Chapter 6.6, Problem 28PPS

To determine

The proof for Theorem 6.21.

Explanation of Solution

Given:

Isosceles Trapezoid Theorem states that, “If a trapezoid is isosceles, then each pair of base angles is congruent”.

Calculation:

So, it is given that, $\overline{BC}\parallel \overline{AD},\overline{AB}\cong \overline{CD}.$

  

Draw the line segments BF and CE so that $\overline{BF}\perp \overline{AD}\text{ and }\overline{CE}\perp \overline{AD}.$

Also, as BC is parallel to AD, BF and CE are congruent to each other.

According to the definition of perpendicular lines, angles BFA and CED are right angles.

Hence, triangles BFA and CED are right triangles.

Given, $\overline{AB}\cong \overline{CD},$ according to the HL congruence theorem, $\Delta BFA\cong \Delta CED.$

Thus, angle A is congruent to angle D and angle CBF is congruent to angle BCE by CPCTC.

Also, angles FBC and ECB are right angles.

So, by Angle Addition Postulate, $\angle ABC\cong \angle DCB$ .