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### Course: High school geometry>Unit 3

Lesson 2: Triangle congruence from transformations

# Justify triangle congruence

## Problem

Below are $\mathrm{△}ABC$ and $\mathrm{△}DEF$. We assume that $AB=DE$, $BC=EF$, and $m\mathrm{\angle }B=m\mathrm{\angle }E$.
Here is a rough outline of a proof that $\mathrm{△}ABC\cong \mathrm{△}DEF$:
1. We can map $\mathrm{△}ABC$ using a sequence of rigid transformations so that ${A}^{\prime }=D$ and ${B}^{\prime }=E$.
2. If ${C}^{\prime }$ and $F$ are on the same side of $\stackrel{↔}{DE}$, then ${C}^{\prime }=F$.
3. If ${C}^{\prime }$ and $F$ are on opposite sides of $\stackrel{↔}{DE}$, then we reflect $\mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }$ across $\stackrel{↔}{DE}$ and then ${C}^{″}=F$, ${A}^{″}=D$ and ${B}^{″}=E$.
What is the justification that ${C}^{\prime }=F$ in step 2?