An inscribed angle is an angle whose vertex lies on the circle and whose sides contain chords of a circle. In circle A, which angle shows an inscribed angle? A. \angle BCE B. \angle BAD C. \angle CAD D. \angle BAC 5. It is an angle whose vertex is the center of a circle and whose sides intersect the circle. A. Acute Angle B. Right Angle C. Central Angle D. Inscribed Angle 6. What is the degree measure of a semicircle? A. 90° B. 120° C. 180° D. 360° 7. It is an arc which is less than 180°. A. Minor Arc B. Major Arc C. Semicircle D. Circumference For item numbers 8-10, refer to the figure below. C E 44° A D 115° B F 8. In circle A, If
. Find the measure of
. A. 39° B. 78° C. 89° D. 115° 9. If the measure of minor arcs
and
are 44° and 115° respectively, what is the measure of
? A. 21° B. 44° C. 115° D. 159° 10. If the measure of minor arcs
and
are 44° and 115° respectively, what is the measure of
? A. 21° B. 44° C. 115° D. 159°
The Correct Answer and Explanation is:
Of course. Here are the correct answers and a detailed explanation for each question on the worksheet.
Correct Answers:
4. A. ∠BCE
5. C. Central Angle
6. C. 180°
7. A. Minor Arc
8. B. 78°
9. D. 159°
10. B. 44°
Explanation
Here is a breakdown of the geometric principles used to find the correct answers for each question.
4. Inscribed Angle: The definition states an inscribed angle has its vertex on the circle and its sides are chords. Looking at the figure for items 8 through 10, the only angle listed that fits this description is A. ∠BCE. Its vertex, C, is on the circle, and its sides, CE and CB, are chords. The other angles (∠BAD, ∠CAD, ∠BAC) have their vertex at the center A, not on the circle.
5. Central Angle: The question asks for the term describing an angle with its vertex at the center of a circle. This is the exact definition of a C. Central Angle. The other terms describe angle measurements (acute, right) or a different type of angle (inscribed).
6. Semicircle Measure: A full circle contains 360 degrees. A semicircle is, by definition, half of a circle. Therefore, its degree measure is half of 360, which is C. 180°.
7. Minor Arc: Arcs are categorized by their measure. An arc measuring less than 180° is called a A. Minor Arc. A major arc is greater than 180°, and a semicircle is exactly 180°.
8. Finding Arc CE: This question uses the Inscribed Angle Theorem, which states that the measure of an inscribed angle is half the measure of its intercepted arc. The inscribed angle is ∠CBE, given as 39°. It intercepts arc CE. So, the measure of arc CE is twice the measure of the angle: 2 * 39° = B. 78°.
9. Finding Arc CDF: This problem applies the Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of their individual measures. Arc CDF is composed of arc CD and arc DF. Adding their given measures: 44° + 115° = D. 159°.
10. Finding Arc BF: In the figure, the chords DF and CB intersect at the center A, forming vertical angles. The angles ∠CAD and ∠BAF are vertical angles, which means they are equal in measure. The measure of a central angle is equal to its intercepted arc. Since the measure of arc CD is 44°, the measure of its central angle ∠CAD is also 44°. Because ∠BAF is equal to ∠CAD, its measure is also 44°. Therefore, the measure of the intercepted arc BF is B. 44°.thumb_upthumb_down
