1. EXAMPLE 1
Use inscribed angles
Find the indicated measure in P. a.
m T
mQR b.
SOLUTION 1 2
M T = mRS = (48o) = 24o a.
mTQ = 2m R = o = 100o. Because TQR is a semicircle, b.
mQR = 180o mTQ = 180o 100o = 80o. So, mQR = 80o. –

2. EXAMPLE 2
Find the measure of an intercepted arc
Find mRS and m STR. What do you notice about STR and RUS?
SOLUTION
From Theorem 10.7, you know that mRS = 2m RUS = 2 (31o) = 62o.
Also, m STR = mRS = (62o) = 31o. So, STR RUS. 1 2

3. EXAMPLE 3
Standardized Test Practice
SOLUTION
Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem Also, KJL and KML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles.
So, by Theorem 10.8, the correct answer is C.

4. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the measure of the red arc or angle. 1.
SOLUTION
m G = mHF = (90o) = 45o 1 2 a.

5. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the measure of the red arc or angle. 2.
SOLUTION
mTV = 2m U = o = 76o. b.

6. GUIDED PRACTICE
for Examples 1, 2 and 3
Find the measure of the red arc or angle. 3.
SOLUTION
Notice that ZYN and ZXN intercept the same arc, and so ZYN by Theorem Also, KJL and KML intercept the same arc, so they must also be congruent.
ZXN
ZYN
ZXN
ZXN
72°

7. EXAMPLE 5
Use Theorem 10.10
Find the value of each variable. a.
SOLUTION
PQRS is inscribed in a circle, so opposite angles are supplementary. a.
m P + m R = 180o
m Q + m S = 180o
75o + yo = 180o
80o + xo = 180o
y = 105
x = 100

8. EXAMPLE 5
Use Theorem 10.10
Find the value of each variable. b.
SOLUTION b.
JKLM is inscribed in a circle, so opposite angles are supplementary.
m J + m L = 180o
m K + m M = 180o
2ao + 2ao = 180o
4bo + 2bo = 180o
4a = 180
6b = 180
b = 30
a = 45

9. GUIDED PRACTICE
for Example 5
Find the value of each variable. 5.
SOLUTION
y = 112
x = 98

10. GUIDED PRACTICE
for Example 5
Find the value of each variable. 6.
SOLUTION
c = 62
x = 10

11. Daily Homework Quiz
Find the value of x . 1.
ANSWER 38

12. Daily Homework Quiz
Find the value of x . 2.
ANSWER 56

13. Daily Homework Quiz
Find the value of x . 3.
ANSWER 44

14. Daily Homework Quiz
Find the value of each variable. 4.
ANSWER
x = 54; y = 20

15. Daily Homework Quiz
Find the value of each variable. 5.
ANSWER
x = 5; y = 10