Download Lesson F5.3

LESSON F5.3 – GEOMETRY III
LESSON F5.3 GEOMETRY III
473
474
TOPIC F5 GEOMETRY
Overview
You have learned how to identify different types of polygons, including triangles, and how
to find their perimeter and their area.
In this lesson, you will learn more about triangles. You will study the properties of
triangles in which two or three lengths or two or three angles have the same measure. You
will then work with similar triangles, triangles with the same shape but different size. Also,
you will study right triangles and the Pythagorean Theorem. Finally, you will learn more
about parallelograms and parallel lines.
Before you begin, you may find it helpful to review the following mathematical ideas
which will be used in this lesson:
To see these Review
problems worked out, go
to the Overview module
of this lesson on the
computer.
Review 1
Use supplementary angles.
∠ABC and ∠DEF are supplementary angles. m∠ABC = 144˚. Find m∠DEF.
Answer: 36°
Review 2
Use vertical angles.
∠XZV and ∠UZV are vertical angles. m∠XZV = 44.6˚. Find m∠UZV.
Answer: 44.6°
Review 3
Square a whole number.
Find the value of this expression: 7 2 + 8 2.
Answer: 113
Review 4
Find the square root of a whole number.
2+
Find the value of this expression: 6
82.
Answer: 10
Review 5
Work with a proportion.
Let’s solve this problem using a proportion.
Mary finishes 3 homework problems in 5 minutes. Working at the same rate, find x,
the number of problems she will finish in 30 minutes. Here’s a proportion we can use
x
3
to find x: = . What is the value of x?
5
30
Answer: 18
LESSON F5.3 GEOMETRY III OVERVIEW
475
Explain
In Concept 1: Triangles and
Parallelograms, you will learn
about:
CONCEPT 1: TRIANGLES AND
PARALLELOGRAMS
• The Sum of the Angle
Measures of a Triangle
The Sum of the Angle Measures of a Triangle
• Congruent Triangles
• Isosceles Triangles and
Equilateral Triangles
• Right Triangles and the
Pythagorean Theorem
In solving problems about angles in triangles, it is useful to know the total number of
degrees in the angles of a triangle.
The angle measures of a triangle add to 180 degrees.
An example is shown in Figure 1.
65ß
• Parallel Lines and
Parallelograms
40ß
75ß
Figure 1
40˚ + 65˚ + 75˚ = 180˚
You may find these
Examples useful while
doing the homework
for this section.
Example 1
1.
In Figure 2, find the missing angle measure in ∆ABC.
A
?
We can name a triangle using its 3 vertices
(corners).
We can list the vertices in any order.
“∆ABC” means “the triangle with vertices
A, B, and C.”
“∠A” means “the angle with vertex A.”
70ß
60ß
B
C
Figure 2
Here’s one way to find the missing angle measure in this triangle:
• Add the given angle measures.
• Subtract that result from 180˚.
So, the measure of ∠A is 50˚.
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TOPIC F5 GEOMETRY
60˚ + 70˚ = 130˚
180˚ – 130˚ = 50°
2.
In Figure 3, find the missing angle measure in ∆DEF.
Example 2
E
1
80 2 ˚
?
F
30˚
D
Figure 3
Here’s one way to find the missing angle measure in this triangle:
1
2
1
2
• Add the given angle measures.
80 ˚ + 30˚ = 110 ˚
• Subtract that result from 180˚.
180˚ – 110 ˚ = 69 ˚
1
2
1
2
1
2
So, the measure of ∠F is 69 ˚.
3.
In Figure 4, the measures of two angles are shown. Find the measure of ∠S.
Example 3
T
30˚
S
70˚
?
A
Figure 4
Here’s one way to find the measure of ∠S.
The measures of supplementary angles add to 180˚.
Substitute 30˚ for m∠ T and 110˚ for m∠A.
Subtract 30˚ and 110˚ from both sides.
Solve for m∠S.
m∠S + m∠ T + m∠A = 180˚
m∠S + 30˚ + 110˚ = 180˚
m∠S = 180˚ – 30˚ – 110˚
m∠S = 40˚
So, m∠S is 40˚.
LESSON F5.3 GEOMETRY III EXPLAIN
477
“≅” means “is congruent to.”
Congruent Triangles
“AB” means “the line segment connecting
vertices A and B.”
Figures with the same shape and size are called congruent figures. See Figure 5.
A
36ß
2 cm
C
J
B
2 cm
D
K
36ß
∠J ∠K
AB CD
Figure 5
For two congruent triangles, we can match all 3 sides and all 3 angles.
In Figure 6, matching marks show the corresponding sides and angles of the two
congruent triangles.
L
˘LMN ˘PQR
P
Q
M
R
N
Figure 6
The order of letters in the congruence relation matches the corresponding angles or vertices
of each triangle. Since ∆LMN ∆PQR,
∠L corresponds to ∠P.
∠M corresponds to ∠Q.
∠N corresponds to ∠R.
You may find these
Examples useful while
doing the homework
for this section.
Example 4
4.
In Figure 7, ∆RUN is congruent to ∆FST. Which angle of ∆FST is congruent
to ∠N?
Figure 6
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TOPIC F5 GEOMETRY
One way is to use the order of the vertices in the given congruence.
