24. find the indicated value.
a. find the measure of ∠b.
b. find the value of the variable.
27. find the indicated value.
a. find the measure of ∠b.
b. find the value of the variable.
25. find the indicated value.
a. find the measure of ∠b.
b. find the value of the variable.

Question

Accepted Answer

### 24a Solution:
# Explanation:
## Step1: Identify angle relationship
The two angles $ \angle b $ and $ 62^\circ $ form a right angle (since there's a right - angle symbol), so their sum is $ 90^\circ $.
## Step2: Solve for $ \angle b $
We know that $ \angle b+ 62^\circ=90^\circ $. To find $ \angle b $, we subtract $ 62^\circ $ from $ 90^\circ $. So $ \angle b = 90^\circ - 62^\circ=28^\circ $.
# Answer:
$ 28^\circ $

### 24b Solution:
# Explanation:
## Step1: Identify angle relationship
The two angles $ (3x + 2)^\circ $ and $ (x + 16)^\circ $ form a right angle (because of the right - angle symbol), so their sum is $ 90^\circ $.
## Step2: Set up the equation
$ (3x + 2)+(x + 16)=90 $
## Step3: Simplify the equation
Combine like terms: $ 3x+x+2 + 16=90 $, which gives $ 4x+18 = 90 $
## Step4: Solve for $ x $
Subtract 18 from both sides: $ 4x=90 - 18=72 $. Then divide both sides by 4: $ x=\frac{72}{4}=18 $
# Answer:
$ x = 18 $

