# 2.5 – Proofs of Congruence

## Objectives

• Explore congruent triangles using the shortcut postulates (SSS, SAS, and ASA), the AAS Theorem, and CPCTC.
• Use Thales’ method and triangle congruence shortcuts to measure the distance of enemy warships from the shore.

## Notes

Congruence Reasoning
Proofs
• Proof: Vertical Angles
• What do I want to prove?
• What do I know? (What is given?)

• Proof: Isosceles SSS
• What do I want to prove?
• What do I know? (What is given?)

• Proof: Parallel Lines & Triangles
• What do I want to prove?
• What do I know? (What is given?)
• Remember
• Theorem: When two parallel lines are cut by a transversal, the resulting alternate interior angles are congruent.
• $\overline{MN}\parallel\overline{QP}$
• $\overline{MP}$ is the transversal
• $\angle{NMP}$ and $\angle{QPM}$ are the two alternate interior angles
• Converse (True in Congruence Proofs): If two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel.

•  Proof: Reflexive Property (Reflection) with ASA
• What do I want to prove?
• What do I know? (What is given?)