Let's solve the questions on the worksheet step-by-step:

Information from the Diagram

• $\angle EOD = 32^\circ$ (Given)
• $AD$, $BE$, and $CF$ are lines that intersect at point $O$.
• $\angle AOF$ is $90^\circ$ since $AD$ is perpendicular to $FC$.

Solving the Problems

a. What is the measure of $\angle FOA$? How do you know?

Since $AD$ is perpendicular to $FC$, the angle between them, $\angle FOA$, is $90^\circ$.

Answer: $\angle FOA = 90^\circ$.

b. Name two adjacent angles that together form $\angle FOD$. What is the sum of their measures?

Adjacent angles that form $\angle FOD$ are $\angle FOA$ and $\angle AOD$.

• $\angle FOA = 90^\circ$ (as determined above),
• $\angle AOD = \angle EOD = 32^\circ$ (vertically opposite angles).

The sum of their measures is: $\angle FOA + \angle AOD = 90^\circ + 32^\circ = 122^\circ.$

Answer: $\angle FOA$ and $\angle AOD$; the sum is $122^\circ$.

c. What is the measure of $\angle FOE$? Explain.

$\angle FOE$ and $\angle EOD$ are adjacent angles that together form a straight angle: $\angle FOE + \angle EOD = 180^\circ.$

Given $\angle EOD = 32^\circ$: $\angle FOE = 180^\circ - 32^\circ = 148^\circ.$

Answer: $\angle FOE = 148^\circ$.

d. Together, $\angle FOE$, $\angle FOA$, and $\angle AOB$ form a line or a straight angle that measures $180^\circ$. Explain how you can find the measure of $\angle AOB$.

The sum of $\angle FOE$, $\angle FOA$, and $\angle AOB$ is: $\angle FOE + \angle FOA + \angle AOB = 180^\circ.$

Substituting known values:

• $\angle FOE = 148^\circ$,
• $\angle FOA = 90^\circ$.

Thus: $148^\circ + 90^\circ + \angle AOB = 180^\circ.$

Solving for $\angle AOB$: $\angle AOB = 180^\circ - (148^\circ + 90^\circ) = 180^\circ - 238^\circ = -58^\circ.$

Correction: **Repeat Calculation Adjusting By adding split sections corrections.