N is the third letter of ∆RUN.
T is the third letter of ∆FST.
So, ∠N is congruent to ∠ T.
5.
In Figure 7, ∆RUN ∆FST. Which side of ∆FST is congruent to UN?
Example 5
N
R
T
U
F
S
˘RUN ˘FST
Figure 7
One way is to use the order of the vertices in the given congruence.
UN is congruent to ST.
∆RUN ∆FST
Here are the other pairs of congruent sides:
RU is congruent to FS.
RN is congruent to FT.
6.
∆RUN ∆FST
∆RUN ∆FST
The triangles in Figure 8 are congruent. Write a congruence relation for these
triangles.
J
E
Example 6
S
T
K
I
Figure 8
Here’s one way to write a congruence relation:
• First, write one pair of corresponding vertices.
• Then, write another pair of corresponding vertices.
• Finally, write the last pair of corresponding vertices.
∆J
∆S
∆JE
∆SK
∆JET ∆SKI
So, one congruence relation for these triangles is ∆JET ∆SKI.
Any congruence relation that matches corresponding vertices is correct. Here are 3
more examples:
∆ETJ ∆KIS
∆JTE ∆SIK
∆TEJ ∆IKS
LESSON F5.3 GEOMETRY III EXPLAIN
479
Example 7
7.
The triangles in Figure 9 are congruent. Find the measure of ∠B.
F
C
50˚
N
? B
30˚
∆FAN ∆CLB
A
L
Figure 9
The congruence relation states that ∠B corresponds to ∠N.
So, ∠B and ∠N have the same measure.
Here’s one way to find the measure of ∠N.
Add 50˚ and 30˚.
Subtract that result from 180˚.
50˚ + 30˚ = 80˚
180˚ – 80˚ = 100˚
So, ∠B and ∠N each have measure 100˚.
Example 8
8.
The triangles in Figure 10 are congruent. Find the length of RK.
K
N
3 ft
5 ft
R
I
4 ft
S
F
∆SRK ∆FIN
Figure 10
The congruence relation states that RK corresponds to IN.
So, RK and IN have the same length.
The length of IN is 3 ft.
So, the length of RK is also 3 ft.
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Isosceles Triangles and Equilateral Triangles
There are several special types of triangles. An Isosceles triangle is one such type.
An isosceles triangle has at least two sides that have the same length. In Figure 11, each
triangle is an isosceles triangle.
The word isosceles comes from two Greek
words:
ISOS
+
“equal”
82 cm
“leg”
So isosceles means having two or three
“equal legs.”
60˚
100 cm
76˚
SKELOS
100 cm
82 cm
52˚
60˚
52˚
60˚
100 cm
100 cm
Isosceles Triangles
Figure 11
In an isosceles triangle, the angles that are opposite the equal-length sides have the same
measure.
If a triangle has two angles with equal measures, then the opposite sides also have equal
lengths, and the triangle is an isosceles triangle.
Equilateral Triangles
An Equilateral triangle is another special type of triangle.
An equilateral triangle has three sides of equal length. In Figure 12, each triangle is an
equilateral triangle.
EQUI
60°
60°
100 cm
100 cm
60°
60°
120 cm
In the word equilateral, the prefix “equi”
means “equal” and “lateral” means
“side.”
+
“equal”
120 cm
LATERAL
“side”
So equilateral means having three “equal
sides.”
60°
60°
100 cm
120 cm
Equilateral Triangles
Figure 12
In an equilateral triangle, all three angle measures are equal. Since the angle measures in a
triangle add to 180˚, each angle in an equilateral triangle measures 180˚ ÷ 3. So, the
measure of each angle is 60˚.
If each angle of a triangle measures 60˚, then the triangle is an equilateral triangle.
LESSON F5.3 GEOMETRY III EXPLAIN
481
You may find these
Examples useful while
doing the homework
for this section.
Example
9
9.
In Figure 13, ∆PAT is an isosceles triangle. Which pair of angles must have the
same measure?
P
4 cm
2 cm
A
4 cm
T
Figure 13
In ∆PAT, PA is the same length as AT.
So the angles opposite these sides have the same measure.
That is, m∠T = m∠P.
Example 10
10.
In Figure 14, ∆TOP is an isosceles triangle. Find the measure of ∠P.
O
T
50˚
?
P
Figure 14
In an isosceles triangle, the angles across from the equal-length sides have equal
measure. The matching marks on TO and OP indicate that these sides have equal
lengths.
So, ∠T and ∠P have the same measure, 50˚.
Example 11
11.
Find the length of SN in ∆SUN, as shown in Figure 15.
N
70˚
S
3 cm
70˚
U
Figure 15
Since two angles, ∠N and ∠U, of ∆SUN have equal measures, the triangle
is isosceles.
In an isosceles triangle, the sides opposite equal-measure angles have the
same length.
So SN has the same length as SU, 3 cm.
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TOPIC F5 GEOMETRY
12.
Sketch an isosceles triangle ∆SAW, with SA and SW of equal lengths.
Figure 16 is one possible answer.
Example 12
S
W
A
Figure 16
13.
In Figure 17, ∆FAX is an equilateral triangle. Find the length of FA.
