Practice worksheet for writing equations to find the value of x in vertical angles.

Here are the step-by-step solutions for each problem on the worksheet.

1.
Vertical angles are equal. The angle labeled $36^\circ$ and the angle labeled $x$ are vertical angles (they are opposite each other).
Therefore, $x = 36$.

2.
The angle labeled $y$ and the angle labeled $(x + 17)^\circ$ are vertical angles, so they are equal:
$y = x + 17$

The angle labeled $x$ and the angle labeled $148^\circ$ form a linear pair (they make a straight line), so they add up to $180^\circ$:
$x + 148 = 180$
$x = 180 - 148$
$x = 32$

Now substitute $x = 32$ into the first equation to find $y$:
$y = 32 + 17$
$y = 49$

3.
The angles $(5x - 10)^\circ$ and $105^\circ$ are vertical angles, so they are equal:
$5x - 10 = 105$
Add 10 to both sides:
$5x = 115$
Divide by 5:
$x = 23$

The angle labeled $y$ and the angle labeled $105^\circ$ form a linear pair (a straight line), so they add up to $180^\circ$:
$y + 105 = 180$
$y = 180 - 105$
$y = 75$

4.
The angle labeled $x$ and the angle labeled $58^\circ$ are vertical angles.
Therefore, $x = 58$.

5.
First, look at the bottom intersection. The angle labeled $y$ and the angle labeled $(2x + 18)^\circ$ are vertical angles, so:
$y = 2x + 18$

The angle labeled $142^\circ$ and the angle labeled $(2x + 18)^\circ$ form a linear pair (straight line), so they add to $180^\circ$:
$(2x + 18) + 142 = 180$
$2x + 160 = 180$
Subtract 160 from both sides:
$2x = 20$
Divide by 2:
$x = 10$

Now find $y$ using the value of $x$:
$y = 2(10) + 18$
$y = 20 + 18$
$y = 38$

6.
Look at the top left intersection. The angle labeled $(2x - 1)^\circ$ and the angle labeled $105^\circ$ form a linear pair (straight line):
$(2x - 1) + 105 = 180$
$2x + 104 = 180$
Subtract 104 from both sides:
$2x = 76$
Divide by 2:
$x = 38$

Look at the bottom right intersection. The angle labeled $y$ and the angle labeled $(x + 31)^\circ$ form a linear pair:
$y + (x + 31) = 180$
Substitute $x = 38$:
$y + (38 + 31) = 180$
$y + 69 = 180$
Subtract 69 from both sides:
$y = 111$

Final Answer:
1. x = 36
2. x = 32, y = 49
3. x = 23, y = 75
4. x = 58
5. x = 10, y = 38
6. x = 38, y = 111