5, 3 yes

yes

Find the range for the measure of the third side of a triangle given the measures of two sides. 7. 1 cm and 6 cm

8. 12 yd and 18 yd

5 cm < n < 7 cm 9. 1.5 ft and 5.5 ft

6 yd < n < 30 yd 10. 82 m and 8 m

4 ft < n < 7 ft

74 m < n < 90 m

11. Suppose you have three different positive numbers arranged in order from least to greatest. What single comparison will let you see if the numbers can be the lengths of the sides of a triangle?

Find the sum of the two smaller numbers. If that sum is greater than the largest number, then the three numbers can be the lengths of the sides of a triangle.

4. Division does not distribute over addition. 5. Addition does not distribute over multiplication. Chapter 5

30

Glencoe Geometry

025_043_GEOCRMC05_890514.indd 30

Chapter 5

4/11/08025_043_GEOCRMC05_890514.indd 8:15:59 AM 31


Glencoe Geometry

8. For the Distributive Property, a(b + c) = ab + ac, it is said that multiplication distributes over addition. Exercises 4 and 5 prove that some operations do not distribute. Write a statement for each exercise that indicates this.


4≠0

31

Lesson 5-5

6-22-2


A14

6 - (4 - 2) (6 - 4) - 2

5. a + (bc) (a + b)(a + c)

a


Let a = 7 and b = 3. Substitute these values in the equation above.

3. a ÷ b b ÷ a

C

a+b>c b+c>a a+c>b

7-33-7 4 ≠ -4

1. a - (b - c) (a - b) - c

c

b

Glencoe Geometry

4/11/08 8:16:02 AM

Chapter 5

NAME

DATE

5-4

PERIOD

NAME

5-5

Enrichment

DATE

PERIOD

Study Guide and Intervention The Triangle Inequality

More Counterexamples


If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. This illustrates the Triangle Inequality Theorem.

Some statements in mathematics can be proven false by counterexamples. Consider the following statement.

A

For any numbers a and b, a - b = b - a. Triangle Inequality Theorem

You can prove that this statement is false in general if you can ﬁnd one example for which the statement is false.

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

B

Example The measures of two sides of a triangle are 5 and 8. Find a range for the length of the third side.

In general, for any numbers a and b, the statement a - b = b - a is false. You can make the equivalent verbal statement: subtraction is not a commutative operation.

By the Triangle Inequality Theorem, all three of the following inequalities must be true. 5+x>8 8+x>5 5+8>x x>3 x > -3 13 > x Therefore x must be between 3 and 13.

In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample.

Sample answers are given. 2. a ÷ (b ÷ c) (a ÷ b) ÷ c

6 ÷ (4 ÷ 2) (6 ÷ 4) ÷ 2

Exercises

6 1.5 − − 2


2

3 ≠ 0.75 4. a ÷ (b + c) (a ÷ b) + (a ÷ c)

6÷44÷6

6 ÷ (4 + 2) (6 ÷ 4) +(6 ÷ 2)

3 2 − ≠− 2 3

6 ÷ 6 1.5 + 3 1 ≠ 4.5 6. a2 + a2 a4

62 + 62 64

6 + (4 . 2) (6 + 4) (6 + 2) 6 + 8 (10) (8)

36 + 36 1296

14 80

72 ≠ 1296

7. Write the verbal equivalents for Exercises 1, 2, and 3.

1. Subtraction is not an associative operation. 2. Division is not an associative operation. 3. Division is not a commutative operation.

1. 3, 4, 6 yes

2. 6, 9, 15 no; 6 + 9 = 15

3. 8, 8, 8 yes

4. 2, 4, 5

5. 4, 8, 16 no; 4 + 8 < 16

6. 1.5, 2.5, 3 yes

yes

Find the range for the measure of the third side of a triangle given the measures of two sides. 7. 1 cm and 6 cm

8. 12 yd and 18 yd

5 cm < n < 7 cm 9. 1.5 ft and 5.5 ft

6 yd < n < 30 yd 10. 82 m and 8 m

4 ft < n < 7 ft

74 m < n < 90 m

11. Suppose you have three different positive numbers arranged in order from least to greatest. What single comparison will let you see if the numbers can be the lengths of the sides of a triangle?

