Chapter 12, Problem 21CR

To determine

To prove: Point P is the midpoint of $\overline{WX}$ .

Explanation of Solution

Given information:

WXYZ is an isosceles trapezoid with $\overline{WZ}\cong \overline{XY}$ .

△PZY is isosceles.

Formula used:

The below properties are used:

In an isosceles triangle, two sides are congruent.

If corresponding sides are congruent then corresponding angles are also congruent.

If a point divides a segment into two congruent segments, it is the midpoint.

Proof:

  

It is given that,

WXYZ is an isosceles trapezoid with $\overline{WZ}\cong \overline{XY}$ .

△PZY is isosceles.

In an isosceles triangle, two sides are congruent.

  $\overline{PZ}\cong \overline{PY}$

If corresponding sides are congruent then corresponding angles are also congruent.

  $\angle PZY \cong \angle PYZ$

The lower base angles of an isosceles trapezoid are congruent.

  $\angle WZY \cong \angle XYZ$

By subtraction property, we get

  $\angle WZP \cong \angle XYP$

Two triangles are similar by SAS congruence rule.

If corresponding sides are congruent then corresponding angles are also congruent.

  $\overline{WP}\cong \overline{XP}$

If a point divides a segment into two congruent segments, it is the midpoint.

Point P is the midpoint of $\overline{WX}$ .