Question

line x is parallel to line y. line z intersects lines x and y. determine whether each statement is always true, sometimes true, or never true.
$m\angle1 + m\angle4=180^{circ}$
$m\angle5 - m\angle7 = 0^{circ}$
$z\perp y$
$\angle2\cong\angle8$
$m\angle2 - m\angle3=m\angle1 - m\angle4$

always true	sometimes true	never true

Explanation:

Step1: Analyze \(m\angle1 + m\angle4=180^{\circ}\)

\(\angle1\) and \(\angle4\) are a linear - pair. By the linear - pair postulate, the sum of the measures of two angles in a linear pair is \(180^{\circ}\). So \(m\angle1 + m\angle4 = 180^{\circ}\) is always true.

Step2: Analyze \(m\angle5 - m\angle7=0^{\circ}\)

\(\angle5\) and \(\angle7\) are vertical angles. Vertical angles are congruent, so \(m\angle5=m\angle7\), and \(m\angle5 - m\angle7 = 0^{\circ}\) is always true.

Step3: Analyze \(z\perp y\)

There is no information indicating that \(z\) is perpendicular to \(y\). The lines \(x\) and \(y\) are parallel, and \(z\) is just an intersecting line. So \(z\perp y\) is sometimes true (when the angles formed are \(90^{\circ}\)).

Step4: Analyze \(\angle2\cong\angle8\)

\(\angle2\) and \(\angle8\) are corresponding angles. Since \(x\parallel y\), corresponding angles are congruent. So \(\angle2\cong\angle8\) is always true.

Step5: Analyze \(m\angle2 - m\angle3=m\angle1 - m\angle4\)

We know \(m\angle1 + m\angle2=180^{\circ}\) and \(m\angle3 + m\angle4 = 180^{\circ}\), so \(m\angle2=180^{\circ}-m\angle1\) and \(m\angle3 = 180^{\circ}-m\angle4\). Then \(m\angle2 - m\angle3=(180 - m\angle1)-(180 - m\angle4)=m\angle4 - m\angle1
eq m\angle1 - m\angle4\) (except when \(m\angle1=m\angle4 = 90^{\circ}\)). So \(m\angle2 - m\angle3=m\angle1 - m\angle4\) is sometimes true.

Answer:

Statement	Always True	Sometimes True	Never True
\(m\angle5 - m\angle7 = 0^{\circ}\)	\(\boxed{\text{Yes}}\)
\(z\perp y\)		\(\boxed{\text{Yes}}\)
\(\angle2\cong\angle8\)	\(\boxed{\text{Yes}}\)
\(m\angle2 - m\angle3=m\angle1 - m\angle4\)		\(\boxed{\text{Yes}}\)