1. EXAMPLE 1
Use congruent chords to find an arc measure
In the diagram, P Q, FG JK , and mJK = 80o. Find mFG
SOLUTION
Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent.
So, mFG = mJK = 80o.

2. GUIDED PRACTICE
for Example 1
Use the diagram of D.
1. If mAB = 110°, find mBC
mBC = 110°
ANSWER

3. GUIDED PRACTICE
for Example 1
Use the diagram of D.
2. If mAC = 150°, find mAB
mAB = 105°
ANSWER

4. EXAMPLE 2
Use perpendicular bisectors
Three bushes are arranged in a garden as shown. Where should you place a sprinkler so that it is the same distance from each bush?
Gardening
SOLUTION
STEP 1
Label the bushes A, B, and C, as shown. Draw segments AB and BC .

5. EXAMPLE 2
Use perpendicular bisectors
STEP 2
Draw the perpendicular bisectors of AB and BC By Theorem 10.4, these are diameters of the circle containing A, B, and C.
STEP 3
Find the point where these bisectors intersect. This is the center of the circle through A, B, and C, and so it is equidistant from each point.

6. EXAMPLE 3
Use a diameter
Use the diagram of E to find the length of AC . Tell what theorem you use.
Diameter BD is perpendicular to AC . So, by Theorem 10.5, BD bisects AC , and CF = AF. Therefore, AC = 2 AF = 2(7) = 14.
ANSWER

7. GUIDED PRACTICE
for Examples 2 and 3
Find the measure of the indicated arc in the diagram.
3. CD
mCD = 72°
ANSWER

8. GUIDED PRACTICE
for Examples 2 and 3
Find the measure of the indicated arc in the diagram.
4. DE
mCD = mDE.
mDE = 72°
ANSWER
5. CE
mCE = mDE + mCD
mCE = 72° + 72° = 144°
ANSWER

9. In the diagram of C, QR = ST = 16. Find CU.
EXAMPLE 4
Use Theorem 10.6
In the diagram of C, QR = ST = 16. Find CU.
SOLUTION
Chords QR and ST are congruent, so by Theorem 10.6 they are equidistant from C. Therefore, CU = CV.
CU = CV
Use Theorem 10.6.
2x = 5x – 9
Substitute.
x = 3
Solve for x.
So, CU = 2x = 2(3) = 6.

10. GUIDED PRACTICE
for Example 4
In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length.
6. QR
QR = 32
ANSWER

11. GUIDED PRACTICE
for Example 4
In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length.
7. QU
QU = 16
ANSWER

12. GUIDED PRACTICE
for Example 4
In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length. 8.
The radius of C
ANSWER
The radius of C = 20

13. Daily Homework Quiz 1.
Find the value of x in C. Explain. .
ANSWER
6; If a diameter of a circle is to the chord, then the diameter bisects the chord and its arc.

14. Daily Homework Quiz
Find the value of x in C. Explain. . 2.
ANSWER
4; In the same circle, if two chords are equidistant from the center, then they are . = ~

15. Daily Homework Quiz 3.
Determine whether RS is a diameter.
ANSWER
Yes. Sample answer: RS is the bisector of TU by Theorem 5.3. Then RS is a diameter of the circle by Theorem 10.4.