Example 13
X
6 cm
F
A
Figure 17
In an equilateral triangle, all 3 sides have the same length.
So FA has the same length as FX, 6 cm. (AX also has length 6 cm.)
14.
In the triangle in Figure 18, find the measure of angle Q.
Example 14
Q
?
Figure 18
Since all 3 sides have matching marks, the 3 sides have equal lengths.
That means the triangle is an equilateral triangle.
All 3 angles have the same measure.
180° ÷ 3 = 60˚
So, m∠Q = 60˚.
“m∠Q” means the measure of angle Q.
LESSON F5.3 GEOMETRY III EXPLAIN
483
Right Triangles and the Pythagorean Theorem
Another special type of triangle is a right triangle.
A right triangle has one right angle that measures 90˚, as shown in Figure 19.
hypotenuse
leg
90°
leg
Right Triangle
Figure 19
In a right triangle, the two sides that form the right angle are called the legs of the triangle.
The side opposite the right angle is called the hypotenuse. It’s the longest side.
In a right triangle, the right angle is the largest angle. The other two angles are acute
angles.
The Pythagorean Theorem
The Pythagorean Theorem describes an important relationship among the sides of a right
triangle, as shown in Figure 21.
Pythagorean Theorem
hypotenuse
c
leg
b
leg
a
a2
+ b2 = c 2
Figure 21
a, b, and c are the lengths of the sides, as
shown in Figure 21.
In a right triangle, the sum of the squares of the legs equals the square of the
hypotenuse: a 2 + b 2 = c 2
The Pythagorean Theorem is true only for right triangles. This gives us a way to test
whether a triangle is a right triangle. See Figure 22.
c
b
a
Figure 22
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TOPIC F5 GEOMETRY
• If a 2 + b 2 = c 2, then the triangle is a right triangle.
• If a 2 + b 2 ≠ c 2, then the triangle is not a right triangle.
(Here, a, b, and c are the lengths of the sides. The length of the longest side, the
hypotenuse, is c.)
A Pythagorean triple is a group of three whole numbers that can be used as the lengths of
the sides of a right triangle. Figure 23 shows some examples of Pythagorean triples.
5
4
13
12
3
5
(3, 4, 5)
(6, 8, 10)
(9, 12, 15)
(12, 16, 20)
(5, 12, 13)
(10, 24, 26)
(15, 36, 39)
(20, 48, 52)
21
29
20
(20, 21, 29)
(40, 42, 58)
(60, 63, 87)
(80, 84, 116)
Figure 23
15.
In a given right triangle, the measure of one angle is 35˚. Find the measures of the
other angles.
Example 15
Figure 24 shows a right triangle with one angle whose measure is 35˚.
You may find these
Examples useful while
doing the homework
for this section.
B
35ß
A
C
Figure 24
One of the angles in a right triangle must be a right angle. It’s labeled ∠B.
Since ∠B is a right angle, m∠B = 90˚.
To find the measure of ∠A, add 90° and 35˚.
90˚ + 35˚ = 125˚
Subtract that result from 180˚.
180˚ – 125˚ = 55˚
So, m∠A = 55˚.
16.
In a certain right triangle, the lengths of the sides are 9 cm, 12 cm, and 15 cm.
Which of these lengths is the length of the hypotenuse?
Example 16
The hypotenuse is the longest side, so the length of the hypotenuse is 15 cm, as
shown in Figure 25.
15 cm
12 cm
9 cm
Figure 25
LESSON F5.3 GEOMETRY III EXPLAIN
485
Example 17
17.
In ∆TRI, the lengths of the sides are 4 ft, 5 ft, and 6 ft. Is ∆TRI a right triangle?
See Figure 26.
T
6 ft
5 ft
R
I
4 ft
Figure 26
Here’s one way to find out whether ∆TRI is a right triangle:
• Square the length of each side.
• Add the two smaller squares and see
if the result equals the largest square.
Since 16 + 25 = 41 (not 36), ∆TRI is not a right triangle.
Example 18
18.
4 2 = 4 4 = 16
5 2 = 5 5 = 25
6 2 = 6 6 = 36
Is 16 + 25 = 36?
Is 41 = 36? No.
In Figure 27, the triangle shown is a right triangle. Find the length of its hypotenuse.
8 ft
6 ft
c
Figure 27
Here’s one way to find the length of the hypotenuse.
Use the Pythagorean Theorem.
Square the given lengths.
Add the squares.
To find c, take the square root of 100.
The length of the hypotenuse is 10 ft.
Example 19
19.
78 cm
a
Figure 28
TOPIC F5 GEOMETRY
=
=
=
=
=
c2
c2
c2
c2
c
In Figure 28, the triangle shown is a right triangle. Find a, the length of one of
its legs.
72 cm
486
a2 + b2
62 + 82
36 + 64
100
10
Here’s one way to find the length of the leg.
Use the Pythagorean Theorem.
Square the given lengths.
Add the squares.
To get a 2 by itself, subtract 5184
from both sides.
a2 +
b2
2
a + 72 2
a 2 + 5184
– 5184
a2
=
=
=
c2
78 2
6084
– 5184
=
900
To find a, take the square root of 900.
