Question

1. find the values of x and y. (18y + 5)° (10x − 61)° (x + 10)°

Explanation:

Step1: Identify Vertical Angles

Vertical angles are equal. So, \(10x - 61 = x + 10\).

Step2: Solve for \(x\)

Subtract \(x\) from both sides: \(10x - x - 61 = 10\) → \(9x - 61 = 10\).
Add 61 to both sides: \(9x = 10 + 61\) → \(9x = 71\)? Wait, no, wait. Wait, maybe I misread. Wait, actually, looking at the diagram, the angle \((10x - 61)^\circ\) and \((x + 10)^\circ\) – wait, no, maybe they are supplementary? Wait, no, the straight line: the angle \((18y + 5)^\circ\) and \((10x - 61)^\circ\) – wait, no, let's re-express. Wait, the two angles \(10x - 61\) and \(x + 10\) – maybe they are alternate interior angles? Wait, no, the diagram shows two intersecting lines, so vertical angles. Wait, no, maybe the angle \(10x - 61\) and \(x + 10\) are equal? Wait, no, maybe the angle \(18y + 5\) and \(x + 10\) are complementary? Wait, no, let's start over.

Wait, the diagram: there's a straight line (horizontal), and another line crossing it, forming angles. So, the angle \((10x - 61)^\circ\) and \((x + 10)^\circ\) – wait, maybe \(10x - 61 = x + 10\) (vertical angles). Let's solve that:

\(10x - x = 10 + 61\)
\(9x = 71\)? No, that can't be. Wait, maybe I made a mistake. Wait, maybe the angle \((10x - 61)^\circ\) and \((x + 10)^\circ\) are supplementary? No, vertical angles are equal. Wait, maybe the other angle: \((18y + 5)^\circ\) and \((x + 10)^\circ\) are complementary? No, let's check again.

Wait, perhaps the angle \(10x - 61\) and \(x + 10\) are equal (vertical angles). So:

\(10x - 61 = x + 10\)
\(10x - x = 10 + 61\)
\(9x = 71\) → \(x = 71/9\)? That's not an integer. Maybe I misread the angles. Wait, maybe the angle is \(10x - 6\) instead of \(61\)? No, the user wrote \(10x - 61\). Wait, maybe the angle \((18y + 5)^\circ\) and \((10x - 61)^\circ\) are supplementary? Wait, no, the horizontal line: the sum of angles on a straight line is \(180^\circ\). Wait, maybe the angle \((18y + 5)^\circ\) and \((10x - 61)^\circ\) are vertical angles, and \((x + 10)^\circ\) is complementary? No, let's think again.

Wait, perhaps the angle \(10x - 61\) and \(x + 10\) are equal (vertical angles). So:

\(10x - 61 = x + 10\)
\(10x - x = 10 + 61\)
\(9x = 71\) → \(x = 71/9\) ≈ 7.89. That seems odd. Maybe the problem has a typo, but assuming the angles are vertical, let's proceed. Then, the angle \(18y + 5\) and \(x + 10\) are supplementary? Wait, no, the sum of angles on a straight line is \(180^\circ\). Wait, maybe \(18y + 5 + 10x - 61 = 180\)? No, that's not right. Wait, maybe the angle \(18y + 5\) and \(x + 10\) are equal? No, vertical angles are opposite. Wait, maybe I made a mistake in identifying the angles. Let's re-express:

Looking at the diagram, there are two intersecting lines: one horizontal, one diagonal. So, the angle \((10x - 61)^\circ\) and \((x + 10)^\circ\) are vertical angles (opposite each other), so they should be equal. Then, the angle \((18y + 5)^\circ\) and the angle adjacent to \(x + 10\) are supplementary. Wait, no, let's solve for \(x\) first:

\(10x - 61 = x + 10\)
\(10x - x = 10 + 61\)
\(9x = 71\) → \(x = 71/9\) ≈ 7.89. Then, the angle \(x + 10 = 71/9 + 10 = 71/9 + 90/9 = 161/9 ≈ 17.89^\circ\). Then, the angle \(18y + 5\) should be equal to \(10x - 61\) (vertical angles), so \(18y + 5 = 10x - 61\). Substitute \(x = 71/9\):

\(18y + 5 = 10*(71/9) - 61 = 710/9 - 549/9 = 161/9\)
\(18y = 161/9 - 5 = 161/9 - 45/9 = 116/9\)
\(y = (116/9)/18 = 116/162 = 58/81 ≈ 0.716\). That seems odd. Maybe the problem has a typo, but assuming the angles are correct, this is the solution. Wait, maybe I misread the angle: maybe it's \(10x - 6\) instead of \(61\)? Let's check:

If \(10x - 6 = x + 10\), then \(9x = 16\), \(x = 16/9\), still odd. Maybe \(10x + 61\)? No, the user wrote \(10x - 61\). Alternatively, maybe the angle \((10x - 61)^\circ\) and \((x + 10)^\circ\) are supplementary (sum to \(180^\circ\))? Let's try that:

\(10x - 61 + x + 10 = 180\)
\(11x - 51 = 180\)
\(11x = 231\)
\(x = 21\). Ah, that makes sense! Maybe I misidentified the angles. So, if they are supplementary (adjacent angles on a straight line), then:

Step1: Set up supplementary angles

\(10x - 61 + x + 10 = 180\)
Simplify: \(11x - 51 = 180\)

Step2: Solve for \(x\)

Add 51 to both sides: \(11x = 180 + 51 = 231\)
Divide by 11: \(x = 231 / 11 = 21\)

Step3: Find \(y\)

Now, the angle \((18y + 5)^\circ\) and \((x + 10)^\circ\) are vertical angles (opposite each other), so they are equal.
\(x + 10 = 21 + 10 = 31\)
Thus, \(18y + 5 = 31\)
Subtract 5: \(18y = 31 - 5 = 26\)? No, wait, \(31 - 5 = 26\)? No, \(31 - 5 = 26\)? Wait, \(18y + 5 = 31\) → \(18y = 26\) → \(y = 26/18 = 13/9 ≈ 1.44\). No, that's not right. Wait, maybe \((18y + 5)^\circ\) and \((10x - 61)^\circ\) are vertical angles. Let's check:

\(10x - 61 = 10*21 - 61 = 210 - 61 = 149\)
Then, \(18y + 5 = 149\)
\(18y = 144\)
\(y = 8\). Ah! That makes sense. So, the mistake was in identifying the vertical angles. So, \((10x - 61)^\circ\) and \((18y + 5)^\circ\) are vertical angles, and \((x + 10)^\circ\) is supplementary to \((10x - 61)^\circ\) (since they are adjacent on a straight line). Let's correct:

Step1: Supplementary angles for \(x\)

\(10x - 61 + x + 10 = 180\)
\(11x - 51 = 180\)
\(11x = 231\)
\(x = 21\)

Step2: Vertical angles for \(y\)

\(10x - 61 = 18y + 5\)
Substitute \(x = 21\):
\(10*21 - 61 = 18y + 5\)
\(210 - 61 = 18y + 5\)
\(149 = 18y + 5\)
Subtract 5: \(144 = 18y\)
Divide by 18: \(y = 8\)

Answer:

\(x = 21\), \(y = 8\)