Find the sum of the two smaller numbers. If that sum is greater than the largest number, then the three numbers can be the lengths of the sides of a triangle.

4. Division does not distribute over addition. 5. Addition does not distribute over multiplication. Chapter 5

30

Glencoe Geometry

025_043_GEOCRMC05_890514.indd 30

Chapter 5

4/11/08025_043_GEOCRMC05_890514.indd 8:15:59 AM 31


Glencoe Geometry

8. For the Distributive Property, a(b + c) = ab + ac, it is said that multiplication distributes over addition. Exercises 4 and 5 prove that some operations do not distribute. Write a statement for each exercise that indicates this.


4≠0

31

Lesson 5-5

6-22-2


A14

6 - (4 - 2) (6 - 4) - 2

5. a + (bc) (a + b)(a + c)

a


Let a = 7 and b = 3. Substitute these values in the equation above.

3. a ÷ b b ÷ a

C

a+b>c b+c>a a+c>b

7-33-7 4 ≠ -4

1. a - (b - c) (a - b) - c

c

b

Glencoe Geometry

4/11/08 8:16:02 AM


5-5

DATE

PERIOD


NAME

5-5

(continued)

The Triangle Inequality Proofs Using The Triangle Inequality Theorem

You can use the Triangle %

1. 2 ft, 3 ft, 4 ft

$

#

Given: ABC DEC Prove: AB + DE > AD − BE

yes

3. 4 mm, 8 mm, 11 mm

"

4. 13 in., 13 in., 26 in.

yes

Reasons 1. Given

5. 9 cm, 10 cm, 20 cm

2. Triangle Inequality Theorem

7. 14 yd, 17 yd, 31 yd

no; 13 + 13 ≯ 26 6. 15 km, 17 km, 19 km

yes

no; 9 + 10 ≯ 20

8. 6 m, 7 m, 12 m

yes

no; 14 + 17 ≯ 31


3. Subtraction 4. 5. 6. 7.

Find the range for the measure of the third side of a triangle given the measures of two sides.

Addition Commutative Distributive Segment Addition Postulate

9. 5 ft, 9 ft

10. 7 in., 14 in.

4 ft < n < 14 ft

7 in. < n < 21 in.

11. 8 m, 13 m

.

1

PROOF Write a two column proof.

,

−− −−− Given: PL MT −− K is the midpoint of PT. Prove: PK + KM > PL

-

5

Reasons

Glencoe Geometry

Given Alternate Interior Angles Theorem Given Definition of midpoint Vertical Angles Theorem ASA Triangle Inequality Theorem CTC Substitution

32

025_043_GEOCRMC05_890514.indd 32

Glencoe Geometry


Exercises

2 mm < n < 22 mm

13. 12 yd, 15 yd

14. 15 km, 27 km

12 km < n < 42 km

3 yd < n < 27 yd 15. 17 cm, 28 cm,

16. 18 ft, 22 ft

11 cm < n < 45 cm 17. Proof Complete the proof.

