Lesson 10.2

Arcs and Chords

Arcs of Circles

• Central Angle-angle whose vertex is the center of the circle.

P

A

BC

central angle

Minor Arc

• formed from a central angle less than 180°

P

A

BC

minor arc

Major Arc

• formed from a central angle that measures between 180 ° - 360 °

P

A

BC

major arc

Semicircle

• formed from an arc of 180 °

• Half circle!

• Endpoints of an arc are endpoints of the diameter

Naming Arcs

• How do we name minor arcs, major arcs, and semicircles??

Minor Arc

• Named by the endpoints of the arc.

P

A

BC

Minor Arc: AB or BA

Major Arc

• Named by the endpoints of the arc and one point in between the arc

P

A

BC

Major Arc: ACB or BCA

Could we name this major arc BAC?

Semicircle

• Named by the endpoints of the diameter and one point in between the arc

CA

B

mABC = 180°

Example

Measuring Arcs

• A Circle measures 360 °

Measure of a Minor Arc

• Measure of its central angle

P

A

BC

95°m AB=95 °

Measure of a Major Arc

• difference between 360° and measure of minor arc

P

A

BC

95°mACB=360°– 95° = 265°

Arc Addition Postulate

• Measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

55

45A C

B

D

What is the measure of BD?

m BD=100 °

Example for #1-10

Congruent Arcs

• Two arcs of the same circle or congruent circles are congruent arcs if they have the same measure.

60

60

C

A

D

B

AB is congruent to DC since their arc measures are the same.

Theorem 10.4

• Two minor arcs are congruent iff their corresponding chords are congruent.

P

A

B

C

Chords are congruent

Example 1Solve for x

E

F

HG

2x X+40

Theorem 10.5

• If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

E

G

D

F

If DE = EF, then DG = GF

Example. Find DC.

40

A ED

B

C

m DC = 40º

Theorem 10.6

• If one chord is a perpendicular bisector to another chord, then the first chord is a diameter.

C

D

A

B

Since AB is perpendicular to CD,

CD is the diameter.

Example. Solve for x.

x

7 x = 7

Theorem 10.7

• Two chords are congruent iff they are equidistant from the center.

Congruent Chords

Example. Solve for x.

15

x x = 15