**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