4 ft < n < 40 ft "

Given: ABC and CDE

#

$ %

Prove: AB + BC + CD + DE > AE

&

Proof: Statements

Reasons

1. AB + BC > AC CD + DE > CE

1. Triangle Inequality Theorem

2. AB + BC + CD + DE > AC + CE

2. Addition Property of Equality

3. AC + CE = AE

3. Seg. Addition Post

4. AB + BC + CD + DE > AE

4. Substitution

Chapter 5

4/11/08025_043_GEOCRMC05_890514.indd 8:16:06 AM 33

Answers

12. 10 mm, 12 mm

5 m < n < 21 m

8. Substitution Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

A15

Statements 1. ABC DEC 2. AB + BC > AC DE + EC > CD 3. AB > AC – BC DE > CD – EC 4. AB + DE > AC - BC + CD - EC 5. AB + DE > AC + CD - BC - EC 6. AB + DE > AC + CD - (BC + EC) 7. AC + CD = AD BC + EC = BE 8. AB + DE > AD - BE

2. 5 m, 7 m, 9 m

yes

&

Proof:

Chapter 5

Skills Practice The Triangle Inequality

Complete the following proof.

1. 2. 3. 4. 5. 6. 7. 8. 9.

PERIOD


Inequality Theorem as a reason in proofs.

Proof: Statements −− −−− 1. PL MT 2. ∠P ∠T −− 3. K is the midpoint of PT. 4. PK = KT 5. ∠PKL ∠MKT 6. PKL TKM 7. PK + KL > PL 8. KL = KM 9. PK + KM > PL

DATE

33

Lesson 5-5

Chapter 5

NAME

Glencoe Geometry

6/6/08 12:50:14 PM

DATE

5-5

PERIOD

NAME

5-5

Practice The Triangle Inequality

1. 9, 12, 18 yes

2. 8, 9, 17 no; 8 + 9 = 17

3. 14, 14, 19 yes

4. 23, 26, 50 no; 23 + 26 < 50

5. 32, 41, 63 yes

6. 2.7, 3.1, 4.3 yes

Use the figure at the right for Exercises 2 and 3.

12. 18 ft and 23 ft

5 ft < n < 41 ft

15. 42 m and 6 m

16. 54 in. and 7 in.

36 m < n < 48 m

47 in. < n < 61 in. %

17. Given: H is the centroid of EDF

: ;

Prove: EY + FY > DE Proof: Statements 1. 2. 3. 4. 5. 6.

Reasons

H is the centroid of EDF −− EY is a median. −− Y is the midpoint of DF

DY = FY EY + DY > DE EY + FY > DE

'

)

1. 2. 3. 4. 5. 6.

&

9 5

Given Definition of centroid Definition of median Definition of midpoint Triangle Inequality Theorem Substitution


8 cm < n < 70 cm


A16

14. 31 cm and 39 cm

13 yd < n < 63 yd

90 mi 5. TRIANGLES The figure shows an equilateral triangle ABC and a point P outside the triangle. P´ B´

Glencoe Geometry

025_043_GEOCRMC05_890514.indd 34

A´

By the Triangle Inequality Theorem, the distance from Tanya’s home to point B and on to the supermarket is greater than the straight distance from Tanya’s home to the Supermarket.

C

a. Draw the figure that is the result of turning the original figure 60° counterclockwise about A. Denote by P', the image of P under this turn.

See figure.

Sample answer: PA is congruent to P'A and m∠PAP' is 60°, So by SAS, triangle PP'A is equilateral. Thus, PP' = PA

Sample answer: Let S be the Supermarket and T be Tanya’s home. Because ∠SAB is 90, m∠SBA < 90, so m∠SBC > 90, making SC > SB. Similarly, CT > BT. Therefore CT + CS > BT + BS.

Chapter 5

A

−− −−− b. Note that P'B is congruent to PC. It is −− −−− also true that PP' is congruent to PA. Why?

3. PATHS While out walking one day Tanya finds a third place to cross the railroad tracks. Show that the path through point C is longer than the path through point B.

4/11/08025_043_GEOCRMC05_890514.indd 8:16:15 AM 35


Glencoe Geometry

34

C´ P

−− −− −− c. Show that PA, PB, and PC satisfy the triangle inequalities.

Sample answer: P'PB is a triangle with side lengths equal to PA, PB, and PC.

