Question

2 in this figure, \\(\overleftrightarrow{pq}\\) intersects \\(\overleftrightarrow{sr}\\) at point \\(t\\). determine each angle measure. explain how you determined the angle measure. \\(p\\) is on the upper part of the vertical line, \\(q\\) is on the lower part. \\(s\\) is on the upper right of the slanted line, \\(r\\) is on the lower left. at point \\(t\\), the angle between \\(pt\\) and \\(st\\) is \\((x + 24)\degree\\), the angle between \\(rt\\) and \\(pt\\) is \\((3x + 4)\degree\\), the angle between \\(rt\\) and \\(qt\\) is \\(z\degree\\), and the angle between \\(st\\) and \\(qt\\) is \\(y\degree\\).

Explanation:

Step1: Identify Linear Pair

Angles \((x + 24)^\circ\) and \((3x + 4)^\circ\) form a linear pair, so their sum is \(180^\circ\).
\((x + 24) + (3x + 4) = 180\)

Step2: Solve for \(x\)

Simplify the equation:
\(4x + 28 = 180\)
Subtract 28: \(4x = 152\)
Divide by 4: \(x = 38\)

Step3: Find \((x + 24)^\circ\)

Substitute \(x = 38\):
\(38 + 24 = 62^\circ\)

Step4: Find \((3x + 4)^\circ\)

Substitute \(x = 38\):
\(3(38) + 4 = 118^\circ\)

Step5: Identify Vertical Angles

\(y^\circ\) is vertical to \((3x + 4)^\circ\)? No, wait: \((x + 24)^\circ\) and \(y^\circ\)? Wait, no—\((x + 24)^\circ\) and \(z^\circ\) are vertical? Wait, correction: \((x + 24)^\circ\) and \(y^\circ\)? No, let's re-examine.
Wait, \(\overrightarrow{PQ}\) is a straight line, so \((x + 24)^\circ + y^\circ = 180^\circ\)? No, wait, \((x + 24)^\circ\) and \((3x + 4)^\circ\) are linear pair. Then, \((x + 24)^\circ\) and \(y^\circ\)? Wait, no—\((3x + 4)^\circ\) and \(y^\circ\) are vertical? Wait, no, the angles at point \(T\): \((x + 24)^\circ\) and \(y^\circ\) are adjacent? Wait, no, let's use vertical angles.
Vertical angles: \((x + 24)^\circ\) and \(z^\circ\) are vertical? No, \((x + 24)^\circ\) and \(y^\circ\)? Wait, no, \(\overrightarrow{PQ}\) and \(\overrightarrow{SR}\) intersect at \(T\), so:
- \((x + 24)^\circ\) and \(y^\circ\): wait, no, \((x + 24)^\circ\) and \(y^\circ\) are adjacent? Wait, no, let's correct.
After finding \(x = 38\), \((x + 24)^\circ = 62^\circ\), \((3x + 4)^\circ = 118^\circ\).
Then, \((x + 24)^\circ\) and \(y^\circ\): since \(\overrightarrow{PQ}\) is straight, \((x + 24)^\circ + y^\circ = 180^\circ\)? No, wait, \((3x + 4)^\circ\) and \(y^\circ\) are vertical? Wait, no, \((3x + 4)^\circ\) and \(y^\circ\) are vertical angles? Wait, no, the angle opposite to \((3x + 4)^\circ\) is \(y^\circ\)? Wait, no, let's look at the diagram:
- \((x + 24)^\circ\) is between \(P\) and \(S\),
- \((3x + 4)^\circ\) is between \(R\) and \(P\),
- \(y^\circ\) is between \(S\) and \(Q\),
- \(z^\circ\) is between \(R\) and \(Q\).

So, \((x + 24)^\circ\) and \(y^\circ\) are adjacent on \(\overrightarrow{PQ}\)? No, \(\overrightarrow{PQ}\) is a straight line, so \((x + 24)^\circ + y^\circ = 180^\circ\)? No, \((x + 24)^\circ\) and \((3x + 4)^\circ\) are linear pair (sum to \(180^\circ\)). Then, \(y^\circ\) is vertical to \((3x + 4)^\circ\)? Wait, no, \(y^\circ\) and \((3x + 4)^\circ\) are vertical angles? Wait, no, vertical angles are opposite each other. So \((x + 24)^\circ\) and \(y^\circ\) are vertical? No, \((x + 24)^\circ\) and \(z^\circ\)? Wait, I think I made a mistake. Let's redo:

1. Linear pair: \((x + 24)^\circ + (3x + 4)^\circ = 180^\circ\) (since they form a straight line on \(\overrightarrow{PQ}\)? Wait, no, \(\overrightarrow{PQ}\) is a straight line, so angles on a straight line sum to \(180^\circ\). So \((x + 24) + (3x + 4) = 180\) (correct, as they are adjacent and form a linear pair).

2. Solve: \(4x + 28 = 180 \implies 4x = 152 \implies x = 38\) (correct).

3. Then, \((x + 24)^\circ = 38 + 24 = 62^\circ\), \((3x + 4)^\circ = 3(38) + 4 = 118^\circ\).

4. Now, vertical angles: \((x + 24)^\circ\) and \(y^\circ\)? No, wait, \((3x + 4)^\circ\) and \(y^\circ\) are vertical? Wait, no, the angle opposite to \((x + 24)^\circ\) is \(y^\circ\)? Wait, no, let's look at the diagram:

- \(\angle PT S = (x + 24)^\circ\)
- \(\angle RT P = (3x + 4)^\circ\)
- \(\angle ST Q = y^\circ\)
- \(\angle RT Q = z^\circ\)

So, \(\angle PT S\) and \(\angle RT Q\) (i.e., \(z^\circ\)) are vertical angles? No, \(\angle PT S\) (62°) and \(\angle QT R\) (z°) are vertical? Wait, no, \(\angle PT S\) and \(\angle QT R\) (z°) are vertical? Wait, no, \(\angle RT P\) (118°) and \(\angle ST Q\) (y°) are vertical angles (opposite each other), so \(y^\circ = 118^\circ\). And \(\angle PT S\) (62°) and \(\angle QT R\) (z°) are vertical angles, so \(z^\circ = 62^\circ\).

Yes, that makes sense: vertical angles are equal. So:

- \(y^\circ = (3x + 4)^\circ = 118^\circ\) (vertical angles with \(\angle RT P\))
- \(z^\circ = (x + 24)^\circ = 62^\circ\) (vertical angles with \(\angle PT S\))

Answer:

• \((x + 24)^\circ = 62^\circ\)
• \((3x + 4)^\circ = 118^\circ\)
• \(y^\circ = 118^\circ\) (vertical to \(118^\circ\))
• \(z^\circ = 62^\circ\) (vertical to \(62^\circ\))

(Explanation: Linear pair gives \(x = 38\); vertical angles are equal.)