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
#### Diagram Description:
- A circle \(O\) with points \(A\), \(B\), \(C\), and \(D\).
- \( \overline{AB} \) and \( \overline{DC} \) are the two bases of the trapezoid \(ABCD\).
- \( \overline{DC} \) is the diameter of circle \(O\).
- An arc \(AB\) (from point \(A\) to point \(B\)) is 80 degrees.
- The trapezoid is inscribed in circle \(O\).
#### Solution Approach:
1. Understand that since \(\overline{DC}\) is the diameter, points \(D\) and \(C\) lie on the circle's circumference, making the circle a semicircle across \(\overline{DC}\).
2. The inscribed angle \(\overset{\frown}{ABC}\) subtended by arc \(\overset{\frown}{AB}\) (80 degrees) and arc \(BC\) must consider the circle's property, where the total angle around a point in a circle is \(360^\circ\).
#### Steps to Solve:
1. Calculate the full circle angle coverage: \(360^\circ\).
2. The arcs \(\overset{\frown}{AB}\) and \(\overset{\frown}{BC}\) should sum up to half the circle since \(\overline{DC}\) is the diameter: \(180^\circ\).
3. Given \(m\overset{\frown}{AB} = 80^\circ\), the remaining angle must be:
\[
180^\circ - 80^\circ = 100^\circ
\]
Thus, \(m\overset{\frown}{BC} = 100^\circ\).
#### Summary:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba437994-7eb2-41d2-bf90-86c7e632409d%2F5da87a7c-ec31-46b7-872f-ca31a4b554db%2Fnd8fuhd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem Explanation
#### Problem Statement:
In the diagram below, trapezoid \(ABCD\), with bases \(\overline{AB}\) and \(\overline{DC}\), is inscribed in circle \(O\), with diameter \(\overline{DC}\). If \(m\overset{\frown}{AB} = 80^\circ\), find \(m\overset{\frown}{BC}\).
#### Diagram Description:
- A circle \(O\) with points \(A\), \(B\), \(C\), and \(D\).
- \( \overline{AB} \) and \( \overline{DC} \) are the two bases of the trapezoid \(ABCD\).
- \( \overline{DC} \) is the diameter of circle \(O\).
- An arc \(AB\) (from point \(A\) to point \(B\)) is 80 degrees.
- The trapezoid is inscribed in circle \(O\).
#### Solution Approach:
1. Understand that since \(\overline{DC}\) is the diameter, points \(D\) and \(C\) lie on the circle's circumference, making the circle a semicircle across \(\overline{DC}\).
2. The inscribed angle \(\overset{\frown}{ABC}\) subtended by arc \(\overset{\frown}{AB}\) (80 degrees) and arc \(BC\) must consider the circle's property, where the total angle around a point in a circle is \(360^\circ\).
#### Steps to Solve:
1. Calculate the full circle angle coverage: \(360^\circ\).
2. The arcs \(\overset{\frown}{AB}\) and \(\overset{\frown}{BC}\) should sum up to half the circle since \(\overline{DC}\) is the diameter: \(180^\circ\).
3. Given \(m\overset{\frown}{AB} = 80^\circ\), the remaining angle must be:
\[
180^\circ - 80^\circ = 100^\circ
\]
Thus, \(m\overset{\frown}{BC} = 100^\circ\).
#### Summary:
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 2 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