Question

apply: finding unknown angle measures
the five angles shown are formed by two lines and a ray.
find the measure of each angle.
click or tap the correct measure from each drop - down menu.
the measure of ∠1 is choose
the measure of ∠2 is choose
the measure of ∠3 is choose
(there is an image on the right side of the text showing angles formed by two lines and a ray, with a 72° angle, and angles labeled 1, 2, 3)

Explanation:

Step1: Analyze ∠3

∠3 is a right angle (marked with the right - angle symbol), so \(m\angle3 = 90^{\circ}\).

Step2: Analyze ∠2

The angle of \(72^{\circ}\), ∠3 (\(90^{\circ}\)) and ∠2 are on a straight line, so their sum is \(180^{\circ}\). Let \(m\angle2=x\), then \(72^{\circ}+90^{\circ}+x = 180^{\circ}\), \(x=180^{\circ}-72^{\circ}-90^{\circ}=18^{\circ}\)? Wait, no, wait. Wait, the vertical line and the horizontal line (the two lines) intersect, and ∠3 is right angle, the angle adjacent to \(72^{\circ}\) and ∠3: Wait, actually, the angle between the left - going ray and the vertical line is \(72^{\circ}\), ∠3 is right angle, so ∠2 and \(72^{\circ}\) are complementary? No, wait, let's look at the right angle. The right angle (∠3) and \(72^{\circ}\) and ∠1: Wait, no, the vertical line is perpendicular to the horizontal line? Wait, ∠3 is a right angle, so the vertical line and the horizontal line (the line with the two arrows) are perpendicular, so \(m\angle3 = 90^{\circ}\). Then, the angle of \(72^{\circ}\), ∠1 and the right angle? Wait, no, the sum of \(72^{\circ}\), ∠1 and the right angle? Wait, no, the three angles \(72^{\circ}\), ∠1 and the right angle (∠3) are on a straight line? No, the straight line is the horizontal line (the one with two arrows). The vertical line is perpendicular to it, so ∠3 is \(90^{\circ}\). Then, the angle between the left - going ray and the vertical line is \(72^{\circ}\), so ∠1 and \(72^{\circ}\) are complementary because they form a right angle (since the vertical line and the ray forming ∠1 and \(72^{\circ}\) are perpendicular? Wait, the right - angle symbol is between the left - going ray and the vertical line? Wait, no, the right - angle symbol is between the left - going ray and the vertical line, so \(72^{\circ}+\angle1 = 90^{\circ}\), so \(m\angle1=90^{\circ}-72^{\circ}=18^{\circ}\)? Wait, no, that can't be. Wait, maybe I got the diagram wrong. Wait, the right - angle symbol is between the vertical line and the ray forming ∠1? Wait, the diagram: left - going ray, then a right - angle symbol, then vertical line, then ∠1, then the other ray. Wait, no, let's re - examine. The two lines are a horizontal line (with two arrows) and a vertical line (with two arrows). A ray is coming from the intersection point. The angle between the left - going ray and the vertical line is \(72^{\circ}\), the angle between the vertical line and the ray forming ∠1 is ∠1, and the angle between the vertical line and the horizontal line (downward) is ∠2. Since the vertical line and horizontal line are perpendicular, ∠3 (between horizontal line and vertical line, downward) is \(90^{\circ}\). Then, the angle between the left - going ray and the vertical line is \(72^{\circ}\), so ∠1 and \(72^{\circ}\) are complementary (because they form a right angle, since the vertical line and the ray forming ∠1 and the \(72^{\circ}\) angle are perpendicular? Wait, the right - angle symbol is between the left - going ray and the vertical line? No, the right - angle symbol is between the left - going ray and the vertical line, so \(72^{\circ}+\angle1 = 90^{\circ}\), so \(m\angle1 = 90 - 72=18^{\circ}\)? No, that seems small. Wait, maybe the right - angle symbol is between the vertical line and the ray forming ∠1. So \(72^{\circ}+\angle1=90^{\circ}\), so ∠1 = \(18^{\circ}\). Then, ∠2: since the horizontal line and vertical line are perpendicular, and ∠3 is \(90^{\circ}\), and the angle adjacent to \(72^{\circ}\) and ∠3: Wait, the sum of angles on a straight line is \(180^{\circ}\). The left - going ray, the vertical line, and the downward vertical line (∠2) and the horizontal line. Wait, the angle between the left - going ray and the vertical line is \(72^{\circ}\), ∠3 is \(90^{\circ}\), so ∠2 = \(180^{\circ}-72^{\circ}-90^{\circ}=18^{\circ}\)? No, that's not right. Wait, maybe the right - angle is between the horizontal line (left - going) and the vertical line, so ∠3 is \(90^{\circ}\). Then, the angle of \(72^{\circ}\) is between the left - going ray and the vertical line, so the angle between the left - going ray and the horizontal line (downward) is ∠2, and ∠3 is \(90^{\circ}\), so \(72^{\circ}+\angle2 = 90^{\circ}\), so ∠2 = \(18^{\circ}\), and ∠1: since the vertical line and the ray forming ∠1 are such that ∠1 and ∠2 are complementary? No, wait, let's start over.

