Question

for questions 34 - 36, solve for both x and y

Explanation:

Question 34:

Step1: Find x (supplementary angles)

Angles on a straight line sum to \(180^\circ\). So \(x + 75^\circ + 103^\circ = 180^\circ\).
Simplify: \(x + 178^\circ = 180^\circ\) → \(x = 180^\circ - 178^\circ = 2^\circ\)? Wait, no—wait, vertical angles? Wait, no, the \(103^\circ\) and the angle opposite to \(x + 75^\circ\)? Wait, no, let's re-examine. Wait, when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. Wait, the \(103^\circ\) angle and the angle adjacent to \(x\) and \(75^\circ\) are supplementary? Wait, no, maybe \(x\) and \(103^\circ\) are related? Wait, no, let's correct.

Wait, actually, when two lines intersect, the sum of angles around a point is \(360^\circ\), but adjacent angles on a straight line sum to \(180^\circ\). Let's look at the diagram: the angle \(103^\circ\) and the angle opposite to \(x + 75^\circ\) are vertical angles? No, wait, the angle \(y\) and \(103^\circ\) are vertical angles? Wait, no, let's do step by step.

First, find \(y\): \(y\) and \(103^\circ\) are vertical angles, so \(y = 103^\circ\).

Then, find \(x\): \(x + 75^\circ + y = 180^\circ\) (since they are on a straight line). Wait, \(y = 103^\circ\), so \(x + 75^\circ + 103^\circ = 180^\circ\)? No, that can't be, because \(75 + 103 = 178\), so \(x = 2\)? That seems odd, but maybe. Wait, maybe I misread the diagram. Alternatively, \(x\) and \(103^\circ\) are supplementary to \(75^\circ\)? Wait, no, let's start over.

Step1: Find \(y\) (vertical angles)

Vertical angles are equal. The angle \(103^\circ\) and \(y\) are vertical angles, so \(y = 103^\circ\).

Step2: Find \(x\) (supplementary angles)

Angles \(x\), \(75^\circ\), and \(y\) (wait, no, \(x\) and \(75^\circ\) are adjacent to a straight line with \(y\)? Wait, no, when two lines intersect, adjacent angles on a straight line sum to \(180^\circ\). So \(x + 75^\circ + \text{(angle opposite to 103°)} = 180^\circ\), but angle opposite to 103° is \(y\), so \(x + 75^\circ + y = 180^\circ\)? But \(y = 103^\circ\), so \(x + 75 + 103 = 180\) → \(x + 178 = 180\) → \(x = 2^\circ\).

Step1: Solve for \(x\) (vertical angles or supplementary with right angle)

There's a right angle (\(90^\circ\)) in the diagram. The angles \((5x + 2)^\circ\) and \((8x - 25)^\circ\) are equal? Wait, no—wait, the right angle is between them? Wait, the diagram has a right angle, so the sum of \((5x + 2)^\circ\), \(90^\circ\), and \((8x - 25)^\circ\) is \(180^\circ\)? No, wait, when two lines intersect, and there's a right angle, so the angles \((5x + 2)\) and \((8x - 25)\) are equal? Wait, no, let's see: the two angles \((5x + 2)\) and \((8x - 25)\) are vertical angles? No, maybe they are equal because of the right angle. Wait, actually, the angles \((5x + 2)\) and \((8x - 25)\) are equal (vertical angles), so:

\(5x + 2 = 8x - 25\)

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(2 = 3x - 25\)

Add 25 to both sides: \(27 = 3x\)

Divide by 3: \(x = 9\)

Step3: Find \(y\) (supplementary with right angle and \((8x - 25)^\circ\))

First, find \((8x - 25)^\circ\) when \(x = 9\): \(8(9) - 25 = 72 - 25 = 47^\circ\)

Now, the right angle is \(90^\circ\), so \(y + 47^\circ + 90^\circ = 180^\circ\) (straight line). Wait, no—wait, \(y\), \(90^\circ\), and \((8x - 25)^\circ\) are on a straight line? So \(y + 90 + 47 = 180\) → \(y + 137 = 180\) → \(y = 43^\circ\). Wait, or maybe \(y\) is complementary? Wait, no, let's check again.

Wait, when \(x = 9\), \((5x + 2) = 47^\circ\), \((8x - 25) = 47^\circ\). The right angle is \(90^\circ\), so the angle \(y\) is adjacent to the right angle and \(47^\circ\), so \(y + 47 + 90 = 180\) → \(y = 43^\circ\).

