Question

what is the sum of opposite angles in any quadrilateral inscribed in a circle? a. 90 degrees b. 270 degrees c. 360 degrees d. 180 degrees if the measure of one intercepted arc in a circle is 140°, what is the measure of the other intercepted arc? a. 140° b. 220° c. 90° d. 180° if arc abc measures 150 degrees in a circle where quadrilateral abcd is inscribed, what is the measure of angle d? a. 30 degrees b. 75 degrees c. 15 degrees d. 115 degrees

Explanation:

Step1: Recall cyclic - quadrilateral property

The sum of opposite angles in a cyclic quadrilateral (a quadrilateral inscribed in a circle) is 180 degrees.

Step2: Recall circle - arc property

The sum of the measures of two intercepted arcs of a circle is 360 degrees. If one arc measures 140 degrees, then the other arc measures 360 - 140=220 degrees.

Step3: Use inscribed - angle and arc relationship

The measure of an inscribed angle is half the measure of its intercepted arc. In cyclic quadrilateral ABCD, if arc ABC measures 150 degrees, then the inscribed angle D intercepts arc ABC. So the measure of angle D is $\frac{1}{2}\times150 = 75$ degrees.

Answer:

1. d. 180 degrees
2. b. 220°
3. b. 75 degrees