Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![### Understanding Vertical Angles
The image features two intersecting lines, forming four angles at the intersection point B. The points of intersection are labeled A, B, C, D, and E. The angle at A-B-C is given as \(2x + 7\) degrees, while the angle at D-B-E is given as \(4x - 14\) degrees.
**Explanation of Vertical Angles:**
Vertical angles are the angles opposite each other when two lines cross. They are always equal.
In this scenario, since ∠ABC and ∠DBE are vertical angles, they must be equal.
Set the two expressions for the angles equal to each other and solve for \(x\):
\[ 2x + 7 = 4x - 14 \]
**Steps:**
1. Subtract \(2x\) from both sides:
\[ 7 = 2x - 14 \]
2. Add 14 to both sides:
\[ 21 = 2x \]
3. Divide by 2:
\[ x = 10.5 \]
**Finding the Measures:**
- To find the measure of ∠ABC:
\[ \text{∠ABC} = 2x + 7 \]
\[ \text{∠ABC} = 2(10.5) + 7 \]
\[ \text{∠ABC} = 21 + 7 \]
\[ \text{∠ABC} = 28 \text{ degrees} \]
- To find the measure of ∠ABD:
\[ \text{∠ABD} = 4x - 14 \]
\[ \text{∠ABD} = 4(10.5) - 14 \]
\[ \text{∠ABD} = 42 - 14 \]
\[ \text{∠ABD} = 28 \text{ degrees} \]
Since ∠ABC and ∠DBE are vertical angles, they are equal:
\[ \text{∠ABC} = 28 \text{ degrees} \]
Thus, the final values are:
- ∠ ABC and ∠DBE are **vertical** angles, so they are **equal**, ∠ABD = **28** degrees.
- \(x\) = **10.5** degrees, ∠ABC = **28** degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab545a03-1d39-4874-800c-bed9ffa7318e%2F194873f1-fe22-47e1-b226-f9ecad14b85d%2Fsetozod_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Vertical Angles
The image features two intersecting lines, forming four angles at the intersection point B. The points of intersection are labeled A, B, C, D, and E. The angle at A-B-C is given as \(2x + 7\) degrees, while the angle at D-B-E is given as \(4x - 14\) degrees.
**Explanation of Vertical Angles:**
Vertical angles are the angles opposite each other when two lines cross. They are always equal.
In this scenario, since ∠ABC and ∠DBE are vertical angles, they must be equal.
Set the two expressions for the angles equal to each other and solve for \(x\):
\[ 2x + 7 = 4x - 14 \]
**Steps:**
1. Subtract \(2x\) from both sides:
\[ 7 = 2x - 14 \]
2. Add 14 to both sides:
\[ 21 = 2x \]
3. Divide by 2:
\[ x = 10.5 \]
**Finding the Measures:**
- To find the measure of ∠ABC:
\[ \text{∠ABC} = 2x + 7 \]
\[ \text{∠ABC} = 2(10.5) + 7 \]
\[ \text{∠ABC} = 21 + 7 \]
\[ \text{∠ABC} = 28 \text{ degrees} \]
- To find the measure of ∠ABD:
\[ \text{∠ABD} = 4x - 14 \]
\[ \text{∠ABD} = 4(10.5) - 14 \]
\[ \text{∠ABD} = 42 - 14 \]
\[ \text{∠ABD} = 28 \text{ degrees} \]
Since ∠ABC and ∠DBE are vertical angles, they are equal:
\[ \text{∠ABC} = 28 \text{ degrees} \]
Thus, the final values are:
- ∠ ABC and ∠DBE are **vertical** angles, so they are **equal**, ∠ABD = **28** degrees.
- \(x\) = **10.5** degrees, ∠ABC = **28** degrees.
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