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
Related questions
Question
![### Problem: Finding an Unknown Interior Angle of a Polygon
#### Question:
The convex polygon below has 5 sides. Find the value of \( x \).
#### Diagram:
The diagram is a convex pentagon with labeled interior angles. Four of the angles are provided:
- \( 135^\circ \)
- \( 106^\circ \)
- \( 81^\circ \)
- \( 108^\circ \)
The fifth angle is labeled as \( x^\circ \).
#### Explanation:
The sum of the interior angles \(S\) of a polygon with \(n\) sides can be calculated using the formula:
\[ S = (n - 2) \times 180^\circ \]
For a pentagon (\(n = 5\)):
\[ S = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
To find the unknown angle \( x \), sum the known angles and subtract from the total sum of the interior angles:
\[ 135^\circ + 106^\circ + 81^\circ + 108^\circ + x^\circ = 540^\circ \]
Combine the known angles:
\[ 430^\circ + x^\circ = 540^\circ \]
Solve for \( x \):
\[ x^\circ = 540^\circ - 430^\circ = 110^\circ \]
#### Answer:
The value of \( x \) is \( 110^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6373081f-17d9-417b-b535-bd8706276caa%2Fa1600328-eda0-4e65-814a-4f458bb2ae78%2F0ecqspt_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem: Finding an Unknown Interior Angle of a Polygon
#### Question:
The convex polygon below has 5 sides. Find the value of \( x \).
#### Diagram:
The diagram is a convex pentagon with labeled interior angles. Four of the angles are provided:
- \( 135^\circ \)
- \( 106^\circ \)
- \( 81^\circ \)
- \( 108^\circ \)
The fifth angle is labeled as \( x^\circ \).
#### Explanation:
The sum of the interior angles \(S\) of a polygon with \(n\) sides can be calculated using the formula:
\[ S = (n - 2) \times 180^\circ \]
For a pentagon (\(n = 5\)):
\[ S = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
To find the unknown angle \( x \), sum the known angles and subtract from the total sum of the interior angles:
\[ 135^\circ + 106^\circ + 81^\circ + 108^\circ + x^\circ = 540^\circ \]
Combine the known angles:
\[ 430^\circ + x^\circ = 540^\circ \]
Solve for \( x \):
\[ x^\circ = 540^\circ - 430^\circ = 110^\circ \]
#### Answer:
The value of \( x \) is \( 110^\circ \).
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