1. Section 12.2 – 12.3
Chords
&
Arcs

2. If mCQD @ mBQA, then AB @ CD.
Theorem 12-4: In the same circle (or congruent circles), congruent central angles create congruent intercepted arcs. If
mBQA, D C A Q
then CD. B

3. If mCQD @ mBQA Then AB @ CD.
Theorem 12-5: In the same circle (or congruent circles), congruent central angles create congruent chords. If
mBQA D C A Q
Then CD. B

4. Arcs can be formed by figures other than central angles
Arcs can be formed by figures other than central angles. Arcs can be formed by chords, inscribed angles, and tangents. Today we will focus on examining relationships between chords and their intercepted arcs.
Chord AB creates intercepted minor arc AB and intercepted major arc ACB. A B C

5. Theorem 12-6: In the same circle (or congruent circles), congruent chords create congruent intercepted arcs.
If CD, D C
then CD. A B

6. A B C C D A B Given: mCB = 140 Given: mAC = 100 mCD = 75 mAC = mAB =
Example 1
Example 2 A B C C D A B
Given: mCB = 140
Given: mAC = 100
mCD = 75
mAC = mAB =
mACB =
mABC =
mBAC =
110
110
mAB = mBD =
mACD =
mBAD = 75
110
250
175
250
250
220

7. Theorem 12-8 – A diameter that is perpendicular to a chord, bisects the chord and its intercepted arc.:
If AB ^ CD,
then FD and DB. B C A D F
Also: AC.

8. Given: AB is a diameter of circle Q; AB = 10, LM = 8. B
Example
Given: AB is a diameter of circle Q; AB = 10, LM = 8. B
Find CA.
CA = 2 Q C M L A
If mML = 118, find mBL. Q L C
mBL = 121

9. 1. Chords equally distant from the center are congruent.
Theorem 12-7 – In the same circle circles):
1. Chords equally distant from the center are congruent.
2. Congruent chords are equally distant front the center.
1) If AB = BC, then QS. P
2) If QS, then AB = BC. B A S C
Remember: To measure distances from a point to a segment, you have to measure the perpendicular distance. R Q

10. Find the length of the radius of circle Q.
Example
FQ = QG = 9; CB = 24.
Find the length of the radius of circle Q. D C F G Q A B
= BQ2 9 B Q F
225 = BQ2 12
BQ = 15

11. Inscribed Angles & Corollaries

12. ÐABC is an inscribed angle of Circle O.
Definition: an Inscribed Angle is an angle with its vertex on the circle. A O C B
ÐABC is an inscribed angle of Circle O.

13. This inscribed angle intercepts an arc of Circle O. The intercepted
Theorem 12-11
110°
The measure of the intercepted arc of an inscribed angle is equal to twice the measure of the inscribed angle. O C
55° B
This inscribed angle intercepts an arc of Circle O. The intercepted
arc is AC.

14. ÐABC and ÐADC both intercept AC.
Corollary 1: Inscribed angles that intercept the same arc are congruent. A D C B
100°
ÐABC and ÐADC both intercept AC.
50°
50°
mÐABC = 50
mÐADC = 50

15. Corollary 2: An angle inscribed inside of a semicircle is a right angle.
110°
70°
mÐBAC = 90
35°
55°
Here’s why…

16. Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. A D C B
mÐBAC = 76
76°
92°
mÐACD = 92
88°
104°

17. Theorem 12-12 B D
The measure of an angle formed by a tangent line and a chord is half the measure of the intercepted arc. B E A C
mÐBAC = ½ mAB