Question

Isosceles triangle ABC is folded along BM with M chosen in such a way that it is the midpoint of side AC, the shortest side.

Isosceles triangle A B C is shown. A fold is shown down the middle from point B to midpoint M of side A C. Angles B A M and B C M are congruent.

Which pair of sides are congruent based on the definition of midpoint?


BM and BM

Which pair of sides are congruent based on the reflexive property?



Which pair of sides are congruent based on the definition of isosceles triangles?

Answer 1

To analyze the properties of the isosceles triangle ABC after folding along the line BM, let’s break down the congruencies based on the criteria provided:

1. **Which pair of sides are congruent based on the definition of midpoint?**
- By the definition of a midpoint, **AM and CM are congruent (AM = CM)** because M is the midpoint of side AC.

2. **Which pair of sides are congruent based on the reflexive property?**
- According to the reflexive property, which states that a geometric figure is congruent to itself, the segment **BM = BM** is congruent. This means the line segment BM is equal to itself before and after the fold.

3. **Which pair of sides are congruent based on the definition of isosceles triangles?**
- In isosceles triangle ABC, the two sides opposite the base (the sides that are equal in length) are congruent. Therefore, based on the definition of an isosceles triangle, we have **AB = BC**, where AB and BC are the lengths of the equal sides of triangle ABC.

This analysis demonstrates the application of the definitions and properties effectively to identify congruent sides within the context of folding the triangle.