1. For ∠3: The symbol indicates a right angle, so \(m\angle3 = 90^{\circ}\).

2. For ∠1: The angle of \(72^{\circ}\), ∠1, and the right angle (∠3) are not on the same line. Wait, the two lines are the horizontal line (with two arrows) and the vertical line (with two arrows). The ray is another line. The angle between the left - going ray and the vertical line is \(72^{\circ}\), and ∠1 is between the vertical line and the upper ray. Since the vertical line and the upper ray form a right angle with the left - going ray? Wait, no, the right - angle symbol is between the left - going ray and the vertical line, so the angle between left - going ray and vertical line is \(72^{\circ}\), so the angle between vertical line and upper ray (∠1) is \(90^{\circ}-72^{\circ}=18^{\circ}\)? No, that can't be. Wait, maybe I made a mistake. Wait, the sum of \(72^{\circ}\), ∠1 and \(90^{\circ}\) is \(180^{\circ}\)? No, \(72 + 90+18 = 180\), so ∠1 = \(18^{\circ}\), ∠2: since ∠2 and \(72^{\circ}\) are vertical angles? No, ∠2 and the angle of \(72^{\circ}\) are not vertical. Wait, no, the vertical line and horizontal line are perpendicular, so ∠3 is \(90^{\circ}\). The angle between the left - going ray and the vertical line is \(72^{\circ}\), so the angle between the left - going ray and the horizontal line (downward) is ∠2, and \(72^{\circ}+\angle2 = 90^{\circ}\), so ∠2 = \(18^{\circ}\), and ∠1 is equal to ∠2? No, that doesn't make sense. Wait, maybe the right - angle is between the upper ray and the vertical line. So \(72^{\circ}+\angle1 = 90^{\circ}\), so ∠1 = \(18^{\circ}\), ∠2: since the horizontal line and vertical line are perpendicular, ∠2 is \(90^{\circ}\)? No, that's not. Wait, I think I messed up the diagram. Let's assume that the vertical line is perpendicular to the horizontal line, so ∠3 = \(90^{\circ}\). The angle of \(72^{\circ}\) and ∠1 are complementary (since they form a right angle), so \(m\angle1=90^{\circ}-72^{\circ}=18^{\circ}\). Then, ∠2: since ∠2 and \(72^{\circ}\) are supplementary to ∠3? No, ∠2 is equal to \(72^{\circ}\)? No, that's not. Wait, no, the sum of angles on a straight line is \(180^{\circ}\). The horizontal line is a straight line, so the sum of the angle of \(72^{\circ}\), ∠3 (\(90^{\circ}\)) and ∠2 is \(180^{\circ}\)? No, \(72 + 90+\angle2=180\), so ∠2 = \(18^{\circ}\), and ∠1 is equal to ∠2? No, ∠1 is \(18^{\circ}\), ∠2 is \(18^{\circ}\), ∠3 is \(90^{\circ}\). Wait, that seems possible.

Step3: Recap

- \(m\angle3 = 90^{\circ}\) (right - angle symbol).
- For ∠1: The angle of \(72^{\circ}\) and ∠1 form a right angle (since the right - angle is between the left - going ray and the vertical line, so \(72^{\circ}+\angle1 = 90^{\circ}\)), so \(m\angle1=90^{\circ}-72^{\circ}=18^{\circ}\).
- For ∠2: The sum of \(72^{\circ}\), ∠3 (\(90^{\circ}\)) and ∠2 is \(180^{\circ}\) (straight line), so \(72^{\circ}+90^{\circ}+m\angle2 = 180^{\circ}\), \(m\angle2=180^{\circ}-72^{\circ}-90^{\circ}=18^{\circ}\). Or, since ∠1 and ∠2 are vertical angles? Wait, no, but in this case, both are \(18^{\circ}\), and ∠3 is \(90^{\circ}\).

Answer:

The measure of \(\angle1\) is \(18^{\circ}\), the measure of \(\angle2\) is \(18^{\circ}\), the measure of \(\angle3\) is \(90^{\circ}\)