Question

1. what is the value of y? (there is a diagram of intersecting lines with angles labeled ((y - 14)^circ), ((-3x + 211)^circ), ((9x - 5)^circ) and multiple - choice options at the bottom: 9, 37, 33, 23, 16)

Explanation:

Step1: Identify Vertical Angles

Vertical angles are equal. So, \(-3x + 211 = 9x - 5\) (since they are vertical angles).

Step2: Solve for \(x\)

Add \(3x\) to both sides: \(211 = 12x - 5\).
Add 5 to both sides: \(216 = 12x\).
Divide by 12: \(x = 18\).

Step3: Find Adjacent Angle to \(y - 14\)

Adjacent angles on a straight line sum to \(180^\circ\). First, find the measure of \(-3x + 211\) (or \(9x - 5\)) when \(x = 18\).
For \(9x - 5\): \(9(18) - 5 = 162 - 5 = 157^\circ\).
So, the angle adjacent to \((y - 14)^\circ\) is \(157^\circ\), so \((y - 14) + 157 = 180\) (since they are supplementary).

Step4: Solve for \(y\)

Simplify: \(y - 14 = 180 - 157 = 23\).
Add 14: \(y = 23 + 14 = 37\). Wait, no—wait, correction: Wait, \((y - 14)\) and \(-3x + 211\) are vertical angles? Wait, no, let's re - check. Wait, the angle \((y - 14)^\circ\) and \(-3x + 211\) are they vertical? Wait, no, the two lines intersect, so \((y - 14)^\circ\) and \(-3x + 211\) are supplementary? Wait, no, vertical angles: when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. Let's re - express the angles. The angle \((y - 14)^\circ\) and the angle \(-3x + 211\) are adjacent? Wait, no, let's look at the diagram again. Wait, the angle \((y - 14)^\circ\) and \((-3x + 211)^\circ\) are adjacent (form a linear pair), so they should sum to \(180^\circ\). Wait, earlier mistake: vertical angles are \(-3x + 211\) and \(9x - 5\), which we solved \(x = 18\). Then, \(-3x + 211=-3(18)+211=-54 + 211 = 157^\circ\). So, \((y - 14)+157 = 180\) (since they are supplementary). So, \(y - 14 = 23\), so \(y = 23 + 14 = 37\)? Wait, but the options have 37? Wait, the options at the bottom: 9, 37, 33, 23, 16. So 37 is an option. Wait, but let's re - check the angle relationships. Alternatively, maybe \((y - 14)\) and \(9x - 5\) are vertical angles? Wait, no, let's re - identify the angles. The two intersecting lines: one line has angles \((y - 14)^\circ\) and the angle opposite to it (vertical angle) is... Wait, maybe I mixed up. Let's start over.

Correct Step1: Vertical angles are equal. So, the angle \(-3x + 211\) and \(9x - 5\) are vertical angles, so \(-3x + 211 = 9x - 5\).
Solving: \(211 + 5 = 9x+3x\) → \(216 = 12x\) → \(x = 18\).

Now, the angle \((y - 14)^\circ\) and \(-3x + 211\) are supplementary (since they form a linear pair). So, \((y - 14)+(-3x + 211)=180\).

Substitute \(x = 18\): \((y - 14)+(-3(18)+211)=180\) → \((y - 14)+(-54 + 211)=180\) → \((y - 14)+157 = 180\) → \(y - 14 = 180 - 157 = 23\) → \(y = 23 + 14 = 37\).

Answer:

37