Day 7: Other Methods of Proving Triangles Congruent

Objective: Use the AAS Theorem to prove two triangles congruent. Use the HL Theorem to prove two right triangles congruent.

So far we have learned three methods for proving triangles congruent; the SSS, SAS and ASA Postulates. In this lesson, we will add the AAS Theorem and the HL Theorem.

AAS Theorem

If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

Given: Angle T congruent to Angle G; Angle C congruent to Angle D; DG congruent to AT

Prove: Triangle CAT congruent to triangle DOG

 Statements Reasons 1. Angle T congruent Angle G; Angle C congruent Angle D 1. Given 2. DG congruent AT 2. Given 3. Angle A congruent to Angle D 3. If 2 angles in a triangle are congruent to 2 angles in another triangle, then third angles congruent. 4. Triangle CAT congruent to triangle DOG 4. ASA

Example Proof using overlapping triangles:

Given: GJ congruent to GK; Angle H congruent to Angle I

Prove: Triangle GHJ congruent to Triangle GIK

 1. Angle H congruent to Angle I 1. Given 2, Angle G congruent to Angle G 2. Reflexive Property 3. GJ congruent to GK 3. Given 4. Triangle GHJ congruent to Triangle GIK 4. AAS

HL Theorem

If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

Given: XY perpendicular to AB; XA congruent to XB

Prove: Angle 1 congruent to Angle 2

 1. XY perpendicular to AB; XA congruent to XB 1. Given 2. Angle 3 and Angle 4 are right angles 2. Definition of perpendicular 3. XY congruent XY 3. Reflexive 4. Triangle XYA congruent to Triangle XYB 4. HL Theorem 5. Angle 1 congruent Angle 2 5. CPCTC

Summary of Ways to Prove Two Triangles Congruent

All triangles: SSS, SAS, ASA, AAS

Right triangles: HL

Exercise Proofs (Write in 2 column form)

1. Given: EF perpendicular to EG; HG perpendicular to EG; EH congruent to GF

Prove: Angle H congruent to Angle F

2. Given: RT congruent to AS; RS congruent to AT

Prove: Angle TSA congruent to Angle STR