The length of the leg is 30 cm.
a
=
30
Parallel Lines and Parallelograms
You have learned about parallel lines. Now you will study some properties of the angles
formed by a line that crosses two parallel lines.
In Figure 29, line k and line n are parallel. Line t crosses lines k and n, and forms eight
angles, which are shown.
When line k is parallel to line n, we write
k || n.
t
k
1
2
3
4
n
5
7
6
8
Figure 29
As shown in Figure 30, there are 4 pairs of vertical angles.
Vertical angles are opposite angles.
t
k
1
2
34
n
56
7 8
∠1 and ∠4
∠2 and ∠3
are vertical angles
are vertical angles
∠5 and ∠8
∠6 and ∠7
are vertical angles
are vertical angles
Vertical angles have the same measure
whether lines k and n are parallel or not.
Figure 30
LESSON F5.3 GEOMETRY III EXPLAIN
487
Vertical Angles
Vertical angles have the same measure.
m∠1 = m∠4
m∠5 = m∠8
m∠2 = m∠3
m∠6 = m∠7
Corresponding angles are two angles on the same side of line t that are both “above” or
both “below” lines k and n.
There are 4 pairs of corresponding angles, as shown in Figure 31.
t
k
1 2
34
n
56
78
Corresponding angles form an “F.” This
“F” may face in any direction.
k
k
n
n
Caution! Line k and line n must be
parallel for corresponding angles to have
the same measure.
∠1 and ∠5
∠2 and ∠6
are corresponding angles
are corresponding angles
∠3 and ∠7
∠4 and ∠8
are corresponding angles
are corresponding angles
Figure 31
Corresponding Angles
Because line k || line n, corresponding angles
have the same measure.
m∠1 = m∠5
m∠3 = m∠7
m∠2 = m∠6
m∠4 = m∠8
Alternate interior angles are two angles on opposite sides of line t that are both “inside”
line k and line n.
488
TOPIC F5 GEOMETRY
There are 2 pairs of alternate interior angles, as shown in Figure 32.
Alternate interior angles form a “Z.” This
“Z” may face in any direction.
t
k
k
n
k
1 2
34
n
∠3 and ∠6
n
56
78
∠4 and ∠5
are alternate interior
angles
are alternate interior
angles
Caution! Lines k and n must be parallel
for alternate interior angles to have the
same measure.
Figure 32
Alternate Interior Angles
Because line k || line n, alternate interior
angles have the same measure.
m∠3 = m∠6
m∠4 = m∠5
Parallelograms
Recall that a parallelogram is a polygon with 4 sides that has two pairs of parallel sides.
See Figure 33. In a parallelogram, the opposite sides have the same length.
Figure 33
LESSON F5.3 GEOMETRY III EXPLAIN
489
In Figure 34, all the marked angles with “(” or “)” have the same measure as angle 1.
That’s because vertical angles have the same measure and corresponding angles formed by
parallel lines have the same measure.
vertical angles
2
1
4
3
corresponding angles
Figure 34
Notice that angle 1 and angle 3 have the same measure. In the same way, angle 2 and
angle 4 have the same measure. It follows that:
Opposite angles of a parallelogram have the same measure.
In Figure 35, since opposite sides of the parallelogram are parallel, the corresponding
angles have the same measure. The measures of supplementary angles add to 180˚.
1
corresponding
angles
2
3
4
supplementary
angles
Figure 35
It follows that:
Consecutive angles of a parallelogram are supplementary angles.
490
TOPIC F5 GEOMETRY
20.
In Figure 36, j || u. The measure of angle 1 is 80˚. Find the measures of the other
angles.
Example 20
t
j
You may find these
Examples useful while
doing the homework
for this section.
80˚
1
2
3
4
u
5
7
6
8
Figure 36
Here’s one way to find the measures of the other angles.
∠4 and ∠1 are vertical angles, so they have
the same measure.
∠2 and ∠1 are supplementary angles,
so their measures add to 180˚. 180˚ – 80˚ = 100˚
∠3 and ∠2 are vertical angles, so they have
the same measure.
∠5 and ∠1 are corresponding angles, so they
have the same measure.
∠6 and ∠2 are corresponding angles, so they
have the same measure.
∠7 and ∠3 are corresponding angles, so they
have the same measure.
∠8 and ∠4 are corresponding angles, so they
have the same measure.
m∠4 = m∠1 = 80˚
m∠2 = 100˚
m∠3 = m∠2 = 100˚
m∠5 = m∠1 = 80˚
m∠6 = m∠2 = 100˚
m∠7 = m∠3 = 100˚
m∠8 = m∠4 = 80˚
Here’s another way to find m∠6: Since
∠6 and ∠3 are alternate interior angles,
m∠6 = m∠3 = 100˚.
Similarly, since ∠4 and ∠5 are alternate
interior angles, m∠4 = m∠5 = 80˚.
LESSON F5.3 GEOMETRY III EXPLAIN
491
Example 21
21.
Line segment BE is parallel to line segment CD. Find the missing angle measures in
∆ACD and ∆ABE. See Figure 37.
C
40˚
B
?
?
?
D
E
Figure 37
Here’s a way to find the measure of angle ABE. Since BE || CD, ∠ABE and
∠BCD are corresponding angles. Corresponding angles have the same measure.
