Exercise 45 Page 396

Let's sketch the isosceles trapezoid $ABCD$ and draw segment $\overline{AE}$ parallel to leg $\overline{DC}$ as suggested.

In quadrilateral $AECD$ opposite sides are parallel, so it is a parallelogram. According to Theorem $6\text{-}3$, this means that the opposite sides are congruent. Let's now focus on triangle $△ABE.$ We know that $ABCD$ is an isosceles trapezoid, so $\overline{AB}$ is congruent to $\overline{DC}.$ Since we already know that $\overline{DC}$ is congruent to $\overline{AE},$ this means that $△ABE$ is an isosceles triangle. According to the Isosceles Triangle Theorem, this means that the angles opposite to the congruent sides are congruent. We can also compare $\angle 1$ to $\angle C.$ These are corresponding angles, and since $\overline{AE}$ and $\overline{DC}$ are parallel, these angles are congruent by the Corresponding Angles Theorem. Using the Transitive Property of Congruence, we can conclude that the base angles $\angle B$ and $\angle C$ are congruent. Let's now focus on the other pair of base angles, starting with the angle $\angle D.$ Let's compare this to angle $\angle C.$ Since $\angle D$ and $\angle C$ are same-side interior angles and $\overline{AD}$ is parallel to $\overline{BC},$ the Same-Side Interior Angles Postulate tells us that these are supplementary angles. Similarly, angles $\angle BAD$ and $\angle B$ are also supplementary. Since angles $\angle B$ and $\angle C$ are congruent, their supplements must also be congruent. The relationship between the sides and angles of the isosceles trapezoid $ABCD$ is illustrated below.

We can summarize the process above in a flow proof.

Completed Proof

Proof: