Question

in the figure, line m is parallel to line n, and line q is perpendicular to line p. suppose m ∠1 = 35°. find m ∠7.

Explanation:

Step1: Find \(m\angle2\)

Since \(\angle1\) and \(\angle2\) are supplementary (\(\angle1+\angle2 = 180^{\circ}\)), and \(m\angle1 = 35^{\circ}\), then \(m\angle2=180^{\circ}-35^{\circ}=145^{\circ}\).

Step2: Use the property of parallel lines

Because \(m\parallel n\), \(\angle2\) and \(\angle3\) are alternate - interior angles. So \(m\angle3 = m\angle2=145^{\circ}\).

Step3: Use the right - angle property

Since \(q\perp p\), \(m\angle8 = 90^{\circ}\). And \(\angle3+\angle7+\angle8=180^{\circ}\) (a straight line).
Substitute \(m\angle3 = 145^{\circ}\) and \(m\angle8 = 90^{\circ}\) into the equation: \(145^{\circ}+m\angle7 + 90^{\circ}=180^{\circ}\).
Then \(m\angle7=180^{\circ}-(145^{\circ}+90^{\circ})=180^{\circ}-235^{\circ}\) (This is wrong, we should use another way.
Another way: Since \(m\parallel n\), \(\angle1\) and \(\angle4\) are corresponding angles, so \(m\angle4 = 35^{\circ}\). And \(q\perp p\), so \(\angle5+\angle4 = 90^{\circ}\), then \(\angle5 = 90^{\circ}-35^{\circ}=55^{\circ}\). \(\angle7\) and \(\angle5\) are vertical angles. So \(m\angle7=m\angle5 = 55^{\circ}\).

Answer:

\(55\)