So, m∠ABE = 40˚.
30˚
A
Here’s a way to find the measure of angle AEB. The angle measures of a triangle
add to 180˚. Since 30˚ + 40˚ = 70˚ and 180˚ – 70˚ = 110˚, m∠AEB = 110˚.
Here’s a way to find the measure of angle D. Since BE || CD, ∠D and ∠AEB
are corresponding angles. Corresponding angles have the same measure.
So, m∠D = 110˚.
Example 22
22.
Find the missing angle measures in parallelogram GRAM. See Figure 38.
G
R
125˚
?
?
?
A
M
Figure 38
In a parallelogram, opposite angles have the same measure. Since angle M and
angle R are opposite angles, m∠M = 125˚.
In a parallelogram, consecutive angles are supplementary. Angle G and angle R are
consecutive angles. Since 180˚ – 125˚ = 55˚, m∠G = 55˚.
Since angle A and angle G are opposite angles, the measure of angle A is also 55˚.
Example 23
23.
Find the missing side lengths in parallelogram RUSH. See Figure 39.
?
R
U
?
H
2 cm
5 cm
Figure 39
S
In a parallelogram, opposite sides have the same length.
Since US and RH are opposite sides, the length of RH is 2 cm.
Since SH and RU are opposite sides, the length of RU is 5 cm.
492
TOPIC F5 GEOMETRY
Explain
In Concept 2: Similar
Polygons, you will find a
section on each of the
following:
CONCEPT 2: SIMILAR POLYGONS
• Recognizing Similar
Polygons
You have learned that congruent objects have the same shape and the same size. Similar
objects must also have the same shape, but they may or may not be the same size. An
architect’s scale model is similar to the full-size building. The street map of a city is similar
to the actual streets and blocks of the city. If you shrink a picture when you photocopy it,
the two images are similar. If you enlarge a photograph, the original and the enlarged
version are similar.
• Writing a Similarity
Statement
• Using Shortcuts to
Recognize Similar Triangles
Recognizing Similar Polygons
In Figure 40, the pentagons are similar polygons. They are also congruent.
• Finding the Measures of
Corresponding Angles of
Similar Triangles
• Finding the Lengths of
Corresponding Sides of
Similar Triangles
D
C
A
B
H
F
G
same shape
same size
ABCDE ~ FGHIJ
Congruent objects must be similar. But
not all similar objects are congruent.
“~” means “is similar to”
I
J
E
Figure 40
In Figure 41, the hexagons are similar polygons, but they are not congruent.
O
N
P
U
K
T
V
M
S
Q
L
same shape
different sizes
R
KLMNOP ~ QRSTUV
Figure 41
For two polygons to be similar, both of the following must be true:
• Corresponding angles have the same measure.
• Lengths of corresponding sides have the same ratio.
LESSON F5.3 GEOMETRY III EXPLAIN
493
In Figure 42, the quadrilaterals STAR and FIND are similar, since they satisfy both of the
conditions above.
D
15 cm
R
10 cm
S
F
6 cm
4 cm
9 cm
6 cm
T
A
8 cm
I
12 cm
STAR ~ FIND
N
Figure 42
ST is the length of line segment ST.
FI is the length of line segment FI,
and so on.
We also say that the lengths of
corresponding sides are proportional.
You may find these
Examples useful while
doing the homework
for this section.
Example 24
Corresponding angles
have the same measure:
Corresponding lengths
have the same ratio:
m∠S = m∠F
m∠T = m∠I
ST
4
2
= = FI
6
3
2
TA
8
= = IN
12
3
m∠A = m∠N
m∠R = m∠D
AR
6
2
= = ND
9
3
2
RS
10
= = DF
15
3
24.
Are the polygons ABCD and EFGH in Figure 43 similar?
B
3 cm
2 cm
A
C
2 cm
3 cm
D
3 cm
F
G
140ß
40ß
2 cm
2 cm
40ß
140ß
E
H
3 cm
Figure 43
The polygons are not similar. Corresponding lengths have the same ratio, but
corresponding angles do not have the same measure:
2
2
• =1
3
=1
3
2
=1
2
3
=1
3
• All the angles in the rectangle measure 90˚.
Two of the angles in the parallelogram measure 40˚, and two of the angles
measure 140˚.
Example 25
25.
Are the polygons ABCD and EFGH in Figure 44 similar?
B
3 cm
2 cm
A
C
2 cm
3 cm
D
Figure 44
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TOPIC F5 GEOMETRY
F 2 cm G
2 cm
2 cm
E 2 cm H
The polygons in Figure 44 are not similar. Corresponding angles have the same
measure, but lengths of corresponding sides do not have the same ratio:
• All the angles have the same measure, 90˚.
2
2
• =1
3
= 1.5
2
2
=1
2
3
= 1.5
2
Some lengths of corresponding sides have ratio 1 and some have ratio 1.5.
Example 26
26. In Figure 45, triangle SUM is similar to triangle CAR. Find the missing angle
measures.
C
R
M
S
20˚
60˚
U
∆SUM ~ ∆CAR
A
Figure 45
Corresponding angles have the same measure.
So, m∠C = m∠S = 20˚ and m∠U = m∠A = 60˚.
