Question

find the measure of the missing angle.
answer attempt 1 out of 5
a =

Explanation:

Step1: Identify the angle type

The two angles (63° and \(a\)) and the straight line form a linear pair? Wait, no, actually, looking at the diagram, the angle between the two lines (the vertical line and the other line) and angle \(a\) – wait, actually, the sum of angles around a point? No, wait, the diagram shows that the angle between the upper arrow and the middle arrow is 63°, and the middle arrow and the lower arrow form angle \(a\), and together they should be a straight angle? Wait, no, actually, the three arrows: the upper, middle, and lower. The middle arrow is between upper and lower. So the angle between upper and middle is 63°, and between middle and lower is \(a\), and together upper to lower is a straight line (180°)? Wait, no, the upper and lower arrows are in a straight line? Wait, the upper arrow is going up, lower is going down, so they are a straight line (180°). The middle arrow is between them, creating two angles: 63° and \(a\). So the sum of these two angles should be 180°? Wait, no, wait, if the upper and lower are a straight line (180°), then the angle between middle and upper is 63°, so angle \(a\) is 180° - 63°? Wait, no, wait, maybe it's a vertical angle? No, wait, let's think again. The diagram: upper arrow (up), middle arrow (left-up), lower arrow (down). The angle between upper and middle is 63°, and between middle and lower is \(a\). Wait, actually, the upper and lower arrows are colinear (forming a straight line), so the angle between upper and lower is 180°. The middle arrow splits this into two angles: 63° and \(a\). So \(63^\circ + a = 180^\circ\)? Wait, no, that would be if they are adjacent angles on a straight line. Wait, no, maybe the middle arrow is such that the angle between upper and middle is 63°, and between middle and lower is \(a\), and since upper and lower are a straight line, the sum of 63° and \(a\) is 180°? Wait, no, that can't be, because 180 - 63 is 117, but wait, maybe I got it wrong. Wait, no, maybe the angle between the middle arrow and the lower arrow is \(a\), and the angle between middle and upper is 63°, and the upper and lower are perpendicular? No, the upper and lower are vertical (up and down), so they are a straight line (180°). So the two angles (63° and \(a\)) are adjacent and form a linear pair, so their sum is 180°? Wait, no, that would mean \(a = 180 - 63 = 117\)? Wait, but that seems off. Wait, maybe the angle is a vertical angle? No, wait, maybe the diagram is of a vertical line (upper and lower arrows) and a line coming out at 63° from the upper arrow, so the angle between the vertical line (upper to lower) and the middle line is 63°, so the other angle (a) is 90°? No, no, the diagram: let's re-express. The upper arrow is along the positive y-axis, lower along negative y-axis (so straight line, 180°). The middle arrow is at 63° from the upper arrow (positive y-axis) towards the left. So the angle between positive y-axis (upper) and middle arrow is 63°, so the angle between middle arrow and negative y-axis (lower) is \(a\). So the sum of 63° and \(a\) should be 180°? Wait, no, that would be if they are adjacent angles on a straight line. Wait, yes, because the upper (positive y) and lower (negative y) are a straight line (180°), and the middle arrow is between them, so the two angles (63° and \(a\)) are adjacent and their sum is 180°. So \(a = 180^\circ - 63^\circ\).

Step2: Calculate \(a\)

\(a = 180^\circ - 63^\circ = 117^\circ\)? Wait, no, wait, that can't be. Wait, maybe I made a mistake. Wait, maybe the angle is a right angle? No, the diagram doesn't show a right angle. Wait, maybe the two angles (63° and \(a\)) are complementary? No, complementary is 90°. Wait, no, let's look again. Wait, the user's diagram: upper arrow (up), middle arrow (left-up), lower arrow (down). The angle between upper and middle is 63°, and between middle and lower is \(a\). Wait, maybe the upper and lower arrows are not a straight line? Wait, no, the upper is going up, lower down, so they are colinear (180°). So the middle arrow is between them, so the two angles (63° and \(a\)) add up to 180°? Wait, no, that would be if they are adjacent. Wait, maybe the angle is actually a vertical angle? No, vertical angles are equal. Wait, maybe I misinterpret the diagram. Wait, another approach: the angle between the middle arrow and the upper arrow is 63°, and the angle between the middle arrow and the lower arrow is \(a\), and since the upper and lower arrows are in a straight line, the sum of 63° and \(a\) is 180°? Wait, no, that would be 63 + a = 180, so a = 180 - 63 = 117. But wait, maybe the diagram is such that the angle between upper and middle is 63°, and the angle between middle and lower is \(a\), and they are supplementary? Wait, supplementary angles add to 180°, so yes. So:

Step2: Calculate \(a\)

\(a = 180^\circ - 63^\circ\)
\(a = 117^\circ\)

Wait, but that seems large. Wait, maybe the diagram is of a different configuration. Wait, maybe the angle is a linear pair, but actually, maybe the angle is 180 - 63 = 117. Let's confirm. If two angles are adjacent and form a straight line, their sum is 180°. So if one is 63°, the other is 180 - 63 = 117. So that's the calculation.

Answer:

\(a = \boxed{117^\circ}\)