Test on Central and inscribed angle

Question 1: In the proof for the inscribed angle theorem where the center of the circle lies outside the inscribed angle, the inscribed angle ∠ABC is expressed as the sum of ∠DBC and ∠DBA.

A. Yes

B. No

Explanation: The study materials state that when the center of the circle lies outside the inscribed angle, the inscribed angle ∠ABC is expressed as the difference: ∠ABC = ∠DBC - ∠DBA, not as a sum.

Question 2: According to the study materials, when proving the inscribed angle theorem for the case where the center of the circle lies inside the inscribed angle (Case 2), which of the following is a key step in relating the inscribed angle to the central angle?

A. The inscribed angle is directly shown to be half of the central angle without any further divisions.

B. A line is drawn from the vertex of the inscribed angle through the center, dividing both the inscribed and central angles into two smaller angles, to which Case 1 is applied.

C. The inscribed angle is expressed as the difference between two other inscribed angles, which are then related to central angles.

D. The theorem is proven by constructing a tangent line at the vertex of the inscribed angle.

Explanation: Case 2 of the proof states: 'Draw a line from the vertex of the inscribed angle (B) through the center (O) to a point D on the circle. This divides the inscribed angle ∠ABC into two smaller inscribed angles, ∠ABD and ∠DBC. Similarly, it divides the central angle ∠AOC into two smaller central angles, ∠AOD and ∠DOC. Applying Case 1 to each smaller angle, we have ∠ABD = (1/2)∠AOD and ∠DBC = (1/2)∠DOC.' This describes how Case 2 leverages Case 1 by dividing the angles.

Question 3: The angle between a tangent and a chord through the point of contact is equal to the central angle subtended by the chord.

A. Yes

B. No

Explanation: According to the study materials, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment (i.e., any inscribed angle subtended by the chord in the opposite segment of the circle), not the central angle.

Question 4: The angle between a tangent and a chord through the point of contact is equal to the central angle subtended by the chord.

A. Yes

B. No

Explanation: The study materials state that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment (i.e., any inscribed angle subtended by the chord in the opposite segment of the circle), not the central angle subtended by the chord.

Question 5: Based on the provided study materials, which statement accurately describes inscribed angles that subtend the same arc?

A. They are all equal in measure.

B. They are all right angles.

C. They are equal to the measure of the central angle subtending the same arc.

D. They are formed by a tangent and a chord.

Explanation: The study materials explicitly state, 'All inscribed angles that subtend the same arc are equal in measure.'