18. GARDENING Ha Poong has 4 lengths of wood from which he plans to make a border for a triangular-shaped herb garden. The lengths of the wood borders are 8 inches, 10 inches, 12 inches, and 18 inches. How many different triangular borders can Ha Poong make? 3

Chapter 5

B


22 km < n < 36 km

14 in. < n < 40 in.

Supermarket

Railroad

10. 7 km and 29 km

13ft < n < 25ft

4. CITIES The distance between New York City and Boston is 187 miles and the distance between New York City and Hartford is 97 miles. Hartford, Boston, and New York City form a triangle on a map. What must the distance between Boston and Hartford be greater than?

2. PATHS To get A B C to the nearest supermarket, Tanya must walk over a railroad track. There are Tanya’s home two places where she can cross the track (points A and B). Which path is longer? Explain.

Find the range for the measure of the third side of a triangle given the measures of two sides.

13. 25 yd and 38 yd


1. STICKS Jamila has 5 sticks of lengths 2, 4, 6, 8, and 10 inches. Using three sticks at a time as the sides of triangles, how many triangles can she make? 3

8. 12.3, 13.9, 25.2 yes

7. 0.7, 1.4, 2.1 no; 0.7 + 1.4 = 2.1

11. 13 in. and 27 in.

PERIOD


Is it possible to form a triangle with the given side lengths? If not explain why not.

9. 6 ft and 19 ft

DATE

35

Lesson 5-5

Chapter 5

NAME

Glencoe Geometry

4/11/08 8:16:20 AM


Chapter 5

NAME

5-5

DATE

PERIOD

NAME

DATE

5-6

Enrichment

PERIOD

Study Guide and Intervention Inequalities in Two Triangles

Constructing Triangles

Hinge Theorem The following theorem and its converse involve the relationship between the sides of two triangles and an angle in each triangle.

The measurements of the sides of a triangle are given. If a triangle having sides with these measurements is not possible, then write impossible. If a triangle is possible, draw it and measure each angle with a protractor.

If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Hinge Theorem

m∠A = 30

2. PI = 8 cm

m∠P =

m∠R = 94

IN = 3 cm

m∠I =

AT = 6 cm

m∠T = 56

PN = 2 cm

m∠N =

A

Converse of the Hinge Theorem

impossible

R

If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the third side in the second, then the included angle in the first triangle is greater than the included angle in the second triangle.

Compare the measures Example 1 −− −− G of GF and FE.

S 80°

T B

60°

N

S 36

M

33

P R

T

m∠M > m∠R

Example 2

Compare the measures

of ∠ABD and ∠CBD. C

T

m∠T =

112

WO = 7 cm

m∠W =

15

OE = 4.6 cm

m∠E =

TO = 2 cm

m∠O =

53

A17

W

impossible

T

O

5. BA = 3.l cm

m∠B = 162

AT = 8 cm

m∠A = 11

BT = 5 cm

m∠T = 7

m∠A =

90

RM = 5 cm

m∠R =

37

AM = 3 cm

m∠M =

53

6. AR = 4 cm

A

T

Glencoe Geometry

B A

F

16

A

R

M

Two sides of HGF are congruent to two sides of HEF, and m∠GHF > m∠EHF. By the Hinge Theorem, GF > FE.

Two sides of ABD are congruent to two sides of CBD, and AD > CD. By the Converse of the Hinge Theorem, m∠ABD > m∠CBD.

Exercises Compare the given measures. M

1. MR and RP

2. AD and CD

R

21° 19°

P

N

AD > CD 4. m∠XYW and m∠WYZ

X

A 48

24

m∠C < m∠Z

Z C

30

40

36

025_043_GEOCRMC05_890514.indd 36

Glencoe Geometry

Chapter 5

4/11/08025_043_GEOCRMC05_890514.indd 8:16:24 AM 37

Answers

X

50

24

Y

28 30

Y

B

W

m∠XYW < m∠WYZ

Write an inequality for the range of values of x. 5. (4x - 10) 6.

