Sides contain chords of the circle. An angle whose vertex lies on the circle can be an obtuse angle, a right angle, a central angle, or an inscribed angle. The measure of an inscribed angle is equal to half the measure of its intercepted arc. The measure of an inscribed angle is also equal to half the measure of a central angle that intercepts the same arc. The measure of an inscribed angle is less than twice the measure of a right angle. The measure of an inscribed angle is also less than the measure of a central angle that intercepts the same arc. Which of the following angles is an inscribed angle? In the figure on the right, which of the angles is AQST if AQRT is 90 degrees? In circle O on the right, what is the measure of angle A458? 55 degrees, 60 degrees, or 65 degrees
The Correct Answer and Explanation is:
Here are the correct answers and explanations for the questions in the image.
11. An angle whose vertex lies on the circle and whose sides contain chords of the circle.
The correct answer is B. inscribed angle. By definition, an inscribed angle is an angle formed by two chords in a circle that have a common endpoint on the circle. This common endpoint is the vertex of the angle. A central angle has its vertex at the center of the circle, while obtuse and right angles describe the measure of an angle, not its specific position within a circle.
12. The measure of an inscribed angle is ______ the measure of its intercepted arc.
The correct answer is C. one-half. This is a fundamental concept in geometry known as the Inscribed Angle Theorem. The theorem states that the measure of an angle inscribed in a circle is exactly half the measure of its intercepted arc, which is the portion of the circle that lies in the interior of the inscribed angle.
13. The measure of the intercepted arc is ______ the measure of the inscribed angle.
The correct answer is D. twice. This is the direct corollary of the Inscribed Angle Theorem mentioned above. If the inscribed angle is one-half the measure of the intercepted arc, then it logically follows that the measure of the intercepted arc must be two times, or twice, the measure of the inscribed angle.
14. In the figure on the right, which of the following angles is an inscribed angle?
The correct answer is ∠QMV. An inscribed angle must have its vertex on the circle. In the given figure, the center of the circle is R. Angles like ∠MRN have their vertex at the center, so they are central angles. The angle ∠QMV has its vertex, M, located on the circle, and its sides, QM and MV, are chords of the circle. This fits the definition of an inscribed angle perfectly.
15. In ⊙R on the right, what is ∠QST if ∠QRT = 90°.
The correct answer is A. 45°. The problem states that R is the center of the circle, making ∠QRT a central angle. The measure of a central angle is equal to the measure of its intercepted arc. Therefore, the measure of arc QT is 90°. The angle we need to find, ∠QST, is an inscribed angle that intercepts the same arc, QT. According to the Inscribed Angle Theorem, the measure of ∠QST is one-half the measure of its intercepted arc.
Calculation:
Measure of ∠QST = (1/2) × Measure of arc QT
Measure of ∠QST = (1/2) × 90°
Measure of ∠QST = 45°thumb_upthumb_down
