SECTION 14-1

• Angles and Their Measures

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ANGLES AND THEIR MEASURES• Basic Terminology• Degree Measure• Angles in a Coordinate System

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BASIC TERMINOLOGY

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A line may be drawn through the two distinct points A and B. This line is called line AB. The portion of the line between A and B, including points A and B themselves, is segment AB. The portion of the line AB that starts at A and continues through B, and on past B, is called ray AB. Point A is the endpoint of the ray.

A B A B A B

Line AB Segment AB Ray AB

BASIC TERMINOLOGY

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Initial sideVertex A

Terminal side

An angle is formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle, while the ray in its location after rotation is the terminal side of the angle. The endpoint of the ray is the vertex of the angle.

BASIC TERMINOLOGY

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Positive angleNegative angle

If the rotation of the terminal side is counterclockwise, the angle measure is positive. If the rotation of the terminal side is clockwise, the angle measure is negative.

BASIC TERMINOLOGY

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B

A

An angle can be named by using the name of its vertex. Alternatively, an angle can be named using three letters, with the vertex in the middle.

C

Name: angle C, angle ACB, or angle BCA.

DEGREE MEASURE

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The most common unit of measure for angles is the degree. (The other common unit of measure is called the radian.)We assign 360 degrees to a complete rotation of a ray.

360°

DEGREE MEASURE

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90°

1° angle

180°

One degree, written 1°, represents 1/360 of a rotation. Therefore, 90° represents 1/4 of a rotation, and 180° represents 1/2 of a rotation

SPECIAL ANGLES

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Name Angle Measure Example

Acute Angle Between 0° and 90°

Right Angle Exactly 90°

Obtuse Angle Between 90° and 180°

Straight Angle Exactly 180°

90°

135°

180°

70°

COMPLEMENT AND SUPPLEMENT

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If the sum of the measures of two angles is 90°, the angles are called complementary. Two angles with measures whose sum is 180° are supplementary.

EXAMPLE: FINDING COMPLEMENT AND SUPPLEMENT

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Give the complement and supplement of 60°.

SolutionThe complement of 60° is 90° – 60° = 30°.

The supplement of 60° is 180° – 60° = 120°.

ANGLE MEASUREMENT

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Portions of a degree have been measured with minutes and seconds.

One minute, written

One second, written

11 , is of a degree.60

11 , is of a minute.60

11 or 60 160

1 11 = or 60 160 3600

ANGLE MEASUREMENT

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The measure represents 42 degrees, 13 minutes, 24 seconds.

42 13 24

Angles can be measured in decimal degrees. For example 12.345° represents

34512.345 12 .1000

EXAMPLE: CALCULATING WITH DEGREE MEASURE

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Perform each calculation.a) 42 23 61 54 b) 32 7 25

Solutiona) 42 23 61 54 103 77

104 17

b) 32 7 25 31 60 7 25 24 35

Use 60 1

Use 32 31 60

EXAMPLE: CONVERTING DEGREES, MINUTES, AND SECONDS TO DECIMAL DEGREES

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Convert to decimal degrees. Round to the nearest thousandth of a degree.

42 13 54

Solution13 5442 13 54 4260 3600

42 .2167 .015

42.232

EXAMPLE: CONVERTING DEGREES, MINUTES, AND SECONDS TO DECIMAL DEGREES

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Convert to degrees, minutes and seconds. Round to the nearest second.

50 31 .86(60 ) 50 31 51.6

Solution50.531 50 (.531)(60 )

50 31.86

50.531

50 31 52

ANGLES IN A COORDINATE SYSTEM

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0

An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis.

Initial side

Terminal side

Vertex0

ANGLES IN A COORDINATE SYSTEM

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An angle in standard position is said to lie in the quadrant in which its terminal side lies. Angles in standard position having their terminal side along the x-axis or y-axis (90°, 180°, 270°, …) are called quadrantal angles.

0 90 Q IQ II

Q III Q IV

90 180

180 270 270 360

0°360°

90°

180°

270°

COTERMINAL ANGLES

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0 50°

410°

Two angles can have the same initial side and same terminal side but different amounts of rotation. Angles that have the same initial side and same terminal side are called coterminal angles.

The angles 50° and 410° shown are coterminal angles.

EXAMPLE: FINDING MEASURES OF COTERMINAL ANGLES

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Find the angle of smallest possible positive measure coterminal with each angle

a) 770° b) –88°

Solutiona) 770° – 2(360°) = 50°

b) –88° + 360° = 272°

GENERATING COTERMINAL ANGLES

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Sometimes it may be necessary to find an expression that will generate all angles coterminal with a given angle. Coterminal angles can be represented by adding integer multiples of 360° to the angle. For example, for an angle measure of 50°, we let n represent any integer and then the expression

50° + n(360°)

will represent all the coterminal angles.