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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![### Example Geometry Problem and Solution
**Problem 4: Solve for \( x \), show your work.**
Below, we have a pentagon with the following labeled angles:
- The pentagon has five interior angles.
- The angles within the pentagon are labeled as \( x \), \( 2x \), \( x + 2^\circ \), a right angle (90°), and the last angle is \( 52° \).
First, recall the sum of interior angles formula for a polygon:
\[ \text{Sum of interior angles} = (n-2) \times 180^\circ \]
where \( n \) is the number of sides of the polygon. For a pentagon:
\[ \text{Sum of interior angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
The labeled angles sum up to:
\[ x + 2x + (x + 2^\circ) + 90^\circ + 52^\circ = 540^\circ \]
Combine like terms to form an equation:
\[ 4x + 2^\circ + 90^\circ + 52^\circ = 540^\circ \]
\[ 4x + 144^\circ = 540^\circ \]
Subtract \( 144^\circ \) from both sides to isolate the term with \( x \):
\[ 4x = 396^\circ \]
Finally, solve for \( x \) by dividing both sides by 4:
\[ x = \frac{396^\circ}{4} = 99^\circ \]
Therefore, the value of \( x \) is \( 99^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94e3dc3a-5138-4de9-a130-9782421a1710%2Fba5ee78d-8fd6-4616-b7ab-7b204c4eedcf%2Flszvgpo_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Geometry Problem and Solution
**Problem 4: Solve for \( x \), show your work.**
Below, we have a pentagon with the following labeled angles:
- The pentagon has five interior angles.
- The angles within the pentagon are labeled as \( x \), \( 2x \), \( x + 2^\circ \), a right angle (90°), and the last angle is \( 52° \).
First, recall the sum of interior angles formula for a polygon:
\[ \text{Sum of interior angles} = (n-2) \times 180^\circ \]
where \( n \) is the number of sides of the polygon. For a pentagon:
\[ \text{Sum of interior angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
The labeled angles sum up to:
\[ x + 2x + (x + 2^\circ) + 90^\circ + 52^\circ = 540^\circ \]
Combine like terms to form an equation:
\[ 4x + 2^\circ + 90^\circ + 52^\circ = 540^\circ \]
\[ 4x + 144^\circ = 540^\circ \]
Subtract \( 144^\circ \) from both sides to isolate the term with \( x \):
\[ 4x = 396^\circ \]
Finally, solve for \( x \) by dividing both sides by 4:
\[ x = \frac{396^\circ}{4} = 99^\circ \]
Therefore, the value of \( x \) is \( 99^\circ \).
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