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 the properties of parallel lines

When two lines are parallel and cut by a transversal, alternate - interior angles are equal, same - side interior angles are supplementary, and corresponding angles are equal.

Step2: Analyze each equation

• For \(x = z\):

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

• For \(x + y=180^{\circ}\):

This equation does not directly relate to the angle - relationships (alternate - interior, corresponding, same - side interior) that guarantee parallel lines.

• For \(x + z = 180^{\circ}\):

If \(x+z = 180^{\circ}\), then by the converse of the same - side interior angles theorem, lines \(p\) and \(q\) are parallel.

• For \(x = y\):

This equation does not directly relate to the angle - relationships (alternate - interior, corresponding, same - side interior) that guarantee parallel lines.

• For \(z = 180^{\circ}\):

An angle of \(180^{\circ}\) is a straight angle. This equation does not imply any relationship between the angles formed by lines \(p\) and \(q\) and a transversal to make \(p\) and \(q\) parallel.

Answer:

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