Question

given $m \parallel n$, find the value of $x$ and $y$.
(7y - 15)° (x - 14)° $m$
(2x + 8)° $n$
answer
$x = \square$ $y = \square$

Explanation:

Step1: Find x using vertical angles and corresponding angles

The angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) are corresponding angles (since \(m\parallel n\) and the transversal cuts them), so they are equal? Wait, no, actually, \((x - 14)^\circ\) and \((2x + 8)^\circ\) – wait, no, first, the angle \((x - 14)^\circ\) and the angle adjacent to \((7y - 15)^\circ\) on line \(m\) are supplementary? Wait, no, let's look again. Wait, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) – actually, \((x - 14)^\circ\) and \((2x + 8)^\circ\) are vertical angles? No, wait, the angle \((x - 14)^\circ\) and the angle \((2x + 8)^\circ\) – wait, no, the two angles \((x - 14)^\circ\) and \((2x + 8)^\circ\) are actually same - side? Wait, no, let's correct. The angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) are supplementary? Wait, no, because \(m\parallel n\), the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) – wait, no, the angle \((x - 14)^\circ\) and the angle \((2x + 8)^\circ\) are actually equal? Wait, no, let's think again. Wait, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) – no, the angle \((x - 14)^\circ\) and the angle \((2x + 8)^\circ\) are vertical angles? No, wait, the two lines \(m\) and \(n\) are parallel, and the transversal creates corresponding angles. Wait, actually, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) are supplementary? Wait, no, let's set up the equation. Wait, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) – wait, no, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) are actually equal? Wait, no, let's do it properly. The angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) – wait, no, the angle \((x - 14)^\circ\) and the angle \((2x + 8)^\circ\) are supplementary? Wait, no, let's see: the angle \((x - 14)^\circ\) and \((7y - 15)^\circ\) are supplementary (since they are adjacent on a straight line), so \((x - 14)+(7y - 15)=180\)? No, wait, no, the two angles \((x - 14)^\circ\) and \((7y - 15)^\circ\) are adjacent and form a linear pair, so they are supplementary: \((x - 14)+(7y - 15)=180\). Also, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) are equal (corresponding angles, since \(m\parallel n\))? Wait, no, corresponding angles are equal. Wait, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) – are they corresponding? Let's see, the transversal cuts \(m\) and \(n\), so the angle \((x - 14)^\circ\) on \(m\) and \((2x + 8)^\circ\) on \(n\) – yes, they are corresponding angles, so they should be equal. So:

\(x - 14=2x + 8\)? Wait, that would give \(x - 2x=8 + 14\), \(-x = 22\), \(x=-22\), which doesn't make sense. So I must have made a mistake. Wait, no, the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) are actually supplementary? Wait, no, let's look at the diagram again. The angle \((x - 14)^\circ\) and \((7y - 15)^\circ\) are adjacent on line \(m\), so they form a linear pair, so \((x - 14)+(7y - 15)=180\). Also, the angle \((2x + 8)^\circ\) and the angle \((7y - 15)^\circ\) – wait, no, the angle \((2x + 8)^\circ\) and \((x - 14)^\circ\) – wait, maybe the angle \((x - 14)^\circ\) and \((2x + 8)^\circ\) are supplementary. Let's try that. So:

\((x - 14)+(2x + 8)=180\)

Step1: Solve for x

Combine like terms: \(x+2x-14 + 8=180\)

\(3x-6 = 180\)

Add 6 to both sides: \(3x=180 + 6=186\)

Divide by 3: \(x=\frac{186}{3}=62\)

Wait, but let's check. If \(x = 62\), then \((x - 14)=62-14 = 48\), and \((2x + 8)=2\times62+8=124 + 8=132\). 48+132 = 180, which is supplementary, that makes sense. Now, the angle \((7y - 15)^\circ\) and \((x - 14)^\circ\) are supplementary (linear pair), so \((7y - 15)+(x - 14)=180\). We know \(x = 62\), so substitute:

Step2: Solve for y

\(7y-15+62 - 14=180\)

Simplify: \(7y+(62 - 15 - 14)=180\)

\(7y+(33)=180\)

Subtract 33: \(7y=180 - 33=147\)

Divide by 7: \(y=\frac{147}{7}=21\)

Wait, let's verify. If \(x = 62\), then \((x - 14)=48\), \((2x + 8)=132\), and \((7y - 15)=7\times21-15=147 - 15 = 132\). Then, on line \(m\), \(48+132 = 180\) (linear pair), and on line \(n\), the angle corresponding to \(132^\circ\) is also \(132^\circ\) (since \(m\parallel n\)), which matches. So that works.

Wait, but earlier I thought \((x - 14)\) and \((2x + 8)\) were corresponding, but they are supplementary. That was my mistake. So the correct equation for \(x\) is \((x - 14)+(2x + 8)=180\) because they are same - side interior angles? Wait, no, same - side interior angles are supplementary. Yes, because \(m\parallel n\), the same - side interior angles are supplementary. So \((x - 14)\) and \((2x + 8)\) are same - side interior angles, so they add up to 180.

Step1: Solve for x

\(x - 14+2x + 8=180\)

\(3x-6 = 180\)

\(3x=180 + 6=186\)

\(x=\frac{186}{3}=62\)

Step2: Solve for y

Now, the angle \((7y - 15)^\circ\) and \((2x + 8)^\circ\) are equal (corresponding angles, since \(m\parallel n\)). So:

\(7y-15=2x + 8\)

We know \(x = 62\), so substitute \(x = 62\):

\(7y-15=2\times62+8\)

\(7y-15=124 + 8=132\)

\(7y=132 + 15=147\)

\(y=\frac{147}{7}=21\)

Yes, that's correct. So that's the right way.

Answer:

\(x = 62\), \(y = 21\)