Question

find the unknown angle measures in the pair of similar triangles.
angle q = □°
angle b = □°
angle p = □°
(simplify your answers. type an integer or a fraction.)

Explanation:

Step1: Recall properties of similar triangles

Similar triangles have corresponding angles equal. Also, in a right triangle, the sum of angles is \(180^\circ\), and one angle is \(90^\circ\).

Step2: Find angle Q

Triangle \(ABC\) and triangle \(PQR\) are similar. Angle \(A\) in \(ABC\) is \(42^\circ\), and angle \(Q\) corresponds to angle \(A\) (since the right angles are at \(C\) and \(R\), so the other acute angles correspond). Thus, \( \angle Q = 42^\circ \).

Step3: Find angle B

In triangle \(ABC\), \( \angle C = 90^\circ \), \( \angle A = 42^\circ \). Using the angle sum property (\( \angle A + \angle B + \angle C = 180^\circ \)):
\( 42^\circ + \angle B + 90^\circ = 180^\circ \)
\( \angle B = 180^\circ - 90^\circ - 42^\circ = 48^\circ \).

Step4: Find angle P

In triangle \(PQR\), \( \angle R = 90^\circ \), \( \angle Q = 42^\circ \). Using the angle sum property:
\( \angle P + \angle Q + \angle R = 180^\circ \)
\( \angle P + 42^\circ + 90^\circ = 180^\circ \)
\( \angle P = 180^\circ - 90^\circ - 42^\circ = 48^\circ \). (Alternatively, since \( \angle P \) corresponds to \( \angle B \) in similar triangles, \( \angle P = \angle B = 48^\circ \))

Answer:

angle \( Q = \boldsymbol{42}^\circ \)
angle \( B = \boldsymbol{48}^\circ \)
angle \( P = \boldsymbol{48}^\circ \)