Exercise 21 Page 427

We are asked to prove that the overlapping triangles shaded in the diagram are congruent in rectangle $QTVW.$ Let's mark the given segment congruence. On the given figure, only one right angle is marked. However, recall that all of the angles of a rectangle are right angles. Let's mark two of these.

First, let's focus on the position of points $Q,$ $R,$ $S,$ and $T.$

It is given that $\overline{QR}$ and $\overline{ST}$ are congruent. When we join segment $\overline{RS}$ to each of these, according to the Segment Addition Postulate, the lengths increase by the same amount. This means that the new segments are also congruent. Let's summarize what we know about triangles $△SWQ$ and $△RVT.$

Congruence	Justification
$\overline{QS}\cong \overline{RT}$	Proven above
$\angle Q\cong \angle T$	The angles of a rectangle are right angles, and all right angles are congruent.
$\overline{QW}\cong \overline{TV}$	A rectangle is a parallelogram, and opposite sides of a parallelogram are congruent (Theorem $6.3$).

We can see that two sides and the included angle of triangle $△SWQ$ is congruent to two sides and the included angle of triangle $△RVT.$ According to the Side-Angle-Side (SAS) Congruence Postulate, these two triangles are congruent. We can summarize the process above in a two-column proof.

Completed Proof

Proof:

0. Statements	0. Reasons
1. $\overline{QR}\cong \overline{ST}$	1. Given.
2. $\overline{RS}\cong \overline{RS}$	2. Reflexive property of congruence.
3. $\overline{QS}\cong \overline{RT}$	3. Segment Addition Posulate.
4. $QTVW$ is a rectangle	4. Given
5. $m\angle Q=m\angle T=90$	5. Definition
6. $\angle Q\cong \angle T$	6. All right angles are congruent.
7. $QTVW$ is a parallelogram	7. Definition
8. $\overline{QW}\cong \overline{TV}$	8. Opposite sides of a parallelogram (Theorem $6.3$)
9. $△SWQ\cong △RVT$	9. SAS