Question

writing algebraic equations
letters x, y, and z are angle measures. which equations would guarantee that lines p and q are parallel? choose two correct answers.
$x + z = 180^{circ}$
$x = y$
$x = z$
$x + y = 180^{circ}$

Explanation:

Step1: Recall parallel line angle - relationships

When two lines are parallel and cut by a transversal:

• If \(x + z=180^{\circ}\), \(x\) and \(z\) are same - side exterior and interior angles. By the same - side interior angles converse (if same - side interior angles are supplementary, then the lines are parallel).
• If \(x = y\), \(x\) and \(y\) are alternate interior angles. By the alternate interior angles converse (if alternate interior angles are equal, then the lines are parallel).

Step2: Analyze other equations

• If \(x = z\), there is no parallel - line converse theorem that directly uses this relationship for the given angle positions.
• If \(x + y=180^{\circ}\), since \(y + z = 180^{\circ}\) (linear pair), then \(x=z\) (by subtraction property of equality), which is not a valid parallel - line converse condition for the given angle positions.

Answer:

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