Question
Question 1
A)There is a pair of vertical angles whose measures are m∠1=106° and m∠2=(3x−75)°. What equation can you write to solve for x?(1 point)
Responses
3x°−75°=106°
3 x minus 75 equals 106 degrees
3x°−75°+106°=180°
3 x minus 75 plus 106 degrees equals 180 degrees
3x°−75°−106°=180°
3 x minus 75 minus 106 degrees equals 180 degrees
3x°−75°+106°=90°
3 x minus 75 plus 106 degrees equals 90 degrees
Question 2
A)Two intersecting lines create four angles; ∠1 and ∠3 are a vertical pair. Their sum of their measures is equal to 68°. What equation solves for the value of each angle?(1 point)
Responses
x°=68°
x equals 68 degrees
2x°=68°
2 x equals 68 degrees
68x°=2°
68 x equals 2
2x°+68°=180°
2 x plus 68 degrees equals 180 degrees
Question 3
A)The sum of the measures of two vertical angles is 300°. Solve for the measure of each angle.(1 point)
Responses
75°
75 degrees
120°
120 degrees
150°
150 degrees
90°
90 degrees
Question 4
A)In a geometric figure two intersecting lines creating four angles. If ∠1 and ∠3 are a vertical pair, m∠1=(4x+26)°, and m∠3=98°, what is the value of x?(1 point)
Responses
18
18
14
14
72
72
98
98
Question 5
A)For two vertical angles with measures m∠2=(5x+23)° and m∠4=(3x+35)°, what is the measure of each angle?(1 point)
Responses
53°
53 degrees
29°
29 degrees
15.3°
15.3 degrees
6°
6 degrees
A)There is a pair of vertical angles whose measures are m∠1=106° and m∠2=(3x−75)°. What equation can you write to solve for x?(1 point)
Responses
3x°−75°=106°
3 x minus 75 equals 106 degrees
3x°−75°+106°=180°
3 x minus 75 plus 106 degrees equals 180 degrees
3x°−75°−106°=180°
3 x minus 75 minus 106 degrees equals 180 degrees
3x°−75°+106°=90°
3 x minus 75 plus 106 degrees equals 90 degrees
Question 2
A)Two intersecting lines create four angles; ∠1 and ∠3 are a vertical pair. Their sum of their measures is equal to 68°. What equation solves for the value of each angle?(1 point)
Responses
x°=68°
x equals 68 degrees
2x°=68°
2 x equals 68 degrees
68x°=2°
68 x equals 2
2x°+68°=180°
2 x plus 68 degrees equals 180 degrees
Question 3
A)The sum of the measures of two vertical angles is 300°. Solve for the measure of each angle.(1 point)
Responses
75°
75 degrees
120°
120 degrees
150°
150 degrees
90°
90 degrees
Question 4
A)In a geometric figure two intersecting lines creating four angles. If ∠1 and ∠3 are a vertical pair, m∠1=(4x+26)°, and m∠3=98°, what is the value of x?(1 point)
Responses
18
18
14
14
72
72
98
98
Question 5
A)For two vertical angles with measures m∠2=(5x+23)° and m∠4=(3x+35)°, what is the measure of each angle?(1 point)
Responses
53°
53 degrees
29°
29 degrees
15.3°
15.3 degrees
6°
6 degrees
Answers
GPT-4o mini
Let's go through each question one by one:
### Question 1
**A)** The equation to solve for \( x \) using the measures of the vertical angles is:
\[ 3x - 75 = 106 \]
- **Correct Response**: **3x°−75°=106°**
### Question 2
**A)** Since ∠1 and ∠3 are vertical angles, their measures are equal. If the sum of their measures is 68°, we can write:
\[ 2x = 68 \]
- **Correct Response**: **2x°=68°**
### Question 3
**A)** For two vertical angles whose sum is 300°, each angle is equal (let's call them \( x \)):
\[ 2x = 300 \]
Thus, \( x = 150° \).
- **Correct Response**: **150°**
### Question 4
**A)** Since ∠1 and ∠3 are vertical angles, we set their measures equal to each other:
\[ 4x + 26 = 98 \]
Solving for \( x \):
\[ 4x = 98 - 26 \]
\[ 4x = 72 \]
\[ x = 18 \]
- **Correct Response**: **18**
### Question 5
**A)** Setting the two expressions for vertical angles equal gives us:
\[ 5x + 23 = 3x + 35 \]
Solving for \( x \):
\[ 5x - 3x = 35 - 23 \]
\[ 2x = 12 \]
\[ x = 6 \]
Substituting back to find the measures of the angles:
- Angle 1: \( 5(6) + 23 = 30 + 23 = 53° \)
- Angle 2: \( 3(6) + 35 = 18 + 35 = 53° \)
- **Correct Response**: **53°**
### Summary of Correct Responses:
1. 3x°−75°=106°
2. 2x°=68°
3. 150°
4. 18
5. 53°
### Question 1
**A)** The equation to solve for \( x \) using the measures of the vertical angles is:
\[ 3x - 75 = 106 \]
- **Correct Response**: **3x°−75°=106°**
### Question 2
**A)** Since ∠1 and ∠3 are vertical angles, their measures are equal. If the sum of their measures is 68°, we can write:
\[ 2x = 68 \]
- **Correct Response**: **2x°=68°**
### Question 3
**A)** For two vertical angles whose sum is 300°, each angle is equal (let's call them \( x \)):
\[ 2x = 300 \]
Thus, \( x = 150° \).
- **Correct Response**: **150°**
### Question 4
**A)** Since ∠1 and ∠3 are vertical angles, we set their measures equal to each other:
\[ 4x + 26 = 98 \]
Solving for \( x \):
\[ 4x = 98 - 26 \]
\[ 4x = 72 \]
\[ x = 18 \]
- **Correct Response**: **18**
### Question 5
**A)** Setting the two expressions for vertical angles equal gives us:
\[ 5x + 23 = 3x + 35 \]
Solving for \( x \):
\[ 5x - 3x = 35 - 23 \]
\[ 2x = 12 \]
\[ x = 6 \]
Substituting back to find the measures of the angles:
- Angle 1: \( 5(6) + 23 = 30 + 23 = 53° \)
- Angle 2: \( 3(6) + 35 = 18 + 35 = 53° \)
- **Correct Response**: **53°**
### Summary of Correct Responses:
1. 3x°−75°=106°
2. 2x°=68°
3. 150°
4. 18
5. 53°