Vertical Angles Worksheet - Find the value of x and the indicated angle in each diagram.

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

1)
* Find $x$: The angles labeled $(2x - 65)^\circ$ and $(3x - 40)^\circ$ are vertical angles, which means they are equal.
$$2x - 65 = 3x - 40$$
Subtract $2x$ from both sides:
$$-65 = x - 40$$
Add $40$ to both sides:
$$-25 = x$$
So, $x = -25$.

* Find $m\angle BOC$: Substitute $x = -25$ into the expression $(2x - 65)$.
$$2(-25) - 65 = -50 - 65 = -115$$
So, $m\angle BOC = -115^\circ$.

* Find $m\angle AOB$: Angles $\angle AOB$ and $\angle BOC$ form a straight line, so they add up to $180^\circ$.
$$m\angle AOB + (-115) = 180$$
$$m\angle AOB = 180 + 115 = 295$$
So, $m\angle AOB = 295^\circ$.

2)
* Find $x$: The angles labeled $(x + 27)^\circ$ and $(2x - 6)^\circ$ are vertical angles.
$$x + 27 = 2x - 6$$
Subtract $x$ from both sides:
$$27 = x - 6$$
Add $6$ to both sides:
$$33 = x$$
So, $x = 33$.

* Find $m\angle QOR$: Substitute $x = 33$ into $(x + 27)$.
$$33 + 27 = 60$$
So, $m\angle QOR = 60^\circ$.

* Find $m\angle SOR$: Angles $\angle QOR$ and $\angle SOR$ are supplementary (add to $180^\circ$).
$$60 + m\angle SOR = 180$$
$$m\angle SOR = 120$$
So, $m\angle SOR = 120^\circ$.

3)
* Find $x$: The angles $(2x + 22)^\circ$ and $(3x - 14)^\circ$ are vertical angles.
$$2x + 22 = 3x - 14$$
Subtract $2x$ from both sides:
$$22 = x - 14$$
Add $14$ to both sides:
$$36 = x$$
So, $x = 36$.

* Find $m\angle VOU$: Substitute $x = 36$ into $(2x + 22)$.
$$2(36) + 22 = 72 + 22 = 94$$
So, $m\angle VOU = 94^\circ$.

* Find $m\angle TOU$: Angles $\angle VOU$ and $\angle TOU$ are supplementary.
$$94 + m\angle TOU = 180$$
$$m\angle TOU = 86$$
So, $m\angle TOU = 86^\circ$.

4)
* Find $x$: The angles $(9x - 11)^\circ$ and $(3x + 7)^\circ$ are vertical angles.
$$9x - 11 = 3x + 7$$
Subtract $3x$ from both sides:
$$6x - 11 = 7$$
Add $11$ to both sides:
$$6x = 18$$
Divide by $6$:
$$x = 3$$
So, $x = 3$.

* Find $m\angle EOF$: Substitute $x = 3$ into $(3x + 7)$.
$$3(3) + 7 = 9 + 7 = 16$$
So, $m\angle EOF = 16^\circ$.

* Find $m\angle GOD$: Angles $\angle EOF$ and $\angle GOD$ are supplementary.
$$16 + m\angle GOD = 180$$
$$m\angle GOD = 164$$
So, $m\angle GOD = 164^\circ$.

5)
* Find $x$: The angles $(2x - 32)^\circ$ and $(x + 12)^\circ$ are vertical angles.
$$2x - 32 = x + 12$$
Subtract $x$ from both sides:
$$x - 32 = 12$$
Add $32$ to both sides:
$$x = 44$$
So, $x = 44$.

* Find $m\angle FOG$: Substitute $x = 44$ into $(x + 12)$.
$$44 + 12 = 56$$
So, $m\angle FOG = 56^\circ$.

* Find $m\angle EOF$: Angles $\angle FOG$ and $\angle EOF$ are supplementary.
$$56 + m\angle EOF = 180$$
$$m\angle EOF = 124$$
So, $m\angle EOF = 124^\circ$.

6)
* Find $x$: The angles $(x + 27)^\circ$ and $(2x - 13)^\circ$ are vertical angles.
$$x + 27 = 2x - 13$$
Subtract $x$ from both sides:
$$27 = x - 13$$
Add $13$ to both sides:
$$40 = x$$
So, $x = 40$.

* Find $m\angle JOM$: Substitute $x = 40$ into $(x + 27)$.
$$40 + 27 = 67$$
So, $m\angle JOM = 67^\circ$.

* Find $m\angle MOL$: Angles $\angle JOM$ and $\angle MOL$ are supplementary.
$$67 + m\angle MOL = 180$$
$$m\angle MOL = 113$$
So, $m\angle MOL = 113^\circ$.

Final Answer:
1) $x = -25$, $m\angle BOC = -115^\circ$, $m\angle AOB = 295^\circ$
2) $x = 33$, $m\angle QOR = 60^\circ$, $m\angle SOR = 120^\circ$
3) $x = 36$, $m\angle VOU = 94^\circ$, $m\angle TOU = 86^\circ$
4) $x = 3$, $m\angle EOF = 16^\circ$, $m\angle GOD = 164^\circ$
5) $x = 44$, $m\angle FOG = 56^\circ$, $m\angle EOF = 124^\circ$
6) $x = 40$, $m\angle JOM = 67^\circ$, $m\angle MOL = 113^\circ$