Develop a transformational proof that the base angles of an isosceles triangle must be congruent.

Proof: Consider an isosceles triangle [math]\triangle ABC[/math]. By definition, [math]\overline{AB}\cong\overline{AC}[/math]. [br]1. Create the perpendicular bisector of [math]\overline{BC}[/math] and call it [math]e[/math]. We know that [math]e[/math] will pass through point [math]A[/math] by construction and definition of an isosceles triangle. [br]2. Reflect the figure around line [math]e[/math]. [br]Recall that a reflection is an isometry, so it preserves angle measurements and distances between points. When the triangle is reflected, notice that [math]\overline{AC}[/math] maps to [math]\overline{AB}[/math] because [math]\overline{CF}\cong\overline{BF}[/math] by definition. This means that [math]\angle B[/math] and [math]\angle C[/math] switch places. [math]\angle A[/math] remains in its place because the perpendicular bisector passes through this point and is a fixed point in the reflection. Since the distances and angles in this triangle are preserved, we know that [math]\triangle ABC\cong\triangle AB'C'[/math]. Therefore, we know that [math]\angle B\cong\angle C[/math].[math]\diamondsuit[/math]