Lesson 9.3 Arcs pp. 381-387

Objectives: 1. To identify and define relationships between arcs of circles, central angles, and inscribed angles. 2. To identify minor arcs, major arcs, and semicircles and express them using correct notation. 3. To prove theorems relating the measure of arcs, central angles, and chords.

Definition A central angle is an angle that is in the same plane as the circle and whose vertex is the center of the circle.

K L M LKM is a central angle.

Definition An inscribed angle is an angle with its vertex on a circle and with sides containing chords of the circle. Arc measure is the same measure as the degree measure of the central angle that intercepts the arc.

LNM is an inscribed angle. K M LNM is an inscribed angle.

B A C 60 Since mABC = 60°, then mAC = 60 also.

Definition A minor arc is an arc measuring less than 180. Minor arcs are denoted with two letters, such as AB, where A and B are the endpoints of the arc.

Definition A major arc is an arc measuring more than 180. Major arcs are denoted with three letters, such as ABC, where A and C are the endpoints and B is another point on the arc.

Definition A semicircle is an arc measuring 180°.

Postulate 9.2 Arc Addition Postulate. If B is a point on AC, then mAB + mBC = mAC.

Theorem 9.8 Major Arc Theorem. mACB = 360 - mAB.

EXAMPLE If mAB = 50, find mACB. mACB = 360 – mAB mACB = 360 – 50 mACB = 310

Definition Congruent Arcs are arcs on congruent circles that have the same measure.

If B  Y and AC  XZ, then AC  XZ Theorem 9.9 Chords on congruent circles are congruent if and only if they subtend congruent arcs. A B C X Y Z If B  Y and AC  XZ, then AC  XZ

If B  Y and AC  XZ, then AC  XZ Theorem 9.9 Chords on congruent circles are congruent if and only if they subtend congruent arcs. A B C X Y Z If B  Y and AC  XZ, then AC  XZ

Theorem 9.10 In congruent circles, chords are congruent if and only if the corresponding central angles are congruent.

Theorem 9.10 X Y Z A B C If B  Y and ABC  XYZ, then AC  XZ

Theorem 9.10 X Y Z A B C If B  Y and AC  XZ, then ABC  XYZ

Theorem 9.11 In congruent circles, minor arcs are congruent if and only if their corresponding central angles are congruent.

Theorem 9.11 X Y Z A B C If B  Y and ABC  XYZ, then AC  XZ

Theorem 9.11 X Y Z A B C If B  Y and AC  XZ, then ABC  XYZ

Theorem 9.12 In congruent circles, two minor arcs are congruent if and only if the corresponding major arcs are congruent.

Theorem 9.12 X Y Z A B C If B  Y and ABC  XYZ, then AC  XZ

Theorem 9.12 X Y Z A B C If B  Y and AC  XZ, then ABC  XYZ

Find mAB. A B C D E M 30° 45° 60°

Find mAE. A B C D E M 30° 45° 60°

Find mDC + mDE. A B C D E M 30° 45° 60°

Given circle M with diameters DB and AC, mAD = 108. Find mAMB. 1. 36 2. 54 3. 72 4. 108 A B C D M 108

Given circle M with diameters DB and AC, mAD = 108. Find mBMC. 1. 36 2. 54 3. 72 4. 108 A B C D M 108

Given circle M with diameters DB and AC, mAD = 108. Find mDAB. 1. 90 2. 180 3. 360 4. Don’t know A B C D M 108

Given circle M with diameters DB and AC, mAD = 108. Find mDC. 1. 36 2. 54 3. 72 4. 108 A B C D M 108

Homework pp. 385-387

Use the diagram for exercises 1-10. In circle O, AC is a diameter. ►A. Exercises Use the diagram for exercises 1-10. In circle O, AC is a diameter. A E G D C B F O 50 40 30 10

Use the diagram for exercises 1-10. In circle O, AC is a diameter. ►A. Exercises Use the diagram for exercises 1-10. In circle O, AC is a diameter. Find each of the following. 5. mAB A E G D C B F O 50 40 30 10 = 130

Use the diagram for exercises 1-10. In circle O, AC is a diameter. ►A. Exercises Use the diagram for exercises 1-10. In circle O, AC is a diameter. Find each of the following. 7. mBOD A E G D C B F O 50 40 30 10 = 90

Use the diagram for exercises 1-10. In circle O, AC is a diameter. ►A. Exercises Use the diagram for exercises 1-10. In circle O, AC is a diameter. Find each of the following. 9. mBC + mBA A E G D C B F O 50 40 30 10 = 180 (Post. 9.2)

►A. Exercises C D P Q A B Use the figure for exercises 11-13. 11. If AB  CD and mBPA = 80, find mCQD. mCQD = 80 (Thm. 9.10)

►A. Exercises C D P Q A B Use the figure for exercises 11-13. 13. If mBPA = 75 and mCQD = 75, what is true about AB and CD? Why?

►B. Exercises Prove the following theorems. 14. Theorem 9.8 Given: mAB + mACB = m☉P Prove: mACB = 360 - mAB C P A B

►B. Exercises Prove the following theorems. 15. Given: ☉U with XY  YZ  ZX Prove: ∆XYZ is an equilateral triangle X Y Z U

►B. Exercises Prove the following theorems. 16. Given: Points M, N, O, and P on ☉L; MO  NP Prove: MP  NO M P N L O

►B. Exercises Prove the following theorems. 17. Given: ☉O; E is the midpoint of BD and AC; BE  AE Prove: MP  NO A B D O C E

■ Cumulative Review 24. State the Triangle Inequality.

■ Cumulative Review 25. State the Exterior Angle Inequality.

■ Cumulative Review 26. State the Hinge Theorem.

■ Cumulative Review 27. State the greater than property.

■ Cumulative Review 28. Prove that the surface area of a cone is always greater than its lateral surface area.