Chapter 5.5, Problem 27WE

To determine

To Prove : The base angles of isosceles trapezoid are congruent.

Explanation of Solution

Given information:

Consider the trapezoid ABXY with $BX\cong AY$

The figure is shown below:

  

Draw line $BP\perp XY$ and line $AQ\perp XY$ at P and Q respectively.

In $\Delta BPX$ and $\Delta AQY$ ,

  $BX\cong AY$ (Congruent sides of trapezoid)

  $\overline{BP}=\overline{AQ}$ (Distance between the parallel lines)

  $\angle BPX=\angle AQY\left(BP\perp XYandAQ\perp XY\right)$

Therefore, $\Delta BPX\cong \Delta AQY$ (Right side, hypotenuse and one side property)

Hence, $\angle BXP\cong \angle AYQ$ (Corresponding parts of congruent triangles are equal)

  $\angle Q+\angle P=180°$ (Interior angles on the same side of transversal are supplementary)

  $\angle YAQ\cong \angle XBP$ (Corresponding parts of congruent triangles are equal)

Hence it is verified that $\angle X\cong \angle Y,\angle B\cong \angle A$ .