Question

line ( \boldsymbol{ell} ) is the perpendicular bisector of segment ( overline{wy} ). ( z ) is a point on ( \boldsymbol{ell} )

image of line ( ell ) intersecting segment ( wy ) at ( x ), with ( z ) on ( ell ), and triangles ( \triangle wxz ) and ( \triangle yxz ) drawn

nicolas noticed that ( \triangle wxz ) and ( \triangle yxz ) are congruent based on the side - angle - side congruency postulate.
what theorem can we prove using these congruent triangles?
choose 1 answer:
a vertical angles are congruent.
b measures of interior angles of a triangle sum to ( 180^circ ).
c a point on the perpendicular bisector of a line segment is equidistant from the segment’s endpoints.
d any line segment is congruent to itself because every point on the segment maps to itself.

Explanation:

1. Analyze Option A: Vertical angles congruence is about angles formed by intersecting lines, not related to perpendicular bisectors and triangle congruence here.
2. Analyze Option B: The triangle angle - sum theorem is proved using different methods (like parallel lines and transversals), not by this triangle congruence related to perpendicular bisectors.
3. Analyze Option C: Since line \( \ell \) is the perpendicular bisector of \( \overline{WY} \), \( WX = YX \) (bisector) and \( \angle WXZ=\angle YXZ = 90^{\circ}\), and \( XZ \) is common. By SAS, \( \triangle WXZ\cong\triangle YXZ \). Then \( WZ = YZ \) (corresponding parts of congruent triangles). So a point \( Z \) on the perpendicular bisector of \( \overline{WY} \) is equidistant from \( W \) and \( Y \), which proves the theorem in option C.
4. Analyze Option D: This is the reflexive property of congruence, which is a basic property and not related to the result from these congruent triangles about perpendicular bisectors.

Answer:

C. A point on the perpendicular bisector of a line segment is equidistant from the segment’s endpoints.