42

Z

60

36

24

120° 115°

38°

C D A

MR > RP 3. m∠C and m∠Z

22°

B

33° (3x - 3)°

24

30

x > 12.5 Chapter 5

D

B

E


4. TW = 6 cm

m∠N =


m∠O =

NE = 5.3 cm

13

28° 22°

H

3. ON = 10 cm

C

RT > AC


RT = 3 cm

A

Lesson 5-5

1. AR = 5 cm

R

60

x < 12 37

Glencoe Geometry

4/11/08 8:16:29 AM


Chapter 5

NAME

5-5

DATE

PERIOD

NAME

DATE

5-6

Enrichment

PERIOD

Study Guide and Intervention Inequalities in Two Triangles

Constructing Triangles

Hinge Theorem The following theorem and its converse involve the relationship between the sides of two triangles and an angle in each triangle.

The measurements of the sides of a triangle are given. If a triangle having sides with these measurements is not possible, then write impossible. If a triangle is possible, draw it and measure each angle with a protractor.

If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Hinge Theorem

m∠A = 30

2. PI = 8 cm

m∠P =

m∠R = 94

IN = 3 cm

m∠I =

AT = 6 cm

m∠T = 56

PN = 2 cm

m∠N =

A

Converse of the Hinge Theorem

impossible

R

If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the third side in the second, then the included angle in the first triangle is greater than the included angle in the second triangle.

Compare the measures Example 1 −− −− G of GF and FE.

S 80°

T B

60°

N

S 36

M

33

P R

T

m∠M > m∠R

Example 2

Compare the measures

of ∠ABD and ∠CBD. C

T

m∠T =

112

WO = 7 cm

m∠W =

15

OE = 4.6 cm

m∠E =

TO = 2 cm

m∠O =

53

A17

W

impossible

T

O

5. BA = 3.l cm

m∠B = 162

AT = 8 cm

m∠A = 11

BT = 5 cm

m∠T = 7

m∠A =

90

RM = 5 cm

m∠R =

37

AM = 3 cm

m∠M =

53

6. AR = 4 cm

A

T

Glencoe Geometry

B A

F

16

A

R

M

Two sides of HGF are congruent to two sides of HEF, and m∠GHF > m∠EHF. By the Hinge Theorem, GF > FE.

Two sides of ABD are congruent to two sides of CBD, and AD > CD. By the Converse of the Hinge Theorem, m∠ABD > m∠CBD.

Exercises Compare the given measures. M

1. MR and RP

2. AD and CD

R

21° 19°

P

N

AD > CD 4. m∠XYW and m∠WYZ

X

A 48

24

m∠C < m∠Z

Z C

30

40

36

025_043_GEOCRMC05_890514.indd 36

Glencoe Geometry

Chapter 5

4/11/08025_043_GEOCRMC05_890514.indd 8:16:24 AM 37

Answers

X

50

24

Y

28 30

Y

B

W

m∠XYW < m∠WYZ

Write an inequality for the range of values of x. 5. (4x - 10) 6.

42

Z

60

36

24

120° 115°

38°

C D A

MR > RP 3. m∠C and m∠Z

22°

B

33° (3x - 3)°

24

30

x > 12.5 Chapter 5

D

B

E


4. TW = 6 cm

m∠N =


m∠O =

NE = 5.3 cm

13

28° 22°

H

3. ON = 10 cm

C

RT > AC


RT = 3 cm

A

Lesson 5-5

1. AR = 5 cm

R

60

x < 12 37

Glencoe Geometry

4/11/08 8:16:29 AM

5-6

DATE

PERIOD


NAME

DATE

5-6

(continued)

Skills Practice

Inequalities Involving Two Triangles PROVE RELATIONSHIPS IN TWO TRIANGLES

Inequalities Involving Two Triangles Compare the given measures.

