Question

quadrilateral abcd is inscribed in circle o below. segment \\( \overline{ac} \\) is a diameter of the circle.
what is the sum of the degree measures of \\( \angle abc \\) and \\( \angle cda \\)?
a. \\( 180^{\circ} \\)
b. \\( 360^{\circ} \\)
c. \\( 270^{\circ} \\)
d. \\( 90^{\circ} \\)
\\( \bigcirc \\) a
\\( \bigcirc \\) b
\\( \bigcirc \\) c
\\( \bigcirc \\) d

Explanation:

Step1: Recall Inscribed Quadrilateral Property

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary, meaning their sum is \( 180^\circ \). Also, the sum of all interior angles of a quadrilateral is \( 360^\circ \). For cyclic quadrilateral \( ABCD \), \( \angle ABC \) and \( \angle CDA \) are opposite angles.

Step2: Apply the Property

By the property of cyclic quadrilaterals, opposite angles sum to \( 180^\circ \). So \( \angle ABC + \angle CDA = 180^\circ \).

Answer:

A. \( 180^\circ \)