21. (overline{mp}) is the angle bisector of (angle lmn). given that (angle lmp) is a right angle, what type of angle is (angle lmn)? explain how you know.
22. state the angle addition postulate in your own words.
23. a protractor is properly aligned with the vertex of (angle klm). (overline{lk}) passes through the mark for (38^{circ}) on the protractor and (overline{lm}) passes through the mark for (126^{circ}). what is (mangle klm)?
4. (angle gfh) and (angle hfj) are adjacent and congruent. what are two conclusions you can draw from this information? support your answers.

Question

Accepted Answer

### 21.
# Explanation:
## Step1: Recall the property of angle bisector
An angle bisector divides an angle into two equal parts. So if $\overline{MP}$ is the angle bisector of $\angle LMN$, then $\angle LMP=\angle PMN$.
## Step2: Use the given right - angle information
Given that $\angle LMP = 90^{\circ}$. Since $\angle LMP=\angle PMN$ (because $MP$ is the angle bisector), then $\angle PMN = 90^{\circ}$.
## Step3: Calculate $\angle LMN$
By the angle - addition property ($\angle LMN=\angle LMP+\angle PMN$), substituting the values we get $\angle LMN=90^{\circ}+90^{\circ}$.

# Answer:
$\angle LMN$ is a straight angle. Because an angle bisector $\overline{MP}$ divides $\angle LMN$ into two equal right - angles ($\angle LMP=\angle PMN = 90^{\circ}$), and $m\angle LMN=m\angle LMP + m\angle PMN=90^{\circ}+90^{\circ}=180^{\circ}$, and an angle with measure $180^{\circ}$ is a straight angle.

### 22.
# Brief Explanations:
If we have two adjacent angles (say $\angle A$ and $\angle B$) that share a common side and a common vertex, then the measure of the larger angle (the angle formed by combining $\angle A$ and $\angle B$) is equal to the sum of the measures of the two adjacent angles. That is, if $\angle C$ is the angle formed by $\angle A$ and $\angle B$ (adjacent), then $m\angle C=m\angle A + m\angle B$.

# Answer:
If two adjacent angles share a common side and a common vertex, the measure of the angle formed by them is the sum of their measures.

### 23.
# Explanation:
## Step1: Recall the formula for the measure of an angle using a protractor
When a protractor is used to measure an angle $\angle KLM$ with sides $\overrightarrow{LK}$ and $\overrightarrow{LM}$, the measure of the angle $m\angle KLM$ is given by the difference of the measures of the two points on the protractor that the sides pass through. The formula is $m\angle KLM=\vert\theta_2-\theta_1\vert$, where $\theta_2$ and $\theta_1$ are the protractor marks.
## Step2: Substitute the given values
Here, $\theta_1 = 38^{\circ}$ and $\theta_2=126^{\circ}$. Then $m\angle KLM=126^{\circ}-38^{\circ}$.

# Answer:
$m\angle KLM = 88^{\circ}$

### 24.
# Brief Explanations:
- **Conclusion 1: They have the same measure**
Since $\angle GFH$ and $\angle HFJ$ are congruent, by the definition of congruent angles ($\angle\alpha\cong\angle\beta$ means $m\angle\alpha=m\angle\beta$), we can say $m\angle GFH=m\angle HFJ$.
- **Conclusion 2: If they are adjacent, the angle formed by them ($\angle GFJ$) has measure $2m\angle GFH$ (or $2m\angle HFJ$)**
By the Angle Addition Postulate ($m\angle GFJ=m\angle GFH + m\angle HFJ$), and since $m\angle GFH=m\angle HFJ$, then $m\angle GFJ = 2m\angle GFH$ (or $2m\angle HFJ$)

# Answer:
1. $m\angle GFH=m\angle HFJ$ (because congruent angles have equal measures).
2. If $\angle GFJ$ is the angle formed by $\angle GFH$ and $\angle HFJ$ (adjacent), then $m\angle GFJ = 2m\angle GFH$ (by the Angle Addition Postulate $m\angle GFJ=m\angle GFH + m\angle HFJ$ and $m\angle GFH=m\angle HFJ$).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

21.

Step1: Recall the property of angle bisector

Step2: Use the given right - angle information

Step3: Calculate \(\angle LMN\)

Step1: Recall the formula for the measure of an angle using a protractor

Step2: Substitute the given values

Answer:

22.