LEARNING TARGET HOMEWORK

Geometry • I can examine the relationship between central and inscribed angles by applying theorems about their measure.• I can solve the unknown measure of arcs and angles in a circle.

Homework on page 607, #12 to #30 (Even Numbers Only)

Arcs and Chords

REVIEWIdentify the following parts of the circle.

1. DC

2. AB

3. AC

4. line E

5. DC

D

C A B

E

• chord

• radius

• diameter

• tangent

• secant

radius diameter chord secant tangent

Note: The following are possible answers.

midpoint

CirclesChapter 10

Sections 10.1 –10.3

Types of Angles

•Central angle

- the vertex is on the center.

• Inscribed angle

- the vertex is on the circle.

Types of Arcs

P

M

O

• Major arc

• Minor arc

• Semicircle

N

Example: MO

Example: MNO

or MN Example: MON

- the measure is more than 180 °

- the measure is less than 180 °

- the measure is equal to 180 °

Solving Unknown Arcs and Angles

• On the next slides… • you will use a white board (or a filler with

printing paper), and a marker to solve and answer the given problems.

• you will be given 30 seconds to solve each of the math problems.

• at the end of each problem , you will raise your white board with your answer on it. Make sure you box your answer.

Have fun!

Measure of Arcs & Angles

• If ∠ ABC is 80°, what is the measure of arc AC?

In a circle, the measure of the central angle is always equal to the measure of its intercepted arc.

x°A

B °

C

n° x = n

m ∠ ABC = m AC

m AC = 80°

Measure of Arcs & Angles

SOLUTION:

a. measure of minor arc

m xyz∠ = m xz (since xyz is a central angle)∠

b. measure of major arc

major arc = 360° – m xz (minor arc)

68°

68°

=360° – 68°

m xz (major arc) = 292°

292°

m xz = 68°

EXAMPLE: In the diagram below, if the m ∠ xyz is 68°, find the measure of a.) minor arc and b.) major arc.

x

y

z

Measure of Arcs & Angles

n°

x°

A

B

C

The measure of inscribed angle is always equal to ½ the measure of its intercepted arc.

x = ½ n or 2x = n m ABC = ½ (m AC)∠

• If ∠ ABC is 35°, what is the measure of arc AC?

m AC = 70°

Measure of Arcs & Angles

SOLUTION:

Inscribed angle = ½ (intercepted arc)

∠ ABC = ½ (68°)

∠ ABC= 34°

68°

34°

A

B

C

EXAMPLE: If the measure of the minor arc below is 68°, find the measure of the inscribe angle, ABC.∠

x

24 °

this is the arc BC

D

Centre of Circle

SOLUTION:

Angle x is a central angle. Therefore, x = arc BC.∠

Arc BC is an intercepted arc of inscribe angle ABC.

Since inscribed angle = ½(intercepted arc),

therefore, the intercepted arc is twice the inscribe angle. n = 2x

∠ x = 2 (24)

∠ x = 48°

PROBLEM: If angle BAC is 24°, solve for x

A

B C

Examine the diagram and solve

105°

x

O

Arc PN

Centre of Circle

SOLVE: If m PON is 105∠ °, what is the measure of (a) arc PN?

(b) m PMN? ∠

M

NP

Examine the diagram and solve

SOLUTION:

Angle PON is a central angle.

Therefore, PON = arc PMN.∠

arc PMN = 105°

a. Arc PN = ?

= 360° – 105°

Arc PN = 255°

b. PMN = ? ∠

∠ PMN an inscribe angle and arc PN is its intercepted arc.

Since inscribed angle = ½(intercepted arc),

therefore,

∠ PMN = ½ (255°)

∠ PMN = 127.5°

Remember…

In a circle, the measure of the central angle is always equal to the measure of its intercepted arc.

x = n

The measure of inscribed angle is always equal to ½ the measure of its intercepted arc. x = ½ n or 2x = n

EXIT SLIP

Use the diagram below to answer the following question:

1. Find m BC

2. Find the m BDC

3. Find m BAC∠A

B

C

D50°