Question

if the $m\angle srw = 85^{circ}$, what are the measures of $\angle vru$ and $\angle urw$?
$m\angle vru = 95^{circ}$
$m\angle urw = \square^{circ}$
angles srw and trv are vertical angles.

Explanation:

Step1: Use vertical angles property

Since angles \(SRW\) and \(TRV\) are vertical angles, \(m\angle TRV=m\angle SRW = 85^{\circ}\). Also, \((2x + 15)^{\circ}\) and \((2x+10)^{\circ}\) form a linear - pair with \(\angle TRV\). So, \((2x + 15)+(2x + 10)+85=180\).

Step2: Simplify the equation

Combine like terms: \(4x+110 = 180\).
Subtract \(110\) from both sides: \(4x=180 - 110\), so \(4x = 70\), then \(x=\frac{70}{4}=17.5\).

Step3: Find \(m\angle VRU\)

\(m\angle VRU=(2x + 15)^{\circ}\). Substitute \(x = 17.5\) into the expression: \(2\times17.5+15=35 + 15=95^{\circ}\).

Step4: Find \(m\angle URW\)

\(m\angle URW=(2x + 10)^{\circ}\). Substitute \(x = 17.5\) into the expression: \(2\times17.5+10=35+10 = 20^{\circ}\).

Answer:

\(m\angle VRU = 95^{\circ}\), \(m\angle URW=20^{\circ}\)