Question

if the arc intercepted by angle x in a quadrilateral inscribed in a circle measures 100 degrees, what is the measure of angle x? a. 100 degrees b. 50 degrees c. 120 degrees d. 25 degrees what must be the measure of the arc that angle c intercepts if angle c = 80 degrees in an inscribed quadrilateral? a. 80 degrees b. 90 degrees c. 160 degrees d. 40 degrees which of the following is true for any quadrilateral inscribed in a circle? a. opposite angles sum to 180 degrees. b. it is always a rectangle. c. it is always a rhombus. d. it is always an isosceles trapezoid

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle in a circle is half the measure of the intercepted arc.

Step2: Solve for angle \(X\) in the first question

Given the intercepted arc is \(100^{\circ}\). Let the measure of angle \(X\) be \(x\). Then \(x=\frac{1}{2}\times100^{\circ}=50^{\circ}\).

Step3: Solve for the intercepted arc of angle \(C\) in the second question

Let the measure of the intercepted arc be \(y\). Given angle \(C = 80^{\circ}\). Since the measure of an inscribed angle is half the measure of the intercepted arc, then \(y = 2\times80^{\circ}=160^{\circ}\).

Step4: Recall properties of a cyclic quadrilateral in the third question

A cyclic quadrilateral (a quadrilateral inscribed in a circle) has the property that opposite angles sum to \(180^{\circ}\).

Answer:

1. B. 50 degrees
2. C. 160 degrees
3. A. Opposite angles sum to 180 degrees.