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
Concept explainers
Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
![**Geometry Problem: Inscribed Quadrilateral**
**Problem Statement:**
As shown in the diagram below, quadrilateral \( DEFG \) is inscribed in a circle and \( m\angle D = 86^\circ \).
*(A diagram is included featuring a circle with four points labeled D, E, F, and G. These points form quadrilateral DEFG. The interior angle at point D is given as 86 degrees.)*
**Tasks:**
1. Determine and state \( m\angle GFE \).
2. Determine and state \( m\angle F \).
**Explanation:**
Given that quadrilateral \( DEFG \) is inscribed in a circle, we can use the property of an inscribed quadrilateral that states: opposite angles of an inscribed quadrilateral sum to \( 180^\circ \).
Therefore:
\[ m\angle D + m\angle F = 180^\circ \]
Since \( m\angle D = 86^\circ \):
\[ 86^\circ + m\angle F = 180^\circ \]
\[ m\angle F = 180^\circ - 86^\circ \]
\[ m\angle F = 94^\circ \]
Next, to determine \( m\angle GFE \), we observe that it is an external angle to triangle \(\Delta GFE\), thus:
\[ m\angle GFE = m\angle F\]
So,
\[ m\angle GFE = 94^\circ \]
**Conclusion:**
1. The measure of \( \angle GFE \) is \( 94^\circ \).
2. The measure of \( \angle F \) is \( 94^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba437994-7eb2-41d2-bf90-86c7e632409d%2F1480f45e-0fa3-4cab-8142-dae4adb9fb4b%2F1io1mx2s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Geometry Problem: Inscribed Quadrilateral**
**Problem Statement:**
As shown in the diagram below, quadrilateral \( DEFG \) is inscribed in a circle and \( m\angle D = 86^\circ \).
*(A diagram is included featuring a circle with four points labeled D, E, F, and G. These points form quadrilateral DEFG. The interior angle at point D is given as 86 degrees.)*
**Tasks:**
1. Determine and state \( m\angle GFE \).
2. Determine and state \( m\angle F \).
**Explanation:**
Given that quadrilateral \( DEFG \) is inscribed in a circle, we can use the property of an inscribed quadrilateral that states: opposite angles of an inscribed quadrilateral sum to \( 180^\circ \).
Therefore:
\[ m\angle D + m\angle F = 180^\circ \]
Since \( m\angle D = 86^\circ \):
\[ 86^\circ + m\angle F = 180^\circ \]
\[ m\angle F = 180^\circ - 86^\circ \]
\[ m\angle F = 94^\circ \]
Next, to determine \( m\angle GFE \), we observe that it is an external angle to triangle \(\Delta GFE\), thus:
\[ m\angle GFE = m\angle F\]
So,
\[ m\angle GFE = 94^\circ \]
**Conclusion:**
1. The measure of \( \angle GFE \) is \( 94^\circ \).
2. The measure of \( \angle F \) is \( 94^\circ \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning