Geometry unit 8.4

UUNNIITT 88..44 AANNGGLLEESS OOFF EELLEEVVAATTIIOONN
Holt Geometry
AANNDD DDEEPPRREESSSSIIOONN

Warm Up
1. Identify the pairs of alternate
interior angles.
Ð2 and Ð7; Ð3 and Ð6
2. Use your calculator to find tan 30° to the
nearest hundredth.
0.58
3. Solve . Round to the nearest
hundredth.
1816.36

Objective
Solve problems involving angles of
elevation and angles of depression.

Vocabulary
angle of elevation
angle of depression

An angle of elevation is the angle formed by a
horizontal line and a line of sight to a point above
the line. In the diagram, Ð1 is the angle of elevation
from the tower T to the plane P.
An angle of depression is the angle formed by a
horizontal line and a line of sight to a point below
the line. Ð2 is the angle of depression from the
plane to the tower.

Since horizontal lines are parallel, Ð1 @ Ð2 by the
Alternate Interior Angles Theorem. Therefore the
angle of elevation from one point is congruent
to the angle of depression from the other point.

Example 1A: Classifying Angles of Elevation and
Depression
Classify each angle as an
angle of elevation or an
angle of depression.
Ð1
Ð1 is formed by a horizontal line and a line of
sight to a point below the line. It is an angle of
depression.

Example 1B: Classifying Angles of Elevation and
Depression
Classify each angle as an
angle of elevation or an
angle of depression.
Ð4
Ð4 is formed by a horizontal line and a line of sight
to a point above the line. It is an angle of elevation.

Check It Out! Example 1
Use the diagram above to
classify each angle as an angle
of elevation or angle of
depression.
1a. Ð5
Ð5 is formed by a horizontal line and a line of
sight to a point below the line. It is an angle of
depression.
1b. Ð6
Ð6 is formed by a horizontal line and a line of sight
to a point above the line. It is an angle of elevation.

Example 2: Finding Distance by Using Angle of
Elevation
The Seattle Space Needle casts a 67-
meter shadow. If the angle of
elevation from the tip of the shadow
to the top of the Space Needle is
70º, how tall is the Space Needle?
Round to the nearest meter.
Draw a sketch to represent the
given information. Let A
represent the tip of the shadow,
and let B represent the top of
the Space Needle. Let y be the
height of the Space Needle.

Example 2 Continued
You are given the side adjacent to
ÐA, and y is the side opposite ÐA.
So write a tangent ratio.
y = 67 tan 70° Multiply both sides by 67.
y » 184 m Simplify the expression.

What if…? Suppose the plane is at an altitude of
3500 ft and the angle of elevation from the airport to
the plane is 29°. What is the horizontal distance
between the plane and the airport? Round to the
nearest foot.
You are given the side opposite
ÐA, and x is the side adjacent to
ÐA. So write a tangent ratio.
Multiply both sides by x and
divide by tan 29°.
3500 ft
x » 6314 ft Simplify the expression.
29°

Example 3: Finding Distance by Using Angle of
Depression
An ice climber stands at the edge of a
crevasse that is 115 ft wide. The angle of
depression from the edge where she stands to
the bottom of the opposite side is 52º. How
deep is the crevasse at this point? Round to
the nearest foot.

Example 3 Continued
Draw a sketch to represent
the given information. Let C
represent the ice climber and
let B represent the bottom of
the opposite side of the
crevasse. Let y be the depth
of the crevasse.

Example 3 Continued
By the Alternate Interior Angles Theorem, mÐB = 52°.
Write a tangent ratio.
y = 115 tan 52° Multiply both sides by 115.
y » 147 ft Simplify the expression.

What if…? Suppose the ranger sees another fire
and the angle of depression to the fire is 3°. What
is the horizontal distance to this fire? Round to the
nearest foot.
3°
By the Alternate Interior Angles Theorem, mÐF = 3°.
Write a tangent ratio.
Multiply both sides by x and
divide by tan 3°.
x » 1717 ft Simplify the expression.

Example 4: Shipping Application
An observer in a lighthouse is 69 ft above the
water. He sights two boats in the water directly
in front of him. The angle of depression to the
nearest boat is 48º. The angle of depression to
the other boat is 22º. What is the distance
between the two boats? Round to the nearest
foot.

Example 4 Application
Step 1 Draw a sketch.
Let L represent the
observer in the
lighthouse and let A
and B represent the
two boats. Let x be the
distance between the
two boats.

Example 4 Continued
Step 2 Find y.
By the Alternate Interior Angles Theorem,
mÐCAL = 58°.
.
In ΔALC,
So

Example 4 Continued
Step 3 Find z.
mÐCBL = 22°.
In ΔBLC,
So

Example 4 Continued
Step 4 Find x.
x = z – y
x » 170.8 – 62.1 » 109 ft
So the two boats are about 109 ft apart.

A pilot flying at an altitude of 12,000 ft sights
two airports directly in front of him. The angle
of depression to one airport is 78°, and the
angle of depression to the second airport is
19°. What is the distance between the two
airports? Round to the nearest foot.

Check It Out! Example 4 Continued
Step 1 Draw a sketch. Let
P represent the pilot and
let A and B represent the
two airports. Let x be the
distance between the two
airports.
78°
19°
78° 19°
12,000 ft

Step 2 Find y.
mÐCAP = 78°.
In ΔAPC,
So

Step 3 Find z.
mÐCBP = 19°.
In ΔBPC,
So

Step 4 Find x.
x = z – y
x » 34,851 – 2551 » 32,300 ft
So the two airports are about 32,300 ft apart.

Lesson Quiz: Part I
Classify each angle as an angle of elevation
or angle of depression.
1. Ð6
2. Ð9
angle of depression
angle of elevation

Lesson Quiz: Part II
3. A plane is flying at an altitude of 14,500 ft. The
angle of depression from the plane to a control
tower is 15°. What is the horizontal distance
from the plane to the tower? Round to the
nearest foot.
54,115 ft
4. A woman is standing 12 ft from a sculpture.
The angle of elevation from her eye to the top
of the sculpture is 30°, and the angle of
depression to its base is 22°. How tall is the
sculpture to the nearest foot?
12 ft

Geometry unit 8.4

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Geometry unit 8.4