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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![**Title: Finding the Value of x in a Polygon with External Angles**
**Problem:**
Find the value of \( x \). (The figure may not be drawn to scale.)
**Diagram Explanation:**
The provided diagram is a hexagon with angles extended outwards. Each vertex of the hexagon is marked with a point, labeled \( A \), \( B \), \( C \), \( D \), \( E \), and \( F \). There are several external angles given around the hexagon:
- At \( B \), the angle formed between the line segments \( AB \) and \( BC \) is \( 60^\circ \).
- At \( A \), the angle formed between the line segments \( BA \) and \( AF \) is \( 69^\circ \).
- At \( F \), the angle formed between the line segments \( EF \) and \( AF \) is \( 42^\circ \).
- At \( E \), the angle formed between the line segments \( DE \) and \( EF \) is \( (x + 23)^\circ \).
- At \( D \), the angle formed between the line segments \( CD \) and \( DE \) is \( x^\circ \).
- At \( C \), the angle formed between the line segments \( BC \) and \( CD \) is \( 64^\circ \).
**Solution:**
To find the value of \( x \), we will use the concept that the sum of external angles of any polygon is always \( 360^\circ \).
Therefore, we set up the equation:
\[ 64^\circ + 60^\circ + 69^\circ + 42^\circ + (x + 23)^\circ + x^\circ = 360^\circ \]
Combining like terms:
\[ 64 + 60 + 69 + 42 + 23 + x + x = 360 \]
\[ 258 + 2x = 360 \]
Subtract 258 from both sides:
\[ 2x = 102 \]
Divide by 2:
\[ x = 51 \]
**Answer:**
The value of \( x \) is \( 51 \) degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4963a1b7-231e-4415-aa15-591e301cddd8%2F898fce7d-0b9e-407b-b964-f37bde55882a%2Fxmd5rv.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Value of x in a Polygon with External Angles**
**Problem:**
Find the value of \( x \). (The figure may not be drawn to scale.)
**Diagram Explanation:**
The provided diagram is a hexagon with angles extended outwards. Each vertex of the hexagon is marked with a point, labeled \( A \), \( B \), \( C \), \( D \), \( E \), and \( F \). There are several external angles given around the hexagon:
- At \( B \), the angle formed between the line segments \( AB \) and \( BC \) is \( 60^\circ \).
- At \( A \), the angle formed between the line segments \( BA \) and \( AF \) is \( 69^\circ \).
- At \( F \), the angle formed between the line segments \( EF \) and \( AF \) is \( 42^\circ \).
- At \( E \), the angle formed between the line segments \( DE \) and \( EF \) is \( (x + 23)^\circ \).
- At \( D \), the angle formed between the line segments \( CD \) and \( DE \) is \( x^\circ \).
- At \( C \), the angle formed between the line segments \( BC \) and \( CD \) is \( 64^\circ \).
**Solution:**
To find the value of \( x \), we will use the concept that the sum of external angles of any polygon is always \( 360^\circ \).
Therefore, we set up the equation:
\[ 64^\circ + 60^\circ + 69^\circ + 42^\circ + (x + 23)^\circ + x^\circ = 360^\circ \]
Combining like terms:
\[ 64 + 60 + 69 + 42 + 23 + x + x = 360 \]
\[ 258 + 2x = 360 \]
Subtract 258 from both sides:
\[ 2x = 102 \]
Divide by 2:
\[ x = 51 \]
**Answer:**
The value of \( x \) is \( 51 \) degrees.
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