Copyright © 2009 Pearson Addison-Wesley 1.2-1 1 Trigonometric Functions.

Copyright © 2009 Pearson Addison-Wesley 1.2-1

1Trigonometric Functions


1.1 Angles

1.2 Angle Relationships and Similar Triangles

1.3 Trigonometric Functions

1.4 Using the Definitions of the Trigonometric Functions

1 Trigonometric Functions

Copyright © 2009 Pearson Addison-Wesley

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Angle Relationships and Similar Triangles

1.2

Geometric Properties ▪ Triangles


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Vertical Angles

The pair of angles NMP and RMQ are vertical angles.

Vertical angles have equal measures.


Parallel Lines

Parallel lines are lines that lie in the same plane and do not intersect.

When a line q intersects two parallel lines, q, is called a transversal.


Angles and Relationships

Angle measures are equal.2 & 6, 1 & 5, 3 & 7, 4 & 8

Corresponding angles

Angle measures add to 180.4 and 6

3 and 5

Interior angles on the same side of the transversal

Angle measures are equal.1 and 8

2 and 7

Alternate exterior angles

Angles measures are equal.4 and 5

3 and 6

Alternate interior angles

RuleAnglesName


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Find the measure of angles 1, 2, 3, and 4, given that lines m and n are parallel.

Example 1 FINDING ANGLE MEASURES

Angles 1 and 4 are alternate exterior angles, so they are equal.

Subtract 3x.Add 40.Divide by 2.

Angle 1 has measureSubstitute 21 for x.


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Example 1 FINDING ANGLE MEASURES (continued)

Angle 4 has measureSubstitute 21 for x.

Angle 2 is the supplement of a 65° angle, so it has measure .

Angle 3 is a vertical angle to angle 1, so its measure is 65°.


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Angle Sum of a Triangle

The sum of the measures of the angles of any triangle is 180°.


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Example 2 APPLYING THE ANGLE SUM OF A TRIANGLE PROPERTY

The measures of two of the angles of a triangle are 48 and 61. Find the measure of the third angle, x.

The third angle of the triangle measures 71°.

The sum of the angles is 180°.

Add.

Subtract 109°.


Types of Triangles: Angles


Types of Triangles: Sides


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1.Corresponding angles must have the same measure.

2.Corresponding sides must be proportional. (That is, the ratios of the corresponding sides must be equal.)

Conditions for Similar TrianglesFor triangle ABC to be similar to triangle DEF, the following conditions must hold.


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Example 3 FINDING ANGLE MEASURES IN SIMILAR TRIANGLES

In the figure, triangles ABC and NMP are similar.Find the measures of angles B and C.

Since the triangles are similar, corresponding angles have the same measure.

B corresponds to M, so angle B measures 31°.

C corresponds to P, so angle C measures 104°.


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Example 4 FINDING SIDE LENGTHS IN SIMILAR TRIANGLES

In the figure, triangles ABC and NMP are similar.Find the measures of angles B and C.

Since the triangles are similar, corresponding sides are proportional.

DF corresponds to AB, and DE corresponds to AC, so


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Example 4 FINDING SIDE LENGTHS IN SIMILAR TRIANGLES (continued)

Side DF has length 12.

EF corresponds to CB, so

Side EF has length 16.


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Example 5 FINDING THE HEIGHT OF A FLAGPOLE

Firefighters at a station need to measure the height of the station flagpole. They find that at the instant when the shadow of the station is 18 m long, the shadow of the flagpole is 99 ft long. The station is 10 m high. Find the height of the flagpole.

Since the two triangles are similar, corresponding sides are proportional.


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Example 5 FINDING THE HEIGHT OF A FLAGPOLE (continued)

Lowest terms

The flagpole is 55 feet high.

Copyright © 2009 Pearson Addison-Wesley 1.2-1 1 Trigonometric Functions.

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