10.2 angles and arcs

10.2 Angles and Arcs10.2 Angles and Arcs

ObjectivesObjectives Recognize major arcs, minor arcs, Recognize major arcs, minor arcs,

semicircles, and central angles semicircles, and central angles and their measuresand their measures

Find arc lengthFind arc length

Angles and ArcsAngles and Arcs

Sum of Central s and Arcs = 360°

Angles and ArcsAngles and Arcs The sum of the measures of the central The sum of the measures of the central

angles is 360angles is 360°.°. A minor arc is less then 180° and is labeled A minor arc is less then 180° and is labeled

using the two endpoints.using the two endpoints. A major arc is greater than 180° but less A major arc is greater than 180° but less

than 360° and is labeled using the two than 360° and is labeled using the two endpoints and another point on the arc. endpoints and another point on the arc.

A semicircle measures 180° and is labeled A semicircle measures 180° and is labeled using the two endpoints and another point using the two endpoints and another point on the arc. on the arc.

Angles and ArcsAngles and Arcs Theorem 10.2:Theorem 10.2: In the same circle In the same circle

or or circles, two arcs are circles, two arcs are if their if their corresponding central angles are corresponding central angles are ..

Postulate 10.1:Postulate 10.1: The measure of an The measure of an arc formed by two adjacent arcs is arc formed by two adjacent arcs is the sum of the measures of the the sum of the measures of the arcs. arcs.

ALGEBRA Refer to .Find .

Example 1a:Example 1a:

SubstitutionSimplify.Add 2 to each side.Divide each side by 26.

Use the value of x to findGivenSubstitution

Answer: 52

The sum of the measures of

Example 1a:Example 1a:

ALGEBRA Refer to .Find .

Example 1b:Example 1b:

Linear pairs are supplementary.

Substitution

Simplify.

Subtract 140 from each side.

form a linear pair.

Answer: 40

Example 1b:Example 1b:

Answer: 65

Answer: 40

ALGEBRA Refer to .

a. Find m

b. Find m

Your Turn:Your Turn:

Find .

In bisects and

Example 2a:Example 2a:

is a minor arc, so

is a semicircle.

is a right angle.

Arc Addition PostulateSubstitutionSubtract 90 from each side.

Answer: 90

Example 2a:Example 2a:

Find .

In bisects and

Example 2b:Example 2b:

since bisects .

is a semicircle.

Arc Addition Postulate

Subtract 46 from each side.

Answer: 67

Example 2b:Example 2b:

Find .

In bisects and

Example 2c:Example 2c:

Vertical angles are congruent.Substitution.

Substitution.Subtract 46 from each side.

Subtract 44 from each side.Substitution.

Answer: 316

Example 2c:Example 2c:

Answer: 54

Answer: 72

In and are diameters, and bisects Find each measure.

a.

b.

c. Answer: 234

Your Turn:Your Turn:

BICYCLES This graph shows the percent of each type of bicycle sold in the United States in 2001.Find the measurement of the central angle representing each category. List them from least to greatest.

Example 3a:Example 3a:

The sum of the percents is 100% and represents the whole. Use the percents to determine what part of the whole circle each central angle contains.

Answer:

Example 3a:Example 3a:

BICYCLES This graph shows the percent of each type of bicycle sold in the United States in 2001.Is the arc for the wedge named Youth congruent to the arc for the combined wedges named Other and Comfort?

Example 3b:Example 3b:

Answer: no

The arc for the wedge named Youth represents 26% or of the circle. The combined wedges named Other and Comfort represent

. Since º, the arcs are not congruent.

Example 3b:Example 3b:

SPEED LIMITS This graph shows the percent of U.S. states that have each speed limit on their interstate highways.

Your Turn:Your Turn:

Answer: no

b. Is the arc for the wedge for 65 mph congruent to the combined arcs for the wedges for 55 mph and 70 mph?

a. Find the measurement of the central angles representing each category. List them from least to greatest.

Answer:

Your Turn:Your Turn:

Arc LengthArc Length Another way to measure an arc is by its Another way to measure an arc is by its

length. An arc is part of a circle, so its length. An arc is part of a circle, so its length is part of the circumference. We length is part of the circumference. We use proportions to solve for the arc use proportions to solve for the arc length, length, ll..

degree measure of arcdegree measure of arc = = arc lengtharc length

degree measure of circumferencedegree measure of circumference

In and . Find the length of .

In and . Write a proportion to compare each part to its whole.

Example 4:Example 4:

Now solve the proportion for .

Simplify.

Answer: The length of is units or about 3.14 units.

degree measure of arcdegree measure of

whole circle

arc lengthcircumference

Multiply each side by 9 .

Example 4:Example 4:

In and . Find the length of .

Answer: units or about 49.48 units

Your Turn:Your Turn:

AssignmentAssignment GeometryGeometry

Pg. 533 #14 – 37, 40, 47 - 50 Pg. 533 #14 – 37, 40, 47 - 50

Pre-AP GeometryPre-AP GeometryPg. 533 #14 – 43, 47 - 52Pg. 533 #14 – 43, 47 - 52