### 27a Solution:
# Explanation:
## Step1: Identify angle relationship
$ \angle b $ and the $ 58^\circ $ angle form a linear pair (they are adjacent and form a straight line), so their sum is $ 180^\circ $. But also, $ \angle b $ and the $ 58^\circ $ angle are such that $ \angle b=180^\circ - 2\times58^\circ $? Wait, no. Wait, the two non - $ \angle b $ angles: one is $ 58^\circ $, and the other is equal to $ 58^\circ $ (vertical angles or isosceles? Wait, actually, the angle $ \angle b $ and the two $ 58^\circ $ - related angles: since the two lines are a straight line, $ \angle b+58^\circ + 58^\circ=180^\circ $? No, wait, the angle adjacent to $ \angle b $ and the $ 58^\circ $ angle: actually, the angle $ \angle b $ and the $ 58^\circ $ angle are supplementary to the same angle? Wait, no. Let's re - look. The two rays form a straight line, and there is a ray in between creating a $ 58^\circ $ angle. So $ \angle b $ and the $ 58^\circ $ angle are such that $ \angle b = 180^\circ- 2\times58^\circ $? No, wait, the angle opposite to the $ 58^\circ $ angle is also $ 58^\circ $ (vertical angles). Then $ \angle b=180^\circ-(58^\circ + 58^\circ)=180 - 116 = 64^\circ $? Wait, no, actually, the angle $ \angle b $ and the $ 58^\circ $ angle: since the two lines are a straight line, and the angle between them and the middle ray: the angle $ \angle b $ and the $ 58^\circ $ angle are supplementary? Wait, no, the correct relationship is that $ \angle b=180^\circ - 2\times58^\circ $? No, let's think again. The angle $ \angle b $ and the $ 58^\circ $ angle: the sum of $ \angle b $ and the $ 58^\circ $ angle is $ 180^\circ- 58^\circ $? No, I made a mistake. The correct approach: the two angles formed by the middle ray and the two straight - line rays: one is $ 58^\circ $, and the angle adjacent to $ \angle b $ is also $ 58^\circ $ (vertical angles). So $ \angle b=180^\circ - 58^\circ- 58^\circ=64^\circ $? Wait, no, the angle $ \angle b $ and the $ 58^\circ $ angle: actually, $ \angle b = 180^\circ- 2\times58^\circ $? Wait, no, the straight line is $ 180^\circ $, and the two angles on one side of the middle ray: if one is $ 58^\circ $, the other (adjacent to $ \angle b $) is also $ 58^\circ $ (because of vertical angles). So $ \angle b=180^\circ-(58^\circ + 58^\circ)=64^\circ $. Wait, no, maybe a better way: the angle $ \angle b $ and the $ 58^\circ $ angle are supplementary to the same angle? No, let's start over. The figure has a straight line (180 degrees) and a ray coming out, making a $ 58^\circ $ angle with one of the straight - line rays. The angle $ \angle b $ is adjacent to the $ 58^\circ $ angle and forms a linear pair with the angle that is equal to $ 58^\circ $ (vertical angles). So $ \angle b=180^\circ - 58^\circ=122^\circ $? Wait, I think I messed up the diagram. Let's assume that the two angles: $ \angle b $ and the $ 58^\circ $ angle are supplementary (since they are on a straight line with the middle ray). Wait, no, the correct relationship is that $ \angle b = 180^\circ- 2\times58^\circ $ is wrong. Let's look at the diagram again: there is a straight line, and a ray that makes a $ 58^\circ $ angle with one of the straight - line rays. The angle $ \angle b $ is on the other side of the middle ray. So the sum of $ \angle b $ and $ 58^\circ $ is $ 180^\circ- 58^\circ $? No, I think the correct way is: the angle $ \angle b $ and the $ 58^\circ $ angle are vertical to the same angle? No, let's use the fact that a straight line is 180 degrees. If one angle is $ 58^\circ $, and the angle opposite to it (vertical angle) is also $ 58^\circ $, then $ \angle b=180^\circ-(58^\circ + 58^\circ)=64^\circ $. Wait, I'm confused. Let's try a different approach. The angle $ \angle b $ and the $ 58^\circ $ angle: since the two lines are a straight line, and the middle ray creates two angles. If one angle is $ 58^\circ $, then the angle adjacent to $ \angle b $ is $ 58^\circ $ (vertical angles), so $ \angle b = 180^\circ- 58^\circ=122^\circ $. Yes, that makes sense. Because the straight line is 180 degrees, and $ \angle b $ plus the $ 58^\circ $ angle (vertical to the given $ 58^\circ $) equals 180 degrees. So $ \angle b=180 - 58 = 122^\circ $
## Step1: Identify the straight - line (180 degrees)
The two main rays form a straight line, so the sum of angles on one side is 180 degrees.
## Step2: Solve for $ \angle b $
The angle adjacent to $ \angle b $ is $ 58^\circ $ (vertical angles), so $ \angle b=180^\circ - 58^\circ = 122^\circ $
# Answer:
$ 122^\circ $

### 27b Solution:
# Explanation:
## Step1: Identify angle relationship
The two angles $ (3x + 1)^\circ $ and $ (x - 5)^\circ $ form a linear pair (they are adjacent and on a straight line), so their sum is $ 180^\circ $.
## Step2: Set up the equation
$ (3x + 1)+(x - 5)=180 $
## Step3: Simplify the equation
Combine like terms: $ 3x+x+1 - 5=180 $, which gives $ 4x - 4 = 180 $
## Step4: Solve for $ x $
Add 4 to both sides: $ 4x=180 + 4=184 $. Then divide both sides by 4: $ x=\frac{184}{4}=46 $
# Answer:
$ x = 46 $

