Proof 8.34
Develop a transformational proof that the base angles of an isosceles triangle must be congruent.
Proof: Consider an isosceles triangle . By definition, .
1. Create the perpendicular bisector of and call it . We know that will pass through point by construction and definition of an isosceles triangle.
2. Reflect the figure around line .
Recall that a reflection is an isometry, so it preserves angle measurements and distances between points. When the triangle is reflected, notice that maps to because by definition. This means that and switch places. 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 . Therefore, we know that .