You can use the Hinge Theorem

B 6

and its converse to prove relationships in two triangles. 1. m∠BXA and m∠DXA

8

C 3

9

D

BC > DC Compare the given measures.

Proof:

3. m∠STR and m∠TRU Reasons 1. Defn of linear pair

30

P

m∠STR < m∠TRU 9

7

95° 85°

S

7

R

PQ > RQ

'

"

%

−− −−− −−− −−− 5. In the figure, BA, BD, BC, and BE are congruent and AC < DE. How does m∠1 compare with m∠3? Explain your thinking.

4

#

$

Glencoe Geometry

025_043_GEOCRMC05_890514.indd 38


3


m∠1 < m∠3; From the given information and the SSS Inequality Theorem, it follows that in ABC and DBE we have m∠ABC < m∠DBE. Since m∠ABC = m∠1 + m∠2 and m∠DBE = m∠3 + m∠2, it follows that m∠1 + m∠2 < m∠3 + m∠2. Subtract m∠2 from each side of the last inequality to get m∠1 < m∠3. 6. PROOF Write a two-column proof. −− −−− Given: BA DA BC > DC Prove: m∠1 > m∠2 Statements 1. 2. 3. 4.

BA DA BC > DC AC AC m∠1 > m∠3

Chapter 5

B 1

3 2

E

A D

C

B 1

A 2

C

D

Proof:

4/11/08025_043_GEOCRMC05_890514.indd 8:16:37 AM 39


Glencoe Geometry

38

22

T

97°

Complete the proof.

Reasons 1. given 2. reflexive 3. exterior angle 4. Hinge Theorem 5. opp sides in rectangle. & 6. Substitution

S

22

U

4. PQ and RQ Q

31

R

Reasons 1. Given 2. Given 3. Reflexive Property 4. SSS Inequality

39

Glencoe Geometry

6/6/08 12:50:36 PM


A18

2. Defn of supplementary 3. Given 4. Substitution 5. Subtraction 6. Inequality 7. Substitution 8. Given 9. Reflexive 10. Hinge Theorem

5

Exercises

Chapter 5

X

2. BC and DC

Given: RX = XS ∠SXT = 97° Prove: ST > RT

Given: rectangle AFBC ED = DC Prove: AE > FB Proof: Statements 1. rectangle AFBC, ED = DC 2. AD = AD 3. m∠EDA > m∠ADC 4. AE > AC 5. AC = FB 6. AE > FB

3

A

m∠BXA < m∠DXA

Example

Statements 1. ∠SXT and ∠RXT are supplementary 2. m ∠SXT + m∠RXT = 180° 3. m∠SXT = 97° 4. 97 + m∠RXT = 180 5. m∠RXT = 83 6. 97 > 83 7. m∠SXT > m∠RXT 8. RX = XS 9. TX = TX 10. ST > RT

PERIOD

Lesson 5-6

Chapter 5

NAME


DATE

NAME

DATE

5-6

Practice


Inequalities in Two Triangles

Inequalities in Two Triangles

Compare the given measures. 1. AB and BK

Q (x + 3)°

(x - 3)° 10 60°

40°

M

K

AB > BK D 14

12

2. FERRIS WHEEL A Ferris wheel has carriages located at the 10 vertices of a regular decagon.

E

K 20

13

R

10 9

3

8

4 7

E

A19

1 2

Proof: Statements

G

F

Reasons

−− 1. G is the midpoint of DF. −− −− 2. DG FG −− −− 3. EG EG

2. Definition of midpoint

4. m∠1 > m∠2

4. Given

5. ED > EF

5. Hinge Theorem

1. Given

3. Reflexive Property

Glencoe Geometry

6. TOOLS Rebecca used a spring clamp to hold together a chair leg she repaired with wood glue. When she opened the clamp, she noticed that the angle between the handles of the clamp decreased as the distance between the handles of the clamp decreased. At the same time, the distance between the gripping ends of the clamp increased. When she released the handles, the distance between the gripping end of the clamp decreased and the distance between the handles increased. Is the clamp an example of the Hinge Theorem or its converse?


