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. Given that ZDAB and ZDCB are right angles and MZDBC = 42º, what is the measure of CAB? %3D D. B.

. Given that ZDAB and ZDCB are right angles and MZDBC = 42º, what is the measure of CAB? %3D D. 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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**Geometry Problem: Angle Measures** ### Problem Statement Given that ∠DAB and ∠DCB are right angles and m∠DBC = 42°, what is the measure of ∠CAB? ### Diagram Description The diagram consists of a circle with four points labeled A, B, C, and D placed on the circumference. The points are connected forming a quadrilateral inscribed in the circle with the following notable features: - Line segments AD and AB create ∠DAB which is a right angle. - Line segments DC and CB create ∠DCB which is also a right angle. - Line segment DB is drawn through the middle, with ∠DBC given as 42°. ### Solution Explanation To solve for the measure of ∠CAB, we need to utilize the given information that ∠DAB and ∠DCB are right angles and the measure of ∠DBC: 1. ∠DAB is a right angle, so m∠DAB = 90°. 2. ∠DCB is a right angle, so m∠DCB = 90°. 3. m∠DBC is given as 42°. Using the property of angles in a semicircle: - The measure of the arc ADB must be 180° because angle ∠DCB subtends this arc and is a right angle. From the diagram, ∠DAB, ∠DBC, and ∠CAB are angles subtended by arcs in the circle. In a cyclic quadrilateral (quadrilateral inscribed in a circle), opposite angles are supplementary. This means: - ∠DAB + ∠DCB = 180°. As both ∠DAB and ∠DCB are right angles (90° each), the arcs they subtend add up to 180°. To find ∠CAB: - Note that ∠CAB and ∠DAB are subtended by the same arc AB, thus these angles add up to equal the total angle around point A. - m∠DAB + m∠CAB = 90°. Since m∠DAB = 90°: - m∠CAB = 90° - m∠DAB = 90° - 42°. - Therefore, m∠CAB = 48°. Hence, the measure of ∠CAB is 48°.
Transcribed Image Text:**Geometry Problem: Angle Measures** ### Problem Statement Given that ∠DAB and ∠DCB are right angles and m∠DBC = 42°, what is the measure of ∠CAB? ### Diagram Description The diagram consists of a circle with four points labeled A, B, C, and D placed on the circumference. The points are connected forming a quadrilateral inscribed in the circle with the following notable features: - Line segments AD and AB create ∠DAB which is a right angle. - Line segments DC and CB create ∠DCB which is also a right angle. - Line segment DB is drawn through the middle, with ∠DBC given as 42°. ### Solution Explanation To solve for the measure of ∠CAB, we need to utilize the given information that ∠DAB and ∠DCB are right angles and the measure of ∠DBC: 1. ∠DAB is a right angle, so m∠DAB = 90°. 2. ∠DCB is a right angle, so m∠DCB = 90°. 3. m∠DBC is given as 42°. Using the property of angles in a semicircle: - The measure of the arc ADB must be 180° because angle ∠DCB subtends this arc and is a right angle. From the diagram, ∠DAB, ∠DBC, and ∠CAB are angles subtended by arcs in the circle. In a cyclic quadrilateral (quadrilateral inscribed in a circle), opposite angles are supplementary. This means: - ∠DAB + ∠DCB = 180°. As both ∠DAB and ∠DCB are right angles (90° each), the arcs they subtend add up to 180°. To find ∠CAB: - Note that ∠CAB and ∠DAB are subtended by the same arc AB, thus these angles add up to equal the total angle around point A. - m∠DAB + m∠CAB = 90°. Since m∠DAB = 90°: - m∠CAB = 90° - m∠DAB = 90° - 42°. - Therefore, m∠CAB = 48°. Hence, the measure of ∠CAB is 48°.
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