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

2. Apply angle relationships in circles.
Target
Apply angle relationships in circles.
You will…
Use angle measures to find arc measures.

3. Vocabulary
central angle (of a circle) – an angle whose vertex is the center of the circle and whose sides intersect the circle in exactly two points
minor arc
central angle
minor arc – intercepted by a central angle less than 180º and in its interior
arc measure = central angle measure
major arc – intercepted by a central angle less than 180º and in its exterior
arc measure = 360 – related minor arc
semicircle – arc whose endpoints are endpoints of a diameter
semicircle measure = 180º
major arc

4. Vocabulary
adjacent arcs – two arcs of the same circle with a common endpoint
Arc Addition Postulate 23 –
the measure of an arc formed
by two adjacent arcs is the sum
of the measures of the two arcs

5. 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.

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 a.
mAC
SOLUTION a.
mAC
mAB = +
mBC
= 29o + 108o
= 137o

7. 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

8. 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

9. 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

10. 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. 1
. TQ
. QRT 2
. TQR 3
. QS 4
. TS 5
. RST 6
SOLUTIONS
minor arc measures: m TQ = 120o,
m QS = 160o ,
m TS = 80o.
major arc measures: m QRT= 240o .
semicircles: m TQR = 180o,
m RST = 180o.

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. EXAMPLE 3
Identify congruent arcs
Tell whether the red arcs are congruent.
Explain why or why not. 8. 7.
SOLUTIONS
AB CD because they are in congruent circles and mAB = mCD . 7.
MN and PQ have the same measure, but are not congruent because they are arcs of circles that are not congruent. 8.