Exercise 23 Page 631

Let's illustrate two right triangles where we know that an angle and a leg of one right triangle is congruent to an angle and the leg of another right triangle. Notice that there are two possible cases, one where the side is included and another where the side is not included.

Side is included

When the side is included, it will be in between the two angles like in the following diagram.

Now we can write our paragraph proof:

By the Right Angles Congruence Theorem, the two triangles have congruent right angles. There is also another pair of congruent corresponding angles and a pair of congruent corresponding sides. Since the congruent corresponding sides are the included sides, the triangles are congruent by the ASA Congruence Theorem.

Side is not included

When the side is not included, it will not be in between the two angles like in the following diagram.

Now we can write our paragraph proof:

By the Right Angles Congruence Theorem, the two triangles have congruent right angles. There is also another pair of congruent corresponding angles and a pair of congruent corresponding sides. Since the congruent corresponding sides are not the included sides, the triangles are congruent by the AAS Congruence Theorem.