Question

find the measures of angles x, y, and z in the figure.
∠y = 105°
∠x = □°

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal. Angle \(y\) and the given \(105^{\circ}\) angle are vertical angles, so \(y = 105^{\circ}\). Angle \(x\) and the \(105^{\circ}\) angle are supplementary (they form a straight - line pair).

Step2: Use the supplementary - angle formula

The sum of two supplementary angles is \(180^{\circ}\). Let \(x\) be the unknown angle and the known angle be \(105^{\circ}\). Then \(x+105^{\circ}=180^{\circ}\).

Step3: Solve for \(x\)

Subtract \(105^{\circ}\) from both sides of the equation: \(x = 180^{\circ}-105^{\circ}=75^{\circ}\).

Step4: Find angle \(z\)

Angle \(z\) and angle \(x\) are vertical angles. Since vertical angles are equal, \(z=x = 75^{\circ}\).

Answer:

\(\angle x = 75^{\circ}\), \(\angle z=75^{\circ}\)