Step1: Solve for \(x\) (vertical angles or supplementary)

The angles \((10x + 38)^\circ\) and \((15x - 2)^\circ\) are supplementary? Wait, no—wait, the angle \(21^\circ\) and \(y\) are related? Wait, let's see: the sum of angles around a point is \(360^\circ\), but adjacent angles on a straight line sum to \(180^\circ\). Also, vertical angles are equal. Wait, the angle \(21^\circ\) and the angle opposite to \(y\) (or adjacent) – wait, let's look at the diagram: the angles \((10x + 38)^\circ\), \(y\), \((15x - 2)^\circ\), and \(21^\circ\) are around a point, but also, \((10x + 38)\) and \((15x - 2)\) are supplementary? Wait, no, maybe \((10x + 38) + (15x - 2) + 21 + y = 180\)? No, that's not right. Wait, actually, the angle \(21^\circ\) and \(y\) are related, and \((10x + 38)\) and \((15x - 2)\) are vertical angles? No, wait, let's assume that \((10x + 38)\) and \((15x - 2)\) are supplementary to \(21^\circ\) and \(y\), but maybe \(21^\circ\) and \(y\) are vertical angles? No, that doesn't make sense. Wait, let's try:

The sum of angles on a straight line is \(180^\circ\). So \((10x + 38) + y + (15x - 2) = 180\), and also, \(21^\circ\) and \(y\) are vertical angles? No, \(21^\circ\) and \(y\) – wait, maybe \(y = 21^\circ\)? No, that doesn't fit. Wait, maybe the angle \(21^\circ\) and \(y\) are supplementary to the other angles. Wait, let's start with \(x\):

Assume that \((10x + 38)\) and \((15x - 2)\) are equal? No, that would be vertical angles, but maybe they are supplementary. Wait, no, let's set up the equation:

The sum of \((10x + 38)\), \(y\), \((15x - 2)\), and \(21^\circ\) is \(180^\circ\) (straight line). But also, \(y\) and \(21^\circ\) are related? Wait, no, maybe \(y\) is equal to \(21^\circ\) plus something? Wait, no, let's try solving for \(x\) first. Let's assume that \((10x + 38) + (15x - 2) + 21 + y = 180\), but we need another equation. Wait, maybe \(y\) is equal to \(21^\circ\) because of vertical angles? No, that's not. Wait, maybe the angle \(21^\circ\) and \(y\) are vertical angles, so \(y = 21^\circ\), and then \((10x + 38) + (15x - 2) + 21 + 21 = 180\)? No, that sums to \(25x + 78 = 180\) → \(25x = 102\) → \(x = 4.08\), which is not integer. So maybe my assumption is wrong.

Wait, another approach: the angle \((10x + 38)\) and \((15x - 2)\) are supplementary to \(21^\circ\) and \(y\), but actually, the correct way is that the sum of \((10x + 38)\), \(y\), and \((15x - 2)\) is \(180^\circ\), and \(y\) is equal to \(21^\circ\) (vertical angles). Wait, no, let's check the diagram again. The user's diagram: 36 has angles \((10x + 38)^\circ\), \(y\), \((15x - 2)^\circ\), and \(21^\circ\) around a point. So the sum of all angles around a point is \(360^\circ\), but adjacent angles on a straight line sum to \(180^\circ\). So two straight lines intersect, so opposite angles are equal. So \((10x + 38)\) and \((15x - 2)\) are not vertical angles, but \(21^\circ\) and \(y\) are vertical angles? No, \(21^\circ\) and \(y\) – maybe \(y\) is equal to \(21^\circ\) plus something. Wait, let's try:

The angle \((10x + 38)\) and \((15x - 2)\) are supplementary to \(21^\circ\) and \(y\), but actually, the correct equation is:

\((10x + 38) + (15x - 2) + 21 + y = 180\) (since they are on a straight line). But also, \(y\) is equal to \(21^\circ\) (vertical angles), so substitute \(y = 21\):

\(10x + 38 + 15x - 2 + 21 + 21 = 180\)

Simplify: \(25x + 78 = 180\)

\(25x = 102\) → \(x = 4.08\) (not integer, so wrong).

Alternative: Maybe \((10x + 38)\) and \((15x - 2)\) are vertical angles? No, vertical angles are equal, so \(10x +…

Answer:

\(x = 2^\circ\), \(y = 103^\circ\)

Question 35: