Question

line m is parallel to line n. line t intersects lines m and n. determine whether each statement is always true, sometimes true, or never true. $m\angle1 = m\angle3$ $m\angle2 + m\angle8 = 180^{\circ}$ $m\perp n$ $\angle5\cong\angle3$ $m\angle7 - m\angle6 = 0^{\circ}$

Explanation:

Step1: Analyze \(m\angle1 = m\angle3\)

\(\angle1\) and \(\angle3\) are vertical angles. Vertical angles are always equal. So \(m\angle1 = m\angle3\) is Always True. The initial marking of "Sometimes True" for this is wrong.

Step2: Analyze \(m\angle2 + m\angle8=180^{\circ}\)

Since \(m\parallel n\), \(\angle2\) and \(\angle6\) are corresponding angles (\(m\angle2 = m\angle6\)). And \(\angle6\) and \(\angle8\) are supplementary (\(m\angle6 + m\angle8=180^{\circ}\)), so \(m\angle2 + m\angle8 = 180^{\circ}\) is Always True.

Step3: Analyze \(m\perp n\)

Lines \(m\) and \(n\) are parallel. Parallel lines are never perpendicular (\(m\perp n\) means the angle between them is \(90^{\circ}\)). So \(m\perp n\) is Never True.

Step4: Analyze \(\angle5\cong\angle3\)

\(\angle5\) and \(\angle3\) are alternate - interior angles. When \(m\parallel n\), alternate - interior angles are congruent. So \(\angle5\cong\angle3\) is Always True.

Step5: Analyze \(m\angle7 - m\angle6=0^{\circ}\)

\(\angle7\) and \(\angle6\) are vertical angles. Vertical angles are equal (\(m\angle7=m\angle6\)), so \(m\angle7 - m\angle6 = 0^{\circ}\) is Always True.

Answer:

The statement \(m\angle1 = m\angle3\) should be marked as Always True (the initial "Sometimes True" is wrong).