Question

these figures are congruent. what is m∠r?

Explanation:

Step1: Recall triangle congruence

Congruent triangles have equal corresponding angles and sides. So, we can find the missing angle in triangle \( CDE \) first.

Step2: Calculate angle in \( \triangle CDE \)

The sum of angles in a triangle is \( 180^\circ \). In \( \triangle CDE \), we know \( \angle C = 61^\circ \) and \( \angle E = 77^\circ \). So, \( \angle D = 180^\circ - 61^\circ - 77^\circ = 42^\circ \).

Step3: Find \( m\angle R \)

Since the triangles are congruent, corresponding angles are equal. We need to find the corresponding angle to \( \angle R \). Looking at the sides: \( CD = 45 \) ft, \( DE = 34 \) ft; \( ST = 45 \) ft, \( SR = 34 \) ft. So, \( \angle R \) corresponds to \( \angle D \)? Wait, no, let's check angles. Wait, in \( \triangle RST \), angles are \( 42^\circ \), \( 77^\circ \), and we need \( \angle R \). Wait, in \( \triangle CDE \), angles are \( 61^\circ \), \( 77^\circ \), \( 42^\circ \). Wait, maybe I mixed up. Wait, the sides: \( DE = 34 \) ft, \( CD = 45 \) ft; \( SR = 34 \) ft, \( ST = 45 \) ft. So, vertex \( C \) corresponds to \( S \), \( D \) corresponds to \( T \), \( E \) corresponds to \( R \)? Wait, no, let's check angles. Wait, in \( \triangle CDE \), angle at \( C \) is \( 61^\circ \), angle at \( E \) is \( 77^\circ \), angle at \( D \) is \( 42^\circ \). In \( \triangle RST \), angle at \( S \) is \( 42^\circ \), angle at \( T \) is \( 77^\circ \), so angle at \( R \) should correspond to angle at \( C \), which is \( 61^\circ \)? Wait, no, wait the sum. Wait, in \( \triangle RST \), sum of angles is \( 180^\circ \). So \( \angle R + 42^\circ + 77^\circ = 180^\circ \), so \( \angle R = 180 - 42 - 77 = 61^\circ \). Wait, that's it. Because in congruent triangles, corresponding angles are equal. So the angle corresponding to \( \angle C \) (which is \( 61^\circ \)) is \( \angle R \). So \( m\angle R = 61^\circ \).

Answer:

\( 61^\circ \)