### 25a Solution:
# Explanation:
## Step1: Identify angle relationship
$ \angle b $ and the $ 44^\circ $ angle are vertical angles? No, wait, $ \angle b $ and the $ 44^\circ $ angle: since they are on a straight line and are vertical angles? Wait, no, the two angles: $ \angle b $ and the $ 44^\circ $ angle are supplementary? No, vertical angles are equal. Wait, the angle $ \angle b $ and the $ 44^\circ $ angle: if they are vertical angles, then $ \angle b = 180^\circ- 44^\circ $? No, vertical angles are equal. Wait, the diagram shows two intersecting lines. The angle $ \angle b $ and the $ 44^\circ $ angle: if they are adjacent and form a linear pair, no. Wait, the angle $ \angle b $ and the $ 44^\circ $ angle: since they are on a straight line, and the angle opposite to $ 44^\circ $ is equal to $ 44^\circ $ (vertical angles), then $ \angle b=180^\circ - 44^\circ=136^\circ $? No, vertical angles are equal. Wait, I think the angle $ \angle b $ and the $ 44^\circ $ angle are supplementary. Wait, no, when two lines intersect, vertical angles are equal, and linear pairs are supplementary. So if one angle is $ 44^\circ $, the angle adjacent to $ \angle b $ is $ 44^\circ $ (vertical angles), so $ \angle b=180^\circ - 44^\circ = 136^\circ $.
## Step1: Identify angle relationship
The angle $ \angle b $ and the $ 44^\circ $ angle are supplementary (they form a linear pair).
## Step2: Solve for $ \angle b $
$ \angle b=180^\circ - 44^\circ = 136^\circ $
# Answer:
$ 136^\circ $

### 25b Solution:
# Explanation:
## Step1: Identify angle relationship
The two angles $ 50^\circ $ and $ (3x + 2)^\circ $ are vertical angles? No, they are supplementary? Wait, no, when two lines intersect, vertical angles are equal, and linear pairs are supplementary. The angle $ 50^\circ $ and $ (3x + 2)^\circ $ are vertical angles? No, they are adjacent and form a linear pair? Wait, the diagram shows two intersecting lines. The angle $ 50^\circ $ and $ (3x + 2)^\circ $: if they are vertical angles, then $ 3x + 2=180 - 50 $? No, vertical angles are equal. Wait, no, the angle $ 50^\circ $ and $ (3x + 2)^\circ $ are supplementary (linear pair). So $ 50+(3x + 2)=180 $
## Step2: Set up the equation
$ 50+(3x + 2)=180 $
## Step3: Simplify the equation
$ 3x+52 = 180 $
## Step4: Solve for $ x $
Subtract 52 from both sides: $ 3x=180 - 52 = 128 $. Then $ x=\frac{128}{3}\approx42.67 $? No, that can't be. Wait, maybe the angle $ 50^\circ $ and $ (3x + 2)^\circ $ are vertical angles. So $ 3x + 2=180 - 50 $? No, vertical angles are equal. Wait, I think I made a mistake. The correct relationship is that the angle $ 50^\circ $ and $ (3x + 2)^\circ $ are supplementary (linear pair), so $ 50+(3x + 2)=180 $
$ 3x+52 = 180 $
$ 3x=180 - 52=128 $
$ x=\frac{128}{3}\approx42.67 $. But that seems odd. Wait, maybe the angle $ 50^\circ $ and $ (3x + 2)^\circ $ are vertical angles, so $ 3x + 2 = 180 - 50=130 $
$ 3x=130 - 2 = 128 $
$ x=\frac{128}{3}\approx42.67 $. Or maybe the angle $ 50^\circ $ and $ (3x + 2)^\circ $ are equal (vertical angles), so $ 3x+2 = 50 $
## Step1: Assume vertical angles (equal)
If $ 3x + 2=50 $
## Step2: Solve for $ x $
Subtract 2 from both sides: $ 3x=50 - 2 = 48 $
Divide both sides by 3: $ x = 16 $
# Answer:
$ x = 16 $ (assuming vertical angles are equal)

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QUESTION IMAGE

Question

Explanation:

24a Solution:

Step1: Identify angle relationship

Step2: Solve for \( \angle b \)

Step1: Identify angle relationship

Step2: Set up the equation

Step3: Simplify the equation

Step4: Solve for \( x \)

Step1: Identify angle relationship

Answer:

24b Solution: