Name ______________________________ Module 5 Central Angles and Inscribed Angles

Learning Target: I can use relationships among central angles, inscribed angles, and the intercepted arcs.

Opening Exercise

In the diagram below of circle O, chords AD and BC intersect at E. If m∠DEB=120 °, determine the measure of the following angles.

m∠C E A=¿¿

m∠CE D=¿¿

m∠B EA=¿¿

Central Angles and Inscribed Angles

A central angle is an angle whose vertex is the center of a circle. An arc is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.

1. AC is the diameter of circle F. Find the measure of each arc.

m ED = _______

m AE = _______

m ADC = _______

mDC = _______

mEC = _______

Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure ST ≅ UV .

2. Find the measure of each of the following in circle A.

m BC = _______

m BEC = _______

m∠BAC = _______

m∠BAD = _______

mCD = _______

An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. An intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them.

Theorem

3. Find each measure.

m∠PRU = _______

mPS = _______

4. Find each measure.

m∠LJM = _______

m∠LKM = _______

m LM = _______

Theorem

5. Solve for a.

6. Find each measure.

m∠KGH = _______

m∠GHJ = _______

mKJH = _______

m JKG = _______

Name ______________________________ Module 5

Central Angles and Inscribed Angles Problem Set

1. JM and KN is the diameters of circle P. Find the measure of each arc.

m JK = _______

m JKL = _______

mKJN = _______

mKM = _______

mKJM = _______

2. Find each measure.

m∠DAE = _______

m ADC = _______

m AC = _______

3. Find each measure.

m∠EDF = _______

m∠EGF = _______

mEF = _______

4. AB is a diameter of the circle shown. The radius is 12.5 cm, and AC=7 cm. a. Find m∠C .

b. Find AB.

c. Find BC.

5. In the circle shown, BC is a diameter with center A. a. Find m∠DAB.

b. Find m∠BAE.

c. Find m∠DAE.

6. In the figure below, O is the center of the circle, and AD is a diameter.a. Find m∠ AOB.

b. If m∠ AOB ∶m∠COD=3 ∶ 4, what is m∠BOC?

7. ∠CBD is inscribed in CD and ∠CA D is a central angle that intercepts the same arc. Prove that y=2x.

8. Determine the value of x.

9. Determine the value of x.

10. Find the angle measure of angle x.

11. Given circle A, find the following angles of measure.a. m∠BAD

b. m∠CAB

c. m BC

d. mBD

e. m BCD

12. In the diagram below, quadrilateral JUMP is inscribed in a circle.

Opposite angles J and M must be1) right2) complementary3) congruent4) supplementary

13. In the accompanying diagram, quadrilateral ABCD is inscribed in circle O. If m AB=132 and m BC=82, find m∠ ABC .

Name ______________________________ Module 5Central Angles and Inscribed Angles Exit Ticket

1. The center of the circle below is 𝑂. If angle 𝐵 has a measure of 15 degrees, find the values of 𝑥 and 𝑦. Explain how you know.

2. Find the measure of angles 𝑥 and 𝑦. Justify your reasoning.

3. Given circle A with diameters BC and DE and mCD=56 °.

A. What is the measure of ∠CAD?

B. What is the measure of ∠CBD?

C. What is the degree measure of CBD?