Question

find ( mangle 1 ) and ( mangle 2 ).

enter your answer in degrees by entering deg after any value that is in degrees.

a) ( mangle 1 )

b) ( mangle 2 )

Explanation:

Step1: Identify Vertical Angles and Parallel Lines

The two horizontal lines are parallel (since they are both horizontal and cut by a transversal, the vertical line). The angle of \(97^\circ\) and \(\angle 1\) are same - side interior angles? Wait, no, actually, the angle of \(97^\circ\) and \(\angle 1\) are supplementary? Wait, no, looking at the diagram, the two horizontal lines are parallel, and the vertical line is a transversal. Wait, actually, the angle of \(97^\circ\) and \(\angle 1\) are corresponding angles? Wait, no, let's think again. The angle of \(97^\circ\) and \(\angle 1\): since the two horizontal lines are parallel, and the vertical line is a transversal, the angle adjacent to \(97^\circ\) (the linear pair) and \(\angle 1\) would be equal? Wait, no, actually, \(\angle 1\) and the \(97^\circ\) angle: are they supplementary? Wait, no, let's look at the straight line. The angle of \(97^\circ\) and its adjacent angle (on the straight line) form a linear pair, so that adjacent angle is \(180 - 97=83^\circ\)? No, wait, no. Wait, the two horizontal lines are parallel, and the vertical line is a transversal. The angle of \(97^\circ\) and \(\angle 1\): if we consider the vertical line as a transversal, the \(97^\circ\) angle and \(\angle 1\) are same - side interior angles? Wait, no, maybe I made a mistake. Wait, actually, \(\angle 1\) and the angle of \(97^\circ\): since the two horizontal lines are parallel, the angle of \(97^\circ\) and \(\angle 1\) are supplementary? No, wait, no. Wait, the vertical line is perpendicular? No, the vertical line is a transversal. Wait, maybe the two horizontal lines are parallel, and the angle of \(97^\circ\) and \(\angle 1\) are equal? No, that can't be. Wait, no, let's look at the diagram again. The first horizontal line (top) and the second horizontal line (bottom) are parallel. The vertical line is a transversal. The angle of \(97^\circ\) is above the top horizontal line and to the left of the vertical line. \(\angle 1\) is above the bottom horizontal line and to the right of the vertical line. Wait, maybe \(\angle 1\) and the \(97^\circ\) angle are supplementary? No, wait, no. Wait, actually, the angle of \(97^\circ\) and \(\angle 1\): since the two horizontal lines are parallel, the alternate - interior angles or corresponding angles. Wait, no, let's think about linear pairs. Wait, the angle of \(97^\circ\) and \(\angle 1\): if we consider that the vertical line is a straight line, no, the horizontal lines are parallel. Wait, maybe \(\angle 1 = 97^\circ\)? No, that doesn't make sense. Wait, no, wait, the angle of \(97^\circ\) and \(\angle 1\): are they vertical angles? No. Wait, maybe I misread the diagram. Wait, the problem is to find \(m\angle1\) and \(m\angle2\). Let's start with \(\angle1\). The angle of \(97^\circ\) and \(\angle1\): since the two horizontal lines are parallel, and the vertical line is a transversal, the angle of \(97^\circ\) and \(\angle1\) are same - side interior angles? No, same - side interior angles are supplementary. Wait, \(180 - 97 = 83\)? No, that's not right. Wait, no, maybe the two horizontal lines are parallel, so the angle of \(97^\circ\) and \(\angle1\) are equal. Wait, no, that would mean \(\angle1 = 97^\circ\). Then, for \(\angle2\), since \(\angle1\) and \(\angle2\) form a linear pair (they are on a straight line), so \(m\angle2=180 - m\angle1\). Wait, let's check again.

Wait, the diagram: there are two horizontal lines (parallel) and a vertical line (transversal). The top horizontal line has an angle of \(97^\circ\) between the vertical line and the top horizontal line (left - top). The bottom horizontal line has \(\angle1\) between the vertical line and the bottom horizontal line (right - top) and \(\angle2\) between the vertical line and the bottom horizontal line (right - bottom). So, since the two horizontal lines are parallel, the angle of \(97^\circ\) and \(\angle1\) are corresponding angles, so \(m\angle1 = 97^\circ\)? Wait, no, that can't be, because then \(\angle1\) and \(\angle2\) would form a linear pair, so \(m\angle2=180 - 97 = 83^\circ\). Wait, but let's confirm.

Step1: Find \(m\angle1\)

The two horizontal lines are parallel, and the vertical line is a transversal. The angle of \(97^\circ\) and \(\angle1\) are corresponding angles (since they are in the same position relative to the parallel lines and the transversal). So, by the Corresponding Angles Postulate, \(m\angle1=97^\circ\).

