1. 1.
Find x and y.
ANSWER
x = 60; y = 60 2.
ANSWER
x = 35; y = 35

2. EXAMPLE 1
Find measures of arcs
Find the measure of each arc of P, where RT is a diameter. RS a.
RTS b.
RST c.
SOLUTION
RS is a minor arc, so mRS = m RPS = 110o. a.
RTS is a major arc, so mRTS = 360o o = 250o. b. – c.
RT is a diameter, so RST is a semicircle, and mRST = 180o.

3. EXAMPLE 2
Find measures of arcs
A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures.
Survey a.
mAC
SOLUTION a.
mAC
mAB = +
mBC
= 29o + 108o
= 137o

4. EXAMPLE 2
Find measures of arcs
A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures.
Survey b.
mACD
SOLUTION b.
mACD = mAC + mCD
= 137o + 83o
= 220o

5. EXAMPLE 2
Find measures of arcs
A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures.
Survey c.
mADC
SOLUTION
mADC
mAC
= 360o – c.
= 360o – 137o
= 223o

6. EXAMPLE 2
Find measures of arcs
A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures.
Survey d.
mEBD
SOLUTION d.
mEBD = 360o – mED
= 360o – 61o
= 299o

7. GUIDED PRACTICE
for Examples 1 and 2
Identify the given arc as a major arc, minor arc, or
semicircle, and find the measure of the arc.

8. GUIDED PRACTICE
for Examples 1 and 2 1
. TQ
TQ is a minor arc, so m TQ = 120o.
ANSWER
. QRT 2
QRT is a major arc, so m QRT= 240o.
ANSWER

9. GUIDED PRACTICE
for Examples 1 and 2
. TQR 3
TQR is a semicircle, so m TQR = 180o.
ANSWER
. QS 4
QS is a minor arc, so m QS = 160o.
ANSWER

10. GUIDED PRACTICE
for Examples 1 and 2
. TS 5
TS is a minor arc, so m TS = 80o.
ANSWER
. RST 6
RST is a semicircle, so m RST = 180o.
ANSWER

11. EXAMPLE 3
Identify congruent arcs
Tell whether the red arcs are congruent. Explain why or why not. a. b.
SOLUTION a.
CD EF because they are in the same circle and mCD = mEF b.
RS and TU have the same measure, but are not congruent because they are arcs of circles that are not congruent.

12. EXAMPLE 3
Identify congruent arcs
Tell whether the red arcs are congruent. Explain why or why not. c.
SOLUTION c.
VX YZ because they are in congruent circles and mVX = mYZ .

13. GUIDED PRACTICE
for Example 3
Tell whether the red arcs are congruent. Explain why or why not. 7.
AB CD because they are in congruent circles and mAB = mCD .
ANSWER

14. GUIDED PRACTICE
for Example 3
Tell whether the red arcs are congruent. Explain why or why not. 8.
MN and PQ have the same measure, but are not congruent because they are arcs of circles that are not congruent.
ANSWER

15. Daily Homework Quiz
Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. 1. BC
ANSWER
minor arc, 32 o

16. Daily Homework Quiz
Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. 2.
CBE
ANSWER
major arc, 212 o

17. Daily Homework Quiz
Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. 3.
BCE
ANSWER
semicircle, 180 o

18. Daily Homework Quiz 4.
Explain why AE = ~ BC .
ANSWER BC AE = ~
m AFE = m BFC because the angles are vertical angles, so AFE BFC. Then arcs and are arcs that have the same measure in the same circle. By definition

19. Two diameters of P are AB and CD.If m = 50 , find m and m . . AD 5.
Daily Homework Quiz AC
Two diameters of P are AB and CD.If m = 50 , find m and m . AD o 5.
ACD
ANSWER o
310 ; 130