You can use the fact that the angle measures in a triangle add to 180˚ to find the
measure of the third angle in each triangle.
• Add the angle measures you know.
20˚ + 60˚ = 80˚
• Subtract that result from 180˚.
180˚ – 80˚ = 100˚
So, m∠M = m∠R = 100˚.
__
length of CA
27. In Figure 46, ∆SUM is similar to ∆CAR. Find this ratio:
length of SU
Example 27
C
R
M
S
3 cm
2 cm
U
∆SUM ~ ∆CAR
A
Figure 46
In similar triangles, corresponding lengths have the same ratio.
__
__
3
length of AR
length of CA
= = 2
length of UM
length of SU
LESSON F5.3 GEOMETRY III EXPLAIN
495
Writing a Similarity Statement
In Figure 47, triangle FAN and triangle CLB are similar.
F
A
N
L
C
B
Figure 47
The order of letters in the similarity statement ∆FAN ~ ∆CLB matches the
corresponding angles or vertices of each polygon.
∠F corresponds to ∠C.
∠A corresponds to ∠L.
∠N corresponds to ∠B.
∆FAN ~ ∆CLB
∆FAN ~ ∆CLB
∆FAN ~ ∆CLB
The order of the letters in the similarity statement also matches corresponding sides.
∆FAN ~ ∆CLB
∆FAN ~ ∆CLB
∆FAN ~ ∆CLB
FA corresponds to CL.
AN corresponds to LB.
FN corresponds to CB.
Example 28
28. Write a similarity statement for the triangles in Figure 48.
You may find these
Examples useful while
doing the homework
for this section.
T
B
W
C
K
A
Figure 48
Here’s one way to write a similarity statement:
• First, write one pair of corresponding vertices.
• Then, write another pair of corresponding vertices.
• Finally, write the last pair of corresponding vertices.
∆A
∆K
∆AB
∆KT
∆ABC ~ ∆KTW
So, one similarity statement for these triangles is ∆ABC ~ ∆KTW.
Here are other similarity statements:
∆BAC ~ ∆TKW
∆ACB ~ ∆KWT
∆CAB ~ ∆WKT
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TOPIC F5 GEOMETRY
29. ∆SLY ~ ∆DOG
Example 29
In ∆DOG, find the angle that has the same measure as angle Y.
Use the order of the letters in the similarity statement.
∠Y corresponds to ∠G.
∆SLY ~ ∆DOG
So, angle G has the same measure as angle Y.
30. ∆SLY ~ ∆DOG
Example 30
The ratio of SL to DO is 4 to 3. Find the ratio of LY to OG.
SL is the length of line segment SL.
LY is the length of line segment LY,
and so on.
Use the order of the letters in the similarity statement.
SL corresponds to DO.
∆SLY ~ ∆DOG
LY corresponds to OG.
∆SLY ~ ∆DOG
The lengths of corresponding sides have the same ratio. So the ratio of LY to OG is
also 4 to 3.
Shortcuts for Determining Similar Triangles
You know that two polygons are similar if all their corresponding angles have the same
measure and all their corresponding lengths have the same ratio.
However, you can show that two triangles are similar by using less information. Here are
two shortcuts.
Shortcut #1
Two triangles are similar if two angles of one triangle have the same measure as two angles
of the other triangle.
This shortcut is sometimes called “Angle
Angle” or “AA.”
Shortcut #2
Two triangles are similar if all their corresponding lengths have the same ratio.
31. Are the triangles in Figure 49 similar?
P
55ß
Caution! These two shortcuts work only
for triangles.
Example 31
O
You may find these
Examples useful while
doing the homework
for this section.
J
T
A
55ß
M
Figure 49
LESSON F5.3 GEOMETRY III EXPLAIN
497
Each triangle has an angle with measure 55°. Each triangle has an angle with
measure 90˚.
So by Shortcut #1, the triangles are similar.
Example 32
32. Write a similarity statement for the triangles in Figure 49.
∠O corresponds to ∠A.
∠P corresponds to ∠M.
One similarity statement for these triangles is ∆TOP ~ ∆JAM.
Example 33
33. Are the triangles in Figure 50 similar?
16 cm
H
12 cm
T
20 cm
E
15 cm
W
N
9 cm
12 cm
O
Figure 50
TH
16
4
= = WO
12
3
TE
20
4
= = WN
15
3
4
HE
12
= = ON
9
3
All corresponding lengths have the same ratio.
So, by Shortcut #2, the triangles are similar.
Example 34
34. Write a similarity statement for the triangles in Figure 50.
TH corresponds to WO, so T corresponds to W and H corresponds to O.
TE corresponds to WN, so E corresponds to N.
One similarity statement for these triangles is ∆THE ~ ∆WON.
498
TOPIC F5 GEOMETRY
Measures of Corresponding Angles of
Similar Triangles
To find missing angle measures in similar triangles, you can apply what you have learned
about triangles.
• The angle measures of a triangle add to 180˚.
• In similar triangles, corresponding angles have the same measure.
• In similar triangles, the order of the letters in a similarity statement matches the
corresponding angles.
35. The triangles in Figure 51 are similar. ∆BUS ~ ∆DRV. Find the missing angle
measures.
