Question

the endpoints of diameter bd in a circle form an angle with point c. what is the measure of the arc intercepted by ∠bcd? 180° 60° 90° 45°

Explanation:

Step1: Recall Thales' Theorem

Thales' theorem states that if you have a triangle inscribed in a circle where one side is the diameter of the circle, then the angle opposite that side (the angle subtended by the diameter) is a right angle (90°). However, here we need the arc intercepted by ∠BCD. The arc intercepted by an inscribed angle is related to the angle, but first, note that BD is a diameter. A diameter divides the circle into a semicircle, which measures 180°. But ∠BCD intercepts arc BD? Wait, no, ∠BCD is an inscribed angle? Wait, no, BD is the diameter, so arc BD is a semicircle (180°). But wait, ∠BCD: let's see, points B, C, D are on the circle? Wait, C is on the circle, B and D are endpoints of the diameter. So triangle BCD is inscribed with BD as diameter. Then ∠BCD: wait, Thales' theorem says ∠BCD should be 90°, but we need the arc intercepted by ∠BCD. The intercepted arc by an inscribed angle is the arc that is opposite the angle, i.e., the arc that is not containing the angle's vertex. Wait, ∠BCD has vertex at C, so the intercepted arc is arc BD. Since BD is a diameter, arc BD is a semicircle, which is 180°? Wait, no, maybe I made a mistake. Wait, no: the inscribed angle theorem says that the measure of an inscribed angle is half the measure of its intercepted arc. But if ∠BCD is an inscribed angle intercepting arc BD, then m∠BCD = 1/2 m(arc BD). But BD is a diameter, so arc BD is 180°, so m∠BCD would be 90°, which matches Thales' theorem. But the question is asking for the measure of the arc intercepted by ∠BCD. So the arc intercepted by ∠BCD is arc BD, which is a semicircle, so 180°? Wait, but let's check the options. The options include 180°, 60°, 90°, 45°. Wait, maybe I misread the angle. Wait, ∠BCD: vertex at C, sides CB and CD. So the intercepted arc is arc BD. Since BD is a diameter, arc BD is 180°, so the measure of the arc intercepted by ∠BCD is 180°? Wait, but let's confirm. The intercepted arc by an angle is the arc that is between the two points (B and D) that the angle's sides meet on the circle, not containing the vertex (C). So arc BD, which is a semicircle, so 180°.

Step2: Confirm the arc measure

Since BD is a diameter of the circle, the arc BD (the semicircle) has a measure of 180° because a full circle is 360°, so a semicircle is half of that, 180°. The angle ∠BCD intercepts arc BD, so the measure of the arc intercepted by ∠BCD is 180°.

Answer:

180°