The diagram shows two right triangles.

Question

Triangle A-B-C with right angle at C, A-B measuring c units, A-C measuring b units, and B-C measuring a units. Triangle P-Q-R has right angle at R, P-Q measuring r units, Q-R measuring p units, and P-R measuring q units. Angle A and P are 32 degrees.

Is it true that

and

must be equal?

A.
No,

and

may not be equal because a and p may not be equal.

B.
No,

and

may not be equal because the measures of
and
may not be equal.

C.
Yes,


because the triangles are both right triangles.

D.
Yes,


because the triangles are similar.

Answer 1

To determine whether angles \( A \) and \( P \) lead to equal corresponding angle properties in the triangles, we need to analyze the information about the triangles and the fact that they both contain a right angle.

Given:

• Triangle A-B-C is a right triangle with right angle at C.
• Triangle P-Q-R is also a right triangle with right angle at R.
• Angles \( A \) and \( P \) are both 32 degrees.

From the information, we can conclude:

• In a right triangle, the sum of the angles is always 180 degrees.
• Therefore, in Triangle A-B-C, the angles will be \( 32^\circ \) (angle A), \( 90^\circ \) (angle C), and \( 58^\circ \) (angle B; since \( 180 - 32 - 90 = 58 \)).
• In Triangle P-Q-R, one angle is \( 32^\circ \) (angle P), \( 90^\circ \) (angle R), which means angle Q must also be \( 58^\circ \) since \( 180 - 32 - 90 = 58 \).

Since the two triangles have two angles that are equal (32 degrees and 58 degrees), they are similar by the angle-angle (AA) similarity criterion for triangles. Because they are similar triangles, corresponding sides are in proportion, and corresponding angles are equal.

Therefore, the correct interpretation of the statements would be:

D. Yes, \( \angle A \) and \( \angle P \) are equal because the triangles are similar.