Example 35
V
B
You may find these
Examples useful while
doing the homework
for this section.
S
U
˘BUS ~ ˘DRV
D
60ß
R
Figure 51
In ∆DRV, one way to find the measure of angle V is to use what you know about the
sum of the angle measures in a triangle.
• Add the given angle measures.
• Subtract that result from 180˚.
60˚ + 90˚ = 150˚
180˚ – 150˚ = 30˚
So, m∠V = 30˚.
One way to find the measure of angle B in ∆BUS is to use the order of the letters in
the similarity statement, ∆BUS ~ ∆DRV.
Angle B corresponds to angle D.
In similar triangles, corresponding angles have the same measure.
So, m∠B = m∠D = 60˚.
In the same way, angle S has the same measure as angle V.
So, m∠S = m∠V =30˚.
Lengths of Corresponding Sides of Similar Triangles
To find missing lengths in similar triangles, you can apply what you have learned about
similar triangles.
• In similar triangles, lengths of corresponding sides have the same ratio.
• In similar triangles, the order of the letters in a similarity statement matches the
corresponding sides.
LESSON F5.3 GEOMETRY III EXPLAIN
499
You may find these
Examples useful while
doing the homework
for this section.
Example 36
36. The triangles in Figure 52 are similar. ∆WAY ~ ∆OUT. Find x, the length of OT.
Y
12 ft
W
10 ft
14 ft
A
T
x
5 ft
O
U
˘WAY ~ ˘OUT
Figure 52
Here’s one way to find x.
AY is the length of line segment AY.
UT is the length of line segment UT,
and so on.
• Corresponding lengths have the same ratio.
AY
UT
=
WY
OT
• We know AY, UT, and WY.
10
5
=
12
x
10 x = 5 12
10x = 60
• Cross multiply.
• To get x by itself, divide both sides by 10.
10x
60
= 10
10
x=6
So, x, the length of OT is 6 ft.
Example 37
37. In Figure 53, triangle ABC and triangle ADE are similar. Find y, the length of DE.
Notice that the triangles in this figure “overlap.” Look carefully at the similarity
statement to find pairs of corresponding angles and corresponding lengths.
E
˘ABC ~ ˘ADE
y
C
3 cm
A
5 cm
D
B
20 cm
Figure 53
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TOPIC F5 GEOMETRY
Here’s one way to find y.
AB
BC
= AD
DE
5
3
= 20
y
• Corresponding lengths have the same ratio.
•We know AB, AD, and BC.
5 y = 3 20
5y = 60
•Cross multiply.
60
5y
= 5
5
•To get y by itself, divide both sides by 5.
y = 12
So, y, the length of DE is 12 cm.
38. In Figure 54, triangle CRN is similar to triangle COB. Find x, the length of NB.
B
Example 38
∆CRN ~ ∆COB
x
N
12 cm
C
20 cm
R
10 cm
O
Figure 54
Here’s one way to find x.
Notice that here, x is the length of only a part of one side of ∆COB.
• Corresponding lengths have the same ratio.
• We know CR and CN. CO = 20 + 10 = 30
CB = x + 12
CR
CN
= CO
CB
12
20
= 30
x + 12
• Cross multiply.
20(x + 12) = 30 12
20(x + 12) = 360
• To remove the parentheses, use
the distributive property.
• Subtract 240 from both sides.
20x + 240 = 360
• To get x by itself, divide both
sides by 20.
20x + 240 – 240 = 360 – 240
20x = 120
120
20x
= 20
20
x =6
So, x, the length of NB is 6 cm.
LESSON F5.3 GEOMETRY III EXPLAIN
501
Explore
This Explore contains two
investigations.
• Congruent Triangles
• “Door to Door”
Investigation 1: Congruent Triangles
For this investigation, you will need a ruler and a protractor.
You have learned that two triangles are congruent if all corresponding sides have the same
length and all corresponding angles have the same measure.
Often you don’t need to verify all of the above information to know that two triangles are
congruent. In this investigation you will test six ways for using less information to prove
that two triangles are congruent. Some of these ways are true and some are false.
You have been introduced
to these investigations
in the Explore module
of this lesson on the
computer. You can
complete them using the
information given here.
1.
True or False: Two triangles are congruent if they have one pair of
congruent sides.
a. Draw a triangle and label its vertices A, B, and C.
b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler.
Record your measurements on the triangle.
c. See if you can draw another triangle (∆DEF) with both of these features:
• One side of ∆DEF is congruent to one side of ∆ABC, and
• ∆DEF is not congruent to ∆ABC.
d. Based on your investigation in (c), do you think the conjecture is true? Discuss
what you tried in (c) and how you decided whether the conjecture is true.
502
TOPIC F5 GEOMETRY
2.
True or False: Two triangles are congruent if they have two pairs of
congruent sides.
a. Draw a triangle and label its vertices A, B, and C.
b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler.
Record your measurements on the triangle.
c. See if you can draw another triangle (∆DEF) with both of these features:
• Two sides of ∆DEF are congruent to two sides of ∆ABC, and
• ∆DEF is not congruent to ∆ABC.
d. Based on your investigation in (c), do you think the conjecture is true? Discuss
what you tried in (c) and how you decided whether the conjecture is true.
