### 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 $m\overset{\frown}{BC}$.![Trapezoid and Circle Diagram](https://turing.mu/assets/images/geometry.png) #### 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:

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Answered: In the diagram below, trapezoid ABCD,…

In the diagram below, trapezoid ABCD, with bases AB and DC, is inscribed in circle O, with diameter DC. If mAB = 80, find mBC. 80° D

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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### 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 $m\overset{\frown}{BC}$.

![Trapezoid and Circle Diagram](https://turing.mu/assets/images/geometry.png)

#### 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:

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