Question

12 find the value of y.

Explanation:

Step1: Identify parallel lines and alternate interior angles

The two horizontal lines are parallel, so the angles \((7x - 7)^\circ\) and \((3x + 45)^\circ\) are equal (alternate interior angles). So we set up the equation:
\(7x - 7 = 3x + 45\)

Step2: Solve for \(x\)

Subtract \(3x\) from both sides:
\(7x - 3x - 7 = 45\)
\(4x - 7 = 45\)
Add 7 to both sides:
\(4x = 45 + 7\)
\(4x = 52\)
Divide both sides by 4:
\(x = \frac{52}{4} = 13\)

Step3: Find the measure of the angle \((7x - 7)^\circ\)

Substitute \(x = 13\) into \((7x - 7)^\circ\):
\(7(13) - 7 = 91 - 7 = 84^\circ\)

Step4: Use the triangle angle sum or linear pair (assuming a right angle or supplementary angles, but looking at the diagram, the angle with \(63^\circ\) and the angle related to \(y\) and the parallel lines. Wait, actually, the angle \((7x - 7)^\circ = 84^\circ\), and we have a \(63^\circ\) angle, so the remaining angle in the triangle (or the angle related to \(y\)): Wait, maybe the two angles at the top and bottom are related, and the angle \((y - 6)^\circ\) is equal to \((7x - 7)^\circ - 63^\circ\)? Wait, let's re-examine.

Wait, the angle \((7x - 7)^\circ\) is at the top, and the angle \((3x + 45)^\circ\) is at the bottom, which we found \(x = 13\), so \((3x + 45)^\circ = 3(13) + 45 = 39 + 45 = 84^\circ\), so both are \(84^\circ\), so the lines are parallel, so the transversal creates alternate interior angles. Now, the angle with \(63^\circ\) and \((y - 6)^\circ\) should add up to \(84^\circ\)? Wait, no, maybe the angle \((7x - 7)^\circ\) is equal to \(63^\circ + (y - 6)^\circ\), because the angle at the top is split into \(63^\circ\) and the angle equal to \((y - 6)^\circ\) (alternate interior angles). Wait, let's check:

Since the two horizontal lines are parallel, the angle \((y - 6)^\circ\) should be equal to the angle adjacent to \(63^\circ\) that is part of the \((7x - 7)^\circ\) angle. So:

\(63 + (y - 6) = 7x - 7\)

We know \(x = 13\), so \(7x - 7 = 84\), so:

\(63 + y - 6 = 84\)

Simplify:

\(57 + y = 84\)

Subtract 57 from both sides:

\(y = 84 - 57 = 27\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's re-express.

Wait, the angle \((7x - 7)^\circ\) is at the top, and the angle \((3x + 45)^\circ\) is at the bottom, which are equal (alternate interior angles), so \(7x - 7 = 3x + 45\), solving gives \(x = 13\), so both angles are \(84^\circ\). Now, the angle at the bottom right is \((y - 6)^\circ\), and the angle at the top right is equal to \((y - 6)^\circ\) (alternate interior angles). The angle at the top is \(84^\circ\), which is composed of \(63^\circ\) and the angle equal to \((y - 6)^\circ\). So:

\(63 + (y - 6) = 84\)

\(y + 57 = 84\)

\(y = 84 - 57 = 27\)? Wait, that seems low. Wait, maybe the angle \((y - 6)^\circ\) is equal to \(84^\circ - 63^\circ\)? Let's calculate \(84 - 63 = 21\), so \(y - 6 = 21\), so \(y = 27\). Wait, that's the same as before. Wait, maybe that's correct.

Wait, let's verify:

If \(x = 13\), then \(7x - 7 = 84\), \(3x + 45 = 84\). Then, the angle at the top is \(84^\circ\), which is split into \(63^\circ\) and \(84 - 63 = 21^\circ\). So the angle \((y - 6)^\circ\) is \(21^\circ\), so \(y - 6 = 21\), so \(y = 27\).

Answer:

\(y = 27\)