Question

1. find the measure of ∠j
2. find the measure of ∠z
3. find the measure of ∠s

Explanation:

Step1: Identify angle - pair relationship for ∠j

The 50 - degree angle and the angle with measure \(65x\) are vertical angles, so \(65x = 50\), then \(x=\frac{50}{65}=\frac{10}{13}\). ∠j and the angle with measure \(65x\) are also vertical angles, so the measure of ∠j is \(50^{\circ}\).

Step2: Identify angle - pair relationship for ∠z

Assume the angles \((28x + 7)\) and \((29x+3)\) are supplementary (since they are adjacent angles on a straight - line formed by the intersection of lines). So \((28x + 7)+(29x + 3)=180\). Combine like terms: \(28x+29x+7 + 3=180\), \(57x+10 = 180\), \(57x=180 - 10=170\), \(x=\frac{170}{57}\). The angle \((29x + 3)\) and ∠z are vertical angles. Substitute \(x=\frac{170}{57}\) into \(29x + 3\): \(29\times\frac{170}{57}+3=\frac{4930}{57}+3=\frac{4930+171}{57}=\frac{5101}{57}\approx89.5^{\circ}\).

Step3: Identify angle - pair relationship for ∠s

The \(70^{\circ}\) angle and the angle with measure \((9x + 7)\) are vertical angles, so \(9x+7 = 70\), \(9x=70 - 7 = 63\), \(x = 7\). The angle \((9x + 7)\) and the angle adjacent to ∠s are supplementary. The angle adjacent to ∠s is \(180-(9x + 7)=180 - 70=110^{\circ}\). ∠s and this adjacent angle are vertical angles, so the measure of ∠s is \(110^{\circ}\).

Answer:

∠j: \(50^{\circ}\)
∠z: \(\frac{5101}{57}\approx89.5^{\circ}\)
∠s: \(110^{\circ}\)