Question

if the ( mangle srw = 85^circ ), what are the measures of ( angle vru ) and ( angle urw )?
( mangle vru = square^circ )
( mangle urw = square^circ )

Explanation:

Step1: Identify linear pair

Angles \( \angle VRU \) and \( \angle SRW \) with \( \angle VRU=(2x + 15)^\circ+(2x + 10)^\circ \) and \( \angle SRW = 85^\circ \) form a linear pair (sum to \( 180^\circ \)). So, \((2x + 15)+(2x + 10)+85 = 180\).

Step2: Solve for \( x \)

Simplify: \( 4x+110 = 180 \). Subtract 110: \( 4x = 70 \). Divide by 4: \( x = 17.5 \).

Step3: Find \( m\angle VRU \)

Substitute \( x = 17.5 \) into \( (2x + 15)+(2x + 10) \): \( 2(17.5)+15 + 2(17.5)+10 = 35+15+35+10 = 95^\circ \).

Step4: Find \( m\angle URW \)

\( \angle URW=(2x + 10)^\circ \). Substitute \( x = 17.5 \): \( 2(17.5)+10 = 35 + 10 = 45^\circ \)? Wait, no, wait. Wait, \( \angle SRW = 85^\circ \), and \( \angle VRU + \angle SRW=180^\circ \), so \( \angle VRU = 180 - 85=95^\circ \). Then, \( \angle VRU=(2x + 15)+(2x + 10)=4x + 25 \). So \( 4x + 25=95 \), \( 4x = 70 \), \( x = 17.5 \). Then \( \angle URW=(2x + 10)=2(17.5)+10 = 45 \)? Wait, but \( \angle SRW = 85^\circ \), and \( \angle URW \) and \( \angle SRW \): Wait, maybe I messed up. Wait, the diagram: \( \angle VRU \) is composed of two angles: \( (2x + 15) \) and \( (2x + 10) \), and \( \angle SRW = 85^\circ \). So \( (2x + 15)+(2x + 10)+85 = 180 \). So \( 4x + 110 = 180 \), \( 4x = 70 \), \( x = 17.5 \). Then \( \angle VRU=(2x + 15)+(2x + 10)=4x + 25=417.5 + 25=70 + 25=95^\circ \). Then \( \angle URW=(2x + 10)=217.5 + 10=35 + 10=45^\circ \)? Wait, but \( \angle SRW = 85^\circ \), and \( \angle URW \) and \( \angle SRW \): Wait, no, maybe \( \angle URW \) is part of \( \angle SRW \)? No, the diagram: \( R \) is the vertex, \( S \), \( R \), \( W \); \( V \), \( R \), \( U \). So \( \angle VRU \) is adjacent to \( \angle SRW \) forming a linear pair. So \( \angle VRU = 180 - 85 = 95^\circ \). Then, \( \angle VRU=(2x + 15)+(2x + 10)=4x + 25=95 \), so \( x = 17.5 \). Then \( \angle URW=(2x + 10)=217.5 + 10=45^\circ \)? Wait, but \( \angle SRW = 85^\circ \), and \( \angle URW \) and \( \angle SRW \): Wait, maybe I made a mistake. Wait, \( \angle SRW = 85^\circ \), and \( \angle URW \) is \( (2x + 10) \), and \( \angle VRU=(2x + 15)+(2x + 10) \). So \( (2x + 15)+(2x + 10)+85 = 180 \). So that's correct. So \( \angle VRU = 95^\circ \), \( \angle URW=(2x + 10)=217.5 + 10=45^\circ \)? Wait, but 45 + 85=130, no. Wait, no, \( \angle VRU \) is \( (2x + 15)+(2x + 10)=4x + 25 \), and \( 4x + 25 + 85=180 \), so 4x + 110=180, 4x=70, x=17.5. Then \( 2x + 10=45 \), \( 2x + 15=50 \). Then \( 50 + 45=95 \), and 95 + 85=180. Correct. So \( m\angle VRU = 95^\circ \), \( m\angle URW = 45^\circ \)? Wait, but the problem says \( m\angle SRW = 85^\circ \), and \( \angle URW \) is part of \( \angle SRW \)? No, the diagram: \( W \) is on the same line as \( U \)? Wait, maybe \( \angle URW \) is \( (2x + 10) \), and \( \angle SRW = 85^\circ \), so \( \angle URW \) and \( \angle SRU \)? No, I think my initial step was correct. So \( \angle VRU = 95^\circ \), \( \angle URW = 45^\circ \)? Wait, no, 2x + 10 when x=17.5 is 45, and 2x + 15 is 50, 50 + 45=95, 95 + 85=180. Correct.

Answer:

\( m\angle VRU = \boldsymbol{95^\circ} \)
\( m\angle URW = \boldsymbol{45^\circ} \)