for problems 1 and 2, use the diagram that shows $overleftrightarrow{ab}$ intersecting $overleftrightarrow{cd}$ and $overleftrightarrow{ef}$ at point $e$.
1. name an example of each of the following angle pairs:
a. vertical angles
b. adjacent angles
c. complementary angles
d. supplementary angles
2. find the measure of the following angles. then describe the angle relationship used to determine each angle measure.
a. $m\angle bec$
b. $m\angle aec$
c. $m\angle bef$

Question

for problems 1 and 2, use the diagram that shows $overleftrightarrow{ab}$ intersecting $overleftrightarrow{cd}$ and $overleftrightarrow{ef}$ at point $e$.
1. name an example of each of the following angle pairs:
   a. vertical angles
   b. adjacent angles
   c. complementary angles
   d. supplementary angles
2. find the measure of the following angles. then describe the angle relationship used to determine each angle measure.
   a. $m\angle bec$
   b. $m\angle aec$
   c. $m\angle bef$

Accepted Answer

### Problem 1
#### a. Vertical angles
# Brief Explanations:
Vertical angles are formed by two intersecting lines and are opposite each other. In the diagram, $\angle AED$ and $\angle BEC$ are vertical angles (or $\angle AEC$ and $\angle BED$).
# Answer:
$\angle AED$ and $\angle BEC$ (or $\angle AEC$ and $\angle BED$)

#### b. Adjacent angles
# Brief Explanations:
Adjacent angles share a common side and vertex. For example, $\angle AED$ and $\angle DEF$ share vertex $E$ and side $ED$, or $\angle AED$ and $\angle AEC$ (they share side $AE$ and vertex $E$).
# Answer:
$\angle AED$ and $\angle DEF$ (or $\angle AED$ and $\angle AEC$, etc.)

#### c. Complementary angles
# Brief Explanations:
Complementary angles sum to $90^\circ$. Given $\angle AED = 37^\circ$ and there's a right angle (let's say $\angle DEF$ is part of a right angle with $\angle AED$), so $\angle AED$ and $\angle DEF$: $37^\circ + 53^\circ = 90^\circ$ (since the right angle is $90^\circ$, $90 - 37 = 53$). So $\angle AED$ and $\angle DEF$ are complementary.
# Answer:
$\angle AED$ and $\angle DEF$

#### d. Supplementary angles
# Brief Explanations:
Supplementary angles sum to $180^\circ$. $\angle AED$ and $\angle BED$: they form a linear pair (along line $AB$ or $CD$), so their sum is $180^\circ$. Or $\angle AEC$ and $\angle BEC$ (linear pair).
# Answer:
$\angle AED$ and $\angle BED$ (or $\angle AEC$ and $\angle BEC$, etc.)

### Problem 2
#### a. $m\angle BEC$
# Explanation:
## Step1: Identify vertical angles
$\angle BEC$ and $\angle AED$ are vertical angles, so they are equal.
## Step2: Get $\angle AED$ measure
From the diagram, $\angle AED = 37^\circ$.
## Step3: Conclude $m\angle BEC$
Since vertical angles are equal, $m\angle BEC = m\angle AED = 37^\circ$.
# Answer:
$37^\circ$ (vertical angles are equal, $\angle BEC \cong \angle AED$)

#### b. $m\angle AEC$
# Explanation:
## Step1: Identify linear pair
$\angle AED$ and $\angle AEC$ form a linear pair, so they are supplementary ($sum = 180^\circ$).
## Step2: Use $\angle AED = 37^\circ$
$m\angle AEC = 180^\circ - m\angle AED$.
## Step3: Calculate
$m\angle AEC = 180 - 37 = 143^\circ$.
# Answer:
$143^\circ$ (supplementary angles, $\angle AED + \angle AEC = 180^\circ$)

#### c. $m\angle BEF$
# Explanation:
## Step1: Identify right angle and known angle
There's a right angle (let's say $\angle AEF$ is $90^\circ$? Wait, looking at the diagram, the right angle is between $EF$ and, say, the vertical? Wait, actually, the right angle is at $E$ between $EF$ and maybe $AB$? Wait, no: the diagram has a right angle symbol, so let's see: $\angle AED = 37^\circ$, the right angle is, for example, $\angle DEF$ and $\angle BEF$? Wait, no. Wait, the straight line $AB$, and the right angle: let's re - examine.

Wait, the right angle is between $EF$ and, let's say, the angle between $AE$ and $EF$? Wait, no. Let's do it properly.

We know $\angle AED = 37^\circ$, and there's a right angle (so $90^\circ$) between, say, $ED$ and $EF$? Wait, no. Wait, the sum of $\angle AED$, the right angle, and $\angle BEF$ should be $180^\circ$ (since $AB$ is a straight line). Wait, $\angle AED = 37^\circ$, the right angle is $90^\circ$, so $m\angle BEF = 180 - 37 - 90$.

## Step1: Sum of angles on straight line
Angles on a straight line ($AB$) sum to $180^\circ$. So $\angle AED + \angle DEF + \angle BEF = 180^\circ$. $\angle DEF$ is $90^\circ$ (right angle), $\angle AED = 37^\circ$.
## Step2: Calculate $m\angle BEF$
$m\angle BEF = 180 - 37 - 90 = 53^\circ$.
# Answer:
$53^\circ$ (angles on a straight line sum to $180^\circ$, $37^\circ + 90^\circ + m\angle BEF = 180^\circ$)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1

a. Vertical angles

Answer:

b. Adjacent angles