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
![### Understanding Angle Calculations in a Cyclic Quadrilateral
#### Problem Statement:
**What is \( m \angle A \) in quadrilateral \( ABCD \) shown below?**
The provided diagram is a cyclic quadrilateral (a quadrilateral inscribed in a circle). Here are the essential elements and angles marked in the quadrilateral:
- Quadrilateral \( ABCD \) is inscribed in a circle.
- \( \angle D = 82^\circ \)
- \( \angle C = 62^\circ \)
The possible answers to \( m \angle A \) are:
- \( A. \ 28^\circ \)
- \( B. \ 98^\circ \)
- \( C. \ 118^\circ \)
#### Diagram Explanation:
The diagram shows a cyclic quadrilateral inscribed in a circle. A cyclic quadrilateral has the property that the sum of each pair of opposite angles is \( 180^\circ \).
So, given:
\[ \angle D + \angle B = 180^\circ \]
\[ \angle A + \angle C = 180^\circ \]
#### Calculation:
Given:
\[ \angle D = 82^\circ \]
\[ \angle C = 62^\circ \]
Since:
\[ \angle A + \angle C = 180^\circ \]
We plug in the given value of \( \angle C \):
\[ \angle A + 62^\circ = 180^\circ \]
\[ \angle A = 180^\circ - 62^\circ \]
\[ \angle A = 118^\circ \]
Thus, the measure of \( \angle A \) is \( 118^\circ \).
#### Correct Answer:
\[ C. \ 118^\circ \]
**Note:** Understanding how to use the properties of cyclic quadrilaterals can significantly simplify solving such geometric problems. Here, knowing that opposite angles of a cyclic quadrilateral sum to \( 180^\circ \) was the key to finding the correct answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa53c2e40-23d6-4c01-b6e5-37b357bb8cf5%2F402b3b3f-781d-4ba8-befd-82f6e14aa92a%2Fn913i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Angle Calculations in a Cyclic Quadrilateral
#### Problem Statement:
**What is \( m \angle A \) in quadrilateral \( ABCD \) shown below?**
The provided diagram is a cyclic quadrilateral (a quadrilateral inscribed in a circle). Here are the essential elements and angles marked in the quadrilateral:
- Quadrilateral \( ABCD \) is inscribed in a circle.
- \( \angle D = 82^\circ \)
- \( \angle C = 62^\circ \)
The possible answers to \( m \angle A \) are:
- \( A. \ 28^\circ \)
- \( B. \ 98^\circ \)
- \( C. \ 118^\circ \)
#### Diagram Explanation:
The diagram shows a cyclic quadrilateral inscribed in a circle. A cyclic quadrilateral has the property that the sum of each pair of opposite angles is \( 180^\circ \).
So, given:
\[ \angle D + \angle B = 180^\circ \]
\[ \angle A + \angle C = 180^\circ \]
#### Calculation:
Given:
\[ \angle D = 82^\circ \]
\[ \angle C = 62^\circ \]
Since:
\[ \angle A + \angle C = 180^\circ \]
We plug in the given value of \( \angle C \):
\[ \angle A + 62^\circ = 180^\circ \]
\[ \angle A = 180^\circ - 62^\circ \]
\[ \angle A = 118^\circ \]
Thus, the measure of \( \angle A \) is \( 118^\circ \).
#### Correct Answer:
\[ C. \ 118^\circ \]
**Note:** Understanding how to use the properties of cyclic quadrilaterals can significantly simplify solving such geometric problems. Here, knowing that opposite angles of a cyclic quadrilateral sum to \( 180^\circ \) was the key to finding the correct answer.
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