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.

1. m∠1 + m∠4 = 180°
2. m∠5 - m∠7 = 0°
3. ∠2 ≅ ∠8
4. m∠2 - m∠3 = m∠1 - m∠4

Explanation:

Step1: Recall parallel - line angle relationships

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

Step2: Analyze \(m\angle1 + m\angle4=180^{\circ}\)

\(\angle1\) and \(\angle4\) are same - side interior angles. If two lines are parallel, same - side interior angles are supplementary (\(m\angle1 + m\angle4 = 180^{\circ}\)). Conversely, if \(m\angle1 + m\angle4=180^{\circ}\), then the two lines (line \(x\) and line \(y\)) are parallel. So the statement \(m\angle1 + m\angle4 = 180^{\circ}\) is Always True when \(x\parallel y\).

Step3: Analyze \(m\angle5−m\angle7 = 0^{\circ}\)

\(\angle5\) and \(\angle7\) are vertical angles. Vertical angles are always equal (\(m\angle5=m\angle7\)), so \(m\angle5 - m\angle7=0^{\circ}\) is Always True regardless of whether line \(x\) and line \(y\) are parallel or not.

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

\(\angle2\) and \(\angle8\) are corresponding angles. If line \(x\parallel y\), then corresponding angles are equal (\(\angle2\cong\angle8\)). So the statement \(\angle2\cong\angle8\) is Always True when \(x\parallel y\).

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

We know that \(\angle1\) and \(\angle2\) are corresponding angles, \(\angle3\) and \(\angle4\) are corresponding angles. If \(x\parallel y\), \(m\angle1=m\angle2\) and \(m\angle3=m\angle4\), then \(m\angle2−m\angle3=m\angle1 - m\angle4\) (since \(m\angle2 - m\angle3=m\angle1 - m\angle4\) can be rewritten as \(m\angle2 - m\angle1=m\angle3 - m\angle4\), and when \(x\parallel y\), \(m\angle2 - m\angle1 = 0\) and \(m\angle3 - m\angle4=0\)). So the statement \(m\angle2−m\angle3=m\angle1 - m\angle4\) is Always True when \(x\parallel y\).

Answer:

• \(m\angle1 + m\angle4 = 180^{\circ}\): Always True
• \(m\angle5−m\angle7 = 0^{\circ}\): Always True
• \(\angle2\cong\angle8\): Always True
• \(m\angle2−m\angle3=m\angle1 - m\angle4\): Always True