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In the circle below, DAB and DCB are right angles and m BDC = 53°. The figure is not drawn to scale. What is CAD? A. 286° B. 254° C. 233° D. 217°

In the circle below, DAB and DCB are right angles and m BDC = 53°. The figure is not drawn to scale. What is CAD? A. 286° B. 254° C. 233° D. 217°

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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In the circle below, DAB and DCB are right angles and m BDC = 53°. The figure is not drawn to scale.

What is CAD?

A. 286°

B. 254°

C. 233°

D. 217°

### Understanding Cyclic Quadrilaterals In this educational section, we will explore the properties of cyclic quadrilaterals with the help of a diagram. #### Diagram Explanation The diagram represents a cyclic quadrilateral, which is a quadrilateral where all vertices lie on the circumference of a circle. The vertices of the cyclic quadrilateral in the diagram are labeled as A, B, C, and D. Here are the main features of the diagram: - **Circle**: There is a circle encompassing the quadrilateral. - **Vertices**: The quadrilateral is defined by the four points on the circle, labeled A, B, C, and D. - **Sides**: The quadrilateral is made up of four sides connecting these vertices: AB, BC, CD, and DA. - **Diagonals**: The diagonals in the quadrilateral are AD and BC, intersecting each other at point E inside the quadrilateral. #### Properties of Cyclic Quadrilaterals 1. **Opposite Angles**: The most important property of a cyclic quadrilateral is that the sum of its opposite angles is always 180 degrees. That is, ∠A + ∠C = 180° and ∠B + ∠D = 180°. 2. **Equal Angles**: The angles subtended by the same arc are equal. For instance, ∠ACB and ∠ADB are equal because they subtend the same arc AB. 3. **Ptolemy’s Theorem**: For any cyclic quadrilateral, the sum of the products of the lengths of its opposite sides is equal to the product of the lengths of its diagonals. Mathematically, it can be expressed as: \[ AC \cdot BD = AB \cdot CD + AD \cdot BC \] By studying this diagram and understanding these properties, students can deepen their knowledge of cyclic quadrilaterals and their applications in geometry.
Transcribed Image Text:### Understanding Cyclic Quadrilaterals In this educational section, we will explore the properties of cyclic quadrilaterals with the help of a diagram. #### Diagram Explanation The diagram represents a cyclic quadrilateral, which is a quadrilateral where all vertices lie on the circumference of a circle. The vertices of the cyclic quadrilateral in the diagram are labeled as A, B, C, and D. Here are the main features of the diagram: - **Circle**: There is a circle encompassing the quadrilateral. - **Vertices**: The quadrilateral is defined by the four points on the circle, labeled A, B, C, and D. - **Sides**: The quadrilateral is made up of four sides connecting these vertices: AB, BC, CD, and DA. - **Diagonals**: The diagonals in the quadrilateral are AD and BC, intersecting each other at point E inside the quadrilateral. #### Properties of Cyclic Quadrilaterals 1. **Opposite Angles**: The most important property of a cyclic quadrilateral is that the sum of its opposite angles is always 180 degrees. That is, ∠A + ∠C = 180° and ∠B + ∠D = 180°. 2. **Equal Angles**: The angles subtended by the same arc are equal. For instance, ∠ACB and ∠ADB are equal because they subtend the same arc AB. 3. **Ptolemy’s Theorem**: For any cyclic quadrilateral, the sum of the products of the lengths of its opposite sides is equal to the product of the lengths of its diagonals. Mathematically, it can be expressed as: \[ AC \cdot BD = AB \cdot CD + AD \cdot BC \] By studying this diagram and understanding these properties, students can deepen their knowledge of cyclic quadrilaterals and their applications in geometry.
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