Exercise 46 Page 396

We are asked to show the converse of Theorem $6\text{-}19$. We need to show that if two of the base angles of a trapezoid are congruent, then the trapezoid is isosceles. Let's draw a trapezoid and a segment parallel to one of its legs. Let's also indicate the congruent base angles.

We will prove that $ABCD$ is isosceles in two steps. We will show that segment $\overline{AE}$ is congruent to both $\overline{AB}$ and $\overline{DC}.$

Proof of $\overline{AE}\cong \overline{DC}$

Let's focus on quadrilateral $AECD$ first.

By construction, opposite sides of $AECD$ are parallel, so $AECD$ is a parallelogram. According to Theorem $6\text{-}3$, this means that opposite sides are congruent.

Proof of $\overline{AB}\cong \overline{AE}$

Let's shift our focus to triangle $△ABE.$

We will show that the three angles marked on the diagram are congruent.

Congruent Angles	Justification
$\angle B\cong \angle C$	Given
$\angle AEB\cong \angle C$	Segments $\overline{AE}$ and $\overline{DC}$ are parallel, and $\angle AEB$ and $\angle C$ are corresponding angles.

By the Transitive Property of Congruence, this means that two angles of triangle $△ABE$ are congruent. According to the Converse of the Isosceles Triangle Theorem, the sides opposite to the congruent angles are congruent.

Conclusion

Since segment $\overline{AE}$ is congruent to both $\overline{AB}$ and $\overline{DC},$ the two legs of the trapezoid are congruent, so $ABCD$ is an isosceles trapezoid. We can summarize the process above in a flow proof.

Completed Proof

Proof: