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What is m ZA in quadrilateral ABCD shown below? A D 82° 620 C B

What is m ZA in quadrilateral ABCD shown below? A D 82° 620 C B

Elementary Geometry For College Students, 7e
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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### 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.
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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