Question

1. ∠3 and ∠4 are complementary.
2. ∠1 and ∠2 are supplementary.

name an angle or angles in the diagram described by each of the following.

1. supplementary to ∠aod
2. adjacent and congruent to ∠aoe
3. supplementary to ∠eoa
4. complementary to ∠eod
5. a pair of vertical angles

Explanation:

Step1: Recall supplementary - angle definition

Two angles are supplementary if their sum is 180°.

Step2: Recall adjacent - angle definition

Adjacent angles share a common side and a common vertex.

Step3: Recall congruent - angle definition

Congruent angles have the same measure.

Step4: Recall complementary - angle definition

Two angles are complementary if their sum is 90°.

Step5: Recall vertical - angle definition

Vertical angles are opposite each other when two lines intersect.

11.
Since $\angle AOD+\angle AOC = 180^{\circ}$ and $\angle AOD+\angle DOB=180^{\circ}$, the angles supplementary to $\angle AOD$ are $\angle AOC$ and $\angle DOB$.
12.
$\angle DOC$ is adjacent to $\angle AOE$ (shares side $OE$ and vertex $O$) and $\angle AOE = 90^{\circ}$, $\angle DOC=90^{\circ}$, so the angle adjacent and congruent to $\angle AOE$ is $\angle DOC$.
13.
Since $\angle EOA+\angle AOF = 180^{\circ}$ (assuming a full - rotation around point $O$), and from the diagram $\angle EOA+\angle AOC = 180^{\circ}$, the angles supplementary to $\angle EOA$ are $\angle AOC$ and $\angle EOB$.
14.
$\angle EOD = 60^{\circ}$, and $\angle DOC=90^{\circ}-60^{\circ}=30^{\circ}$, $\angle EOD+\angle DOC = 90^{\circ}$, so the angle complementary to $\angle EOD$ is $\angle DOC$.
15.
Vertical angles in the diagram are $\angle AOB$ and $\angle DOC$, $\angle AOC$ and $\angle DOB$.

Answer:

1. $\angle AOC$, $\angle DOB$
2. $\angle DOC$
3. $\angle AOC$, $\angle EOB$
4. $\angle DOC$
5. $\angle AOB$ and $\angle DOC$, $\angle AOC$ and $\angle DOB$