D

Emily 10

mil

es

Topper

T

S

m∠R < m∠T

5. PROOF Write a two-column proof. −−− Given: G is the midpoint of DF. m∠1 > m∠2 Prove: ED > EF

High Point

2

21

F

m∠CDF < m∠EDF

1

Cloud Nine

iles

12 m

6

Topper and Cloud Nine 5. RUNNERS A photographer is taking pictures of three track stars, Amy, Noel, and Beth. The photographer stands on a track, which is shaped like a rectangle with semicircles on both ends.

5

Which carriages are farther away from carriage number 1 than carriage number 4?

Photographer

118˚

5, 6, and 7 146˚

3. WALKWAY Tyree wants to make two slightly different triangles for his walkway. He has three pieces of wood to construct the frame of his triangles. After Tyree makes the first concrete triangle, he adjusts two sides of the triangle so that the angle they create is smaller than the angle in the first triangle. Explain how this changes the triangle.

Amy 36˚ Noel Beth

a. Based on the information in the figure, list the runners in order from nearest to farthest from the photographer.

Amy, Beth, Noel

Sample answer: By the Hinge Theorem, the third side opposite the angle that was made smaller is now shorter than the third side of the first triangle.

b. Explain how to locate the point along the semicircular curve that the runners are on that is farthest away from the photographer.

Extend the line through the photographer and the center of the semicircle to where it intersects the semicircular track.

Hinge Theorem Chapter 5

40

025_043_GEOCRMC05_890514.indd 40

Glencoe Geometry

Chapter 5

4/11/08025_043_GEOCRMC05_890514.indd 8:16:44 AM 41

Answers

41

Glencoe Geometry

4/11/08 8:16:49 AM


C

8:00

4. m∠R and m∠T J

14

4. MOUNTAIN PEAKS Emily lives the same distance from three mountain peaks: High Point, Topper, and Cloud Nine. For a photography class, Emily must take a photograph from her house that shows two of the mountain peaks. Which two peaks would she have the best chance of capturing in one image?

T

S

ST > SR

3. m∠CDF and m∠EDF

14

10

R


A

1. CLOCKS The minute hand of a grandfather clock is 3 feet long and the hour hand is 2 feet long. Is the distance between their ends greater at 3:00 or at 8:00?

2. ST and SR

B 30°

PERIOD

Lesson 5-6

5-6

PERIOD

9 mile s

Chapter 5

NAME

Chapter 5

NAME

DATE

5-6

PERIOD

Enrichment

Hinge Theorem The Hinge Theorem that you studied in this section states that if two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle. In this activity, you will investigate whether the converse, inverse and contrapositive of the Hinge Theorem are also true. X

Q

1

2

Z

R

A20

Hypothesis: XY = QR, YZ = RS, m∠1 > m∠2 Conclusion: XZ > QS 1. What is the converse of the Hinge Theorem?

2. Can you find any counterexamples to prove that the converse is false?

No, it appears to be true. 3. What is the inverse of the Hinge Theorem?

If two sides of a triangle are not congruent to two sides of another triangle or the included angle in one triangle does not have a greater measure than the included angle in the other, then the third side of the first triangle is not longer than the third side of the second triangle. 4. Can you find any counterexamples to prove that the inverse is false?

No, it appears to be true. 5. What is the contrapositive of the Hinge Theorem?

6. Can you find any counterexamples to prove that the contrapositive is false?

No, it appears to be true. Chapter 5

025_043_GEOCRMC05_890514.indd 42

42

Glencoe Geometry

4/11/08 8:16:53 AM


Glencoe Geometry

If the third side of the first triangle is not longer than the third side of the second triangle, then the other two sides are not congruent or the included angle does not have a greater measure.


If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.


S Y

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