Question

The proof that is shown. Select the answer that best completes the proof.

Given: ΔMNQ is isosceles with base , and and bisect each other at S.
Prove:

Square M N Q R is shown with point S in the middle. Lines are drawn from each point of the square to point S to form 4 triangles.

We know that ΔMNQ is isosceles with base . So, by the definition of isosceles triangle. The base angles of the isosceles triangle, and , are congruent by the isosceles triangle theorem. It is also given that and bisect each other at S. Segments _______ are therefore congruent by the definition of bisector. Thus, by SAS.

NS and QS
NS and RS
MS and RS
MS and QS

Answer 1

To complete the proof, the correct option to fill in the blank is:

**MS and QS**.

Here's the reasoning:

- Triangles \( \triangle MNS \) and \( \triangle QRS \) can be compared.
- Since \( \triangle MNQ \) is isosceles with \( MN = MQ \), then by definition, the angles \( \angle MNS \) and \( \angle QRS \) are congruent (base angles of the isosceles triangle).
- It is also given that \( MS \) and \( QS \) are segments formed by the points of the triangle and thus they intersect at the midpoint \( S \), making \( MS \) and \( QS \) congruent.
- Since we have two pairs of congruent sides and one pair of congruent angles between the two triangles, we can apply the SAS (Side-Angle-Side) criterion.

Therefore, the segments **MS and QS** are congruent by the definition of bisector.