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1
Trigonometric
Functions
Copyright © 2009 Pearson Addison-Wesley
1.2-1
1 Trigonometric Functions
1.1 Angles
1.2 Angle Relationships and Similar
Triangles
1.3 Trigonometric Functions
1.4 Using the Definitions of the
Trigonometric Functions
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1.2 Angle Relationships and
Similar Triangles
Geometric Properties ▪ Triangles
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Vertical Angles
Vertical angles have equal measures.
The pair of angles NMP and
RMQ are vertical angles.
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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.
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Angles and Relationships
Name
Angles
Rule
Alternate interior angles
4 and 5
3 and 6
Angles measures are equal.
Alternate exterior angles
1 and 8
2 and 7
Angle measures are equal.
Interior angles on the same
side of the transversal
4 and 6
3 and 5
Angle measures add to 180.
Corresponding angles
2 & 6, 1 & 5,
3 & 7, 4 & 8
Angle measures are equal.
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Example 1
FINDING ANGLE MEASURES
Find the measure of angles 1, 2, 3, and 4, given
that lines m and n are parallel.
Angles 1 and 4 are
alternate exterior angles,
so they are equal.
Subtract 3x.
Add 40.
Divide by 2.
Angle 1 has measure
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Substitute 21
for x.
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Example 1
FINDING ANGLE MEASURES (continued)
Angle 4 has measure
Substitute 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 sum of the
angles is 180°.
Add.
Subtract 109°.
The third angle of the triangle measures 71°.
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Types of Triangles: Angles
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Types of Triangles: Sides
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Conditions for Similar
Triangles
For triangle ABC to be similar to triangle
DEF, the following conditions must hold.
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.)
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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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FINDING THE HEIGHT OF A FLAGPOLE
(continued)
Example 5
Lowest terms
The flagpole is 55 feet high.
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