Step2: Find \(m\angle2\)

\(\angle1\) and \(\angle2\) form a linear pair (they are adjacent angles that form a straight line). The sum of the measures of angles in a linear pair is \(180^\circ\). So, \(m\angle1 + m\angle2=180^\circ\). We know \(m\angle1 = 97^\circ\), so \(m\angle2=180 - 97 = 83^\circ\). Wait, no, that's the opposite. Wait, no, if \(\angle1\) is \(97^\circ\), then \(m\angle2 = 180 - 97=83^\circ\)? Wait, no, maybe I got the corresponding angles wrong. Wait, maybe the angle of \(97^\circ\) and \(\angle1\) are supplementary. Wait, let's think again. The angle of \(97^\circ\) and its adjacent angle (on the top horizontal line) form a linear pair, so that adjacent angle is \(180 - 97 = 83^\circ\). Then, since the two horizontal lines are parallel, that adjacent angle ( \(83^\circ\)) and \(\angle1\) are corresponding angles, so \(m\angle1 = 83^\circ\)? No, that's confusing. Wait, maybe the vertical line is perpendicular to the horizontal lines? No, the diagram doesn't show that. Wait, let's start over.

The problem is to find \(m\angle1\) and \(m\angle2\). Let's look at the straight line (the vertical line) and the horizontal lines. The angle of \(97^\circ\) and \(\angle1\): if we consider that the two horizontal lines are parallel, then the angle of \(97^\circ\) and \(\angle1\) are same - side interior angles, which are supplementary. So, \(m\angle1=180 - 97 = 83^\circ\)? No, that's not right. Wait, no, same - side interior angles are supplementary when the lines are parallel. Wait, the two horizontal lines are parallel, and the vertical line is a transversal. The angle of \(97^\circ\) is above the top horizontal line, left of the vertical line. \(\angle1\) is above the bottom horizontal line, right of the vertical line. So, they are same - side interior angles, so \(m\angle1 + 97^\circ=180^\circ\), so \(m\angle1 = 180 - 97=83^\circ\)? No, that can't be. Wait, maybe I have the direction wrong. Wait, maybe the angle of \(97^\circ\) and \(\angle1\) are equal. Let's assume that the two horizontal lines are parallel, and the vertical line is a transversal, so the corresponding angles are equal. The angle of \(97^\circ\) and \(\angle1\) are corresponding angles, so \(m\angle1 = 97^\circ\). Then, since \(\angle1\) and \(\angle2\) form a linear pair (they are on a straight line), \(m\angle2=180 - 97 = 83^\circ\).

Wait, let's check with the linear pair. If \(\angle1\) is \(97^\circ\), then \(\angle2 = 180 - 97 = 83^\circ\). Alternatively, if \(\angle1\) is \(83^\circ\), then \(\angle2 = 97^\circ\). Wait, maybe the angle of \(97^\circ\) and \(\angle2\) are equal? No, that doesn't make sense. Wait, maybe the vertical line is a transversal, and the angle of \(97^\circ\) and \(\angle2\) are corresponding angles. So \(m\angle2 = 97^\circ\), and then \(m\angle1=180 - 97 = 83^\circ\). Ah, that makes more sense. Let's re - analyze:

The top horizontal line and the bottom horizontal line are parallel. The vertical line is a transversal. The angle of \(97^\circ\) is above the top horizontal line, left of the vertical line. \(\angle2\) is below the bottom horizontal line, right of the vertical line. So, they are corresponding angles, so \(m\angle2 = 97^\circ\). Then, \(\angle1\) and \(\angle2\) form a linear pair, so \(m\angle1=180 - m\angle2=180 - 97 = 83^\circ\).

Yes, that makes sense. So:

Step1: Find \(m\angle2\)

Since the two horizontal lines are parallel and the vertical line is a transversal, the \(97^\circ\) angle and \(\angle2\) are corresponding angles. Corresponding angles are equal when lines are parallel. So \(m\angle2 = 97^\circ\).

Step2: Find \(m\angle1\)

\(\angle1\) and \(\angle2\) form a linear pair, so their measures add up to \(180^\circ\). So \(m\angle1=180^\circ - m\angle2\). Substitute \(m\angle2 = 97^\circ\) into the formula: \(m\angle1=180 - 97 = 83^\circ\).

Answer:

a) \(m\angle1 = 83^\circ\) (or \(83\) deg)
b) \(m\angle2 = 97^\circ\) (or \(97\) deg)