Question

letters x, y, and z are angle measures. which equations would guarantee that lines p and q are parallel? check all that apply. x = z x + y = 180° x + z = 180° x = y z = 180°

Explanation:

Step1: Recall parallel line theorems

For lines \(p\) and \(q\) to be parallel, we can use the same - side interior angles theorem (if the sum of two same - side interior angles is \(180^{\circ}\), the lines are parallel) and the alternate interior angles theorem (if alternate interior angles are equal, the lines are parallel).

Step2: Analyze each option

• Option \(x = z\):

If \(x = z\), by the alternate interior angles theorem, lines \(p\) and \(q\) are parallel.

• Option \(x + y=180^{\circ}\):

This equation does not correspond to any of the parallel - line angle theorems. \(x\) and \(y\) are not in a position (same - side interior, alternate interior, etc.) that would guarantee parallel lines.

• Option \(x + z = 180^{\circ}\):

By the same - side interior angles theorem, if \(x+z = 180^{\circ}\), lines \(p\) and \(q\) are parallel.

• Option \(x = y\):

This equation does not correspond to any of the parallel - line angle theorems. \(x\) and \(y\) are not in a position (same - side interior, alternate interior, etc.) that would guarantee parallel lines.

• Option \(z = 180^{\circ}\):

An angle \(z = 180^{\circ}\) is a straight angle. But this does not relate to the parallel - line angle relationships (same - side interior, alternate interior, etc.) that would guarantee \(p\parallel q\).

Answer:

\(x = z\), \(x + z=180^{\circ}\)