Hl Theorem 2t4lq

Elizabeth Rodriguez Title: HL Theorem Subject: Geometry Common Core: • CCSS.MATH.CONTENT.HSG.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Educational Objectives: • The students will be able to identify the components in a triangle. (hypotenuse, leg, leg, right angle) • The students will then be able to use that knowledge to prove congruency between two triangles. Assessment on Learning: • Pre-test on labeling the components of a triangle. (Hypotenuse, leg, leg, right angle) Central Focus • The students will be able to prove congruency using the hypotenuse leg theorem. Academic Language • Hypotenuse – the side of the triangle opposite from the right angle • Leg – the other two sides of the right triangle • Hypotenuse Leg Theorem – If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another triangle, then the right triangles are congruent. Materials Required for the lesson: Teacher: • Notes • White Board • Markers • Dry eraser Students: • Pencil • Paper Anticipatory Set/ Attention Getter: (1 minute) Begin with prayer. (3 minutes) We’re going to start off with a quick triangle identifier. I have a special triangle on this paper, and there are 4 labels for this triangle. I want everyone to label as many parts as they can and it down to the front of the row. Sequence of Events: (1 minute) Good morning students, today we will be learning another theorem to prove congruency between two triangles. If we look at the triangle in the paper we have: A Leg__

Right Angle

C

Hypotenus e B

Leg__

These components are important to the new theorem we are going to learn today. Take out your notes and follow along. (5 minutes) The hypotenuse leg theorem states that if • There are two right triangles • The triangles have congruent hypotenuses • There is one pair of congruent legs Then the two triangles are congruent. See example of triangles PQR and STU. P

S

Q

R

T

U

The teacher will then go through three examples to show how the proofs for these triangles are done. Hypotenuse Leg Theorem Examples (20 minutes) K

O

G

̅̅̅̅ ⊥ ̅̅̅̅̅ ̅̅̅ Given: 𝐾𝑂 𝑂𝑀 𝑎𝑛𝑑 ̅̅̅̅ 𝐺𝐻 ⊥ ̅𝐻𝐽 ̅̅̅̅ ≅ 𝐺𝐻 ̅̅̅̅ , 𝐾𝑀 ̅̅̅̅̅ ≅ 𝐺𝐽 ̅̅̅ 𝐾𝑂 Prove: 𝛥𝐾𝑂𝑀 ≅ 𝛥𝐺𝐻𝐽

J M

H

Statement ̅̅̅̅ ⊥ ̅̅̅̅̅ ̅̅̅ 1. 𝐾𝑂 𝑂𝑀 𝑎𝑛𝑑 ̅̅̅̅ 𝐺𝐻 ⊥ ̅𝐻𝐽 ̅̅̅̅ ≅ 𝐺𝐻 ̅̅̅̅ , (𝐿𝑒𝑔) 𝐾𝑀 ̅̅̅̅̅ ≅ 𝐺𝐽 ̅̅̅ (𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 𝐾𝑂 2. ∠B and ∠Q are right ∠’s 3. 𝛥𝐾𝑂𝑀 𝑎𝑛𝑑 𝛥𝐺𝐻𝐽 are right triangles 4. 𝛥𝐾𝑂𝑀 ≅ 𝛥𝐺𝐻𝐽

Reason 1. Given 2. Definition of ⊥ Lines 3. Definition of right triangles 4. HL theorem

A

P

Given: ∠B and ∠Q are right ∠’s

Prove: 𝛥𝐴𝐵𝑇 ≅ 𝛥𝑃𝑄𝑇

̅̅̅̅ T is a midpoint of 𝐵𝑄 ̅̅̅̅ ̅̅̅̅ 𝐴𝑇 ≅ 𝑃𝑇 B

Q

T

Statement

Reason

1. ∠B and ∠Q are right ∠’s ̅̅̅̅ T is a midpoint of 𝐵𝑄 ̅̅̅̅ ̅̅̅̅ (Hypotenuse) 𝐴𝑇 ≅ 𝑃𝑇 2. 𝛥𝐴𝐵𝑇 𝑎𝑛𝑑 𝛥𝑃𝑄𝑇 are right triangles ̅̅̅̅ ≅ 𝑃𝑇 ̅̅̅̅ (Leg) 3. 𝐴𝑇

1. Given

4. 𝛥𝐴𝐵𝑇 ≅ 𝛥𝑃𝑄𝑇

4. HL Theorem

2. Definition of right triangles 3. Midpoint → 2 ≅ parts

T

Given: M bisects ̅̅̅̅̅ 𝑆𝑂 𝑎𝑛𝑑 ̅̅̅̅ 𝑇𝐴 ∠𝑆 𝑎𝑛𝑑 ∠O are right ∠’s

S

M

Prove: 𝛥𝑆𝐴𝑀 ≅ 𝛥𝑇𝑂𝑀 O

A

Statement

Reason

1. M bisects ̅̅̅̅̅ 𝑆𝑂 𝑎𝑛𝑑 ̅̅̅̅ 𝑇𝐴

1. Given

∠𝑆 𝑎𝑛𝑑 ∠O are right ∠’s 2. 𝛥𝑆𝐴𝑀 𝑎𝑛𝑑 𝛥𝑇𝑂𝑀 are right triangles

2. Definition of right triangles

3. ̅̅̅̅ 𝑆𝑀 ≅ ̅̅̅̅̅ 𝑂𝑀 (Leg)

3. Bisect → 2 ≅ parts

4. ̅̅̅̅̅ 𝐴𝑀 ≅ ̅̅̅̅̅ 𝑇𝑀 (Hypotenuse)

4. Bisect → 2 ≅ parts

5. 𝛥𝑆𝐴𝑀 ≅ 𝛥𝑇𝑂𝑀

5. HL Theorem

Questions to Ask • Does anyone know why the hypotenuse leg theorem does not work on triangles other than right triangles? – Comprehension • What congruency statements do we see the most in these proofs? – analysis Conclusion/Summary: That is all we are learning today! The hypotenuse leg theorem. So, if you are given two right triangles that have congruent hypotenuses and one pair of congruent legs you can prove the two triangles are congruent! Any last questions?

a

___________________ _________________ _

b

c ________________

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