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. which pair of sides are congruent based on the definition of midpoint? which pair of sides are congruent based on the reflexive property? which pair of sides are congruent based on the definition of isosceles triangles?

Explanation:

Step1: Analyze midpoint property

M is the midpoint of AC, so by definition of midpoint, \( AM = CM \).

Step2: Analyze reflexive property

The reflexive property states that a segment is congruent to itself. So \( BM = BM \) (or other self - congruent sides, but in the context of the triangle, for reflexive property, a side congruent to itself. Also, in the isosceles triangle \( ABC \), by definition of isosceles triangle, the two equal sides (legs) are congruent. Since \( \triangle ABC \) is isosceles with \( \angle A=\angle C \), the sides opposite equal angles are equal, so \( AB = CB \).

For the first question (Which pair of sides are congruent based on the definition of midpoint?):
The definition of a midpoint M of AC means that M divides AC into two equal parts, so \( AM = CM \).

For the second question (Which pair of sides are congruent based on the reflexive property?):
The reflexive property of congruence states that any geometric figure is congruent to itself. So a side like \( BM \) is congruent to itself, i.e., \( BM = BM \), or also, in the triangle, a side such as \( AB \) is congruent to itself (\( AB = AB \)) or \( CB=CB \), but more relevant in the context of the triangle's sides, the reflexive property gives a side congruent to itself.

For the third question (Which pair of sides are congruent based on the definition of isosceles triangles?):
An isosceles triangle has at least two congruent sides. In \( \triangle ABC \), since \( \angle A=\angle C \) (marked with red arcs), the sides opposite these angles are \( CB \) and \( AB \) respectively. By the isosceles triangle theorem (or definition), sides opposite equal angles are congruent, so \( AB = CB \).

Answer:

• Based on the definition of midpoint: \( AM = CM \)
• Based on the reflexive property: \( BM = BM \) (or \( AB = AB \), \( CB = CB \))
• Based on the definition of isosceles triangles: \( AB = CB \)