Question

the two triangles below are similar. also, ( mangle a = 15^circ ) and ( mangle b = 45^circ ) as shown below. find ( mangle p ), ( mangle q ), and ( mangle r ). assume the triangles are accurately drawn.

Explanation:

Step1: Find $\angle C$ in $\triangle ABC$

Sum of angles in a triangle is $180^\circ$.
$$m\angle C = 180^\circ - m\angle A - m\angle B = 180^\circ - 15^\circ - 45^\circ = 120^\circ$$

Step2: Match corresponding angles

Similar triangles have congruent corresponding angles. From the diagram, $\angle P \cong \angle A$, $\angle Q \cong \angle C$, $\angle R \cong \angle B$.
$$m\angle P = 15^\circ, \, m\angle R = 45^\circ, \, m\angle Q = 120^\circ$$

Answer:

$m\angle P = 15^\circ$, $m\angle Q = 120^\circ$, $m\angle R = 45^\circ$