Question

use the figure below to answer the following questions.
find the value of y.
type your answer...

Explanation:

Step1: Find angle 2

Since the sum of angles on a straight line is \(180^\circ\), and we have a right angle (\(90^\circ\)) and the \(72^\circ\) angle, we first find angle 2. Wait, actually, looking at the vertical angles or the triangle. Wait, another approach: the angle \(72^\circ\), angle 2, and the right angle? Wait, no, let's look at the straight line at the top. The sum of \(72^\circ\), angle 2, and the right angle? Wait, no, the triangle has a right angle, so the angles in the triangle: one angle is \(40^\circ\), right angle \(90^\circ\), so the third angle is \(180 - 90 - 40 = 50^\circ\)? Wait, no, maybe using vertical angles or linear pairs. Wait, the angle \((8y - 16)^\circ\) and the angle formed by \(72^\circ\), angle 2, and the \(40^\circ\)? Wait, maybe better: the straight line at the top, the sum of \(72^\circ\), angle 2, and the angle \((8y - 16)^\circ\) and the right angle? No, let's see: the angle \(72^\circ\), angle 2, and the right angle (since there's a right angle symbol) – wait, no, the right angle is on the horizontal line. Wait, maybe the key is that the angle \(72^\circ\), angle 2, and the right angle? No, let's look at the triangle. The triangle has a right angle (\(90^\circ\)), a \(40^\circ\) angle, so the third angle (at the top) is \(180 - 90 - 40 = 50^\circ\). Then, the angle \((8y - 16)^\circ\) and the angle \(72^\circ\) and angle 2: wait, maybe angle 2 is \(180 - 72 - (8y - 16)\)? No, maybe vertical angles. Wait, another way: the angle \(72^\circ\) and the angle \((8y - 16)^\circ\) and angle 2: wait, actually, the sum of \(72^\circ\), angle 2, and the angle \((8y - 16)^\circ\) should be \(180^\circ\) (since they are on a straight line). But angle 2: since there's a right angle in the triangle, angle 2 + \(40^\circ\) + \(90^\circ\)? No, maybe I'm overcomplicating. Wait, let's look at the angle \((8y - 16)^\circ\) and the angle that is vertical or supplementary. Wait, the angle \(72^\circ\), angle 2, and the angle \((8y - 16)^\circ\) are on a straight line, so \(72 + \angle 2 + (8y - 16) = 180\). But also, in the triangle, angle 2 + \(40 + 90 = 180\)? No, angle 2 is part of the triangle? Wait, no, the triangle has a right angle, so angle 2 + \(40^\circ\) + the angle adjacent to \((8y - 16)^\circ\)? Wait, maybe the correct approach is: the angle \((8y - 16)^\circ\) is equal to \(72^\circ + 40^\circ\)? Wait, no, \(72 + 40 = 112\), but that doesn't make sense. Wait, maybe the angle \((8y - 16)^\circ\) and the angle \(72^\circ\) are related to the triangle. Wait, let's try: the sum of angles on a straight line is \(180^\circ\). The angle \(72^\circ\), angle 2, and the angle \((8y - 16)^\circ\) – but angle 2: since there's a right angle in the triangle, angle 2 = \(180 - 90 - 40 = 50^\circ\)? Wait, no, angle 2 is adjacent to the right angle? Wait, the right angle is on the horizontal line, so the vertical line is perpendicular, so angle 2 + \(40^\circ\) + the right angle? No, the vertical line is perpendicular, so angle 2 is \(180 - 72 - 90\)? Wait, \(180 - 72 - 90 = 18\)? No, that can't be. Wait, maybe I made a mistake. Let's start over.

Looking at the figure, there's a straight line at the top (where the angles \(72^\circ\), \(\angle 2\), and \((8y - 16)^\circ\) are) and a right angle in the triangle. The triangle has a right angle (\(90^\circ\)) and a \(40^\circ\) angle, so the third angle (at the vertex where the vertical line and the slant line meet) is \(180 - 90 - 40 = 50^\circ\). Now, the angle \((8y - 16)^\circ\) and the \(72^\circ\) angle: wait, maybe the angle \((8y - 16)^\circ\) is equal to \(72^\circ + 50^\circ\)? Wait, \(72 + 50 = 122\), so \(8y - 16 = 122\)? No, that would be \(8y = 138\), \(y = 17.25\), which doesn't seem right. Wait, maybe the angle \((8y - 16)^\circ\) and the angle \(72^\circ\) are supplementary to the angle in the triangle. Wait, another approach: the angle \(72^\circ\), \(\angle 2\), and the right angle (90°) – no, the right angle is on the horizontal. Wait, the vertical line is perpendicular to the horizontal, so the angle between the vertical line and the slant line is \(40^\circ\), so the angle between the slant line (with \(8y - 16\)) and the other slant line (with \(72^\circ\)): maybe the angle \((8y - 16)^\circ\) is equal to \(72^\circ + 40^\circ\)? Wait, \(72 + 40 = 112\), so \(8y - 16 = 112\). Then, \(8y = 112 + 16 = 128\), so \(y = 16\). Let's check: \(8*16 - 16 = 128 - 16 = 112\). Then, \(72 + 40 = 112\), which matches. So that makes sense.

Step1: Set up the equation

The angle \((8y - 16)^\circ\) is equal to the sum of \(72^\circ\) and \(40^\circ\) (since they are supplementary or form a linear pair with the triangle's angle). So:
\(8y - 16 = 72 + 40\)

Step2: Solve for y

Simplify the right side: \(72 + 40 = 112\)
So, \(8y - 16 = 112\)
Add 16 to both sides: \(8y = 112 + 16 = 128\)
Divide both sides by 8: \(y = \frac{128}{8} = 16\)

Answer:

16