3.
True or False: Two triangles are congruent if they have three pairs of
congruent sides.
a. Draw a triangle and label its vertices A, B, and C.
b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler.
Record your measurements on the triangle.
LESSON F5.3 GEOMETRY III EXPLORE
503
c. See if you can draw another triangle (∆DEF) with both of these features:
• Three sides of ∆DEF are congruent to three sides of ∆ABC, and
• ∆DEF is not congruent to ∆ABC.
d. Based on your investigation in (c), do you think the conjecture is true? Discuss
what you tried in (c) and how you decided whether the conjecture is true.
4.
True or False: Two triangles are congruent if they have three pairs of
congruent angles.
a. Draw a triangle and label its vertices A, B, and C.
b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler.
Record your measurements on the triangle.
c. See if you can draw another triangle (∆DEF) with both of these features:
• Three angles of ∆DEF are congruent to three angles of ∆ABC, and
• ∆DEF is not congruent to ∆ABC.
d. Based on your investigation in (c), do you think the conjecture is true? Discuss
what you tried in (c) and how you decided whether the conjecture is true.
504
TOPIC F5 GEOMETRY
5.
True or False: Two triangles are congruent if they have two pairs of congruent
sides and the angles formed by those sides are also congruent.
a. Draw a triangle and label its vertices A, B, and C.
b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler.
Record your measurements on the triangle.
c. See if you can draw another triangle (∆DEF) with both of these features:
• Two sides of ∆DEF are congruent to two sides of ∆ABC, and
• The angles formed by those sides are congruent.
• ∆DEF is not congruent to ∆ABC.
d. Based on your investigation in (c), do you think the conjecture is true? Discuss
what you tried in (c) and how you decided whether the conjecture is true.
6.
True and False: Two triangles are congruent if they have two pairs of congruent
angles and the sides shared by those angles are also congruent.
a. Draw a triangle and label its vertices A, B, and C.
b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler.
Record your measurements on the triangle.
LESSON F5.3 GEOMETRY III EXPLORE
505
c. See if you can draw another triangle (∆DEF) with both of these features:
• Two angles of ∆DEF are congruent to two angles of ∆ABC, and
• The sides shared by those angles are congruent.
• ∆DEF is not congruent to ∆ABC.
d. Based on your investigation in (c), do you think the conjecture is true? Discuss
what you tried in (c) and how you decided whether the conjecture is true.
Investigation 2: “Door to Door”
1.
First, sketch a copy of the rectangle in Figure 55. Then sketch a larger rectangle
which is similar to the first rectangle and has the property that corresponding lengths
2
of sides are in the ratio .
1
1 cm
3 cm
Figure 55
a. Find the perimeter of each rectangle.
b. What is the ratio of the perimeters? (Put the perimeter of the larger rectangle in
the numerator.) How does the ratio of the perimeters compare to the ratio of the
corresponding lengths?
c. Sketch another larger rectangle similar to the rectangle in Figure 55, so that the
3
ratio of corresponding lengths is .
1
d. Find the perimeter of each rectangle.
506
TOPIC F5 GEOMETRY
e. What is the ratio of the perimeters? (Put the perimeter of the larger rectangle in
the numerator.) How does the ratio of the perimeters compare to the ratio of the
corresponding lengths?
f. Look back at your results in (b) and (e). Do you see a pattern? Test your ideas by
sketching some more similar shapes and finding the ratio of their perimeters.
2.
Sketch a larger rectangle similar to the rectangle in Figure 56, so that the ratio of
2
their corresponding lengths is .
1
2 cm
3 cm
Figure 56
a. Find the area of each rectangle.
b. What is the ratio of the areas? (Put the area of the larger rectangle in the
numerator.) How does the ratio of the areas compare to the ratio of the
corresponding lengths?
c. Sketch another larger rectangle similar to the rectangle in Figure 56, so that the
3
ratio of the corresponding lengths is .
1
d. Find the area of each rectangle.
LESSON F5.3 GEOMETRY III EXPLORE
507
e. What is the ratio of the areas? (Put the area of the larger rectangle in the
numerator.) How does the ratio of the areas compare to the ratio of the
corresponding lengths?
f. Look back at your results in (b) and (e). Do you see a pattern? Test your ideas by
sketching some more similar shapes and finding the ratio of their areas.
3.
Sketch a rectangular prism similar to the one in Figure 57, so that the ratio of their
2
corresponding lengths is .
1
2 cm
1 cm
3 cm
Figure 57
a. Find the volume of each rectangular prism.
b. What is the ratio of the volumes? (Put the volume of the larger prism in the
numerator.) How does the ratio of the volumes compare to the ratio of the
corresponding lengths?
c. Sketch another rectangular prism similar to the one in Figure 57, so that the ratio
3
of corresponding lengths is .
1
d. Find the volume of each prism.
508
TOPIC F5 GEOMETRY
e. What is the ratio of the volumes? (Put the volume of the larger prism in the
numerator.) How does the ratio of the volumes compare to the ratio of the
corresponding lengths?
f. Look back at your results in (b) and (e). Do you see a pattern? Test your ideas by
sketching some more similar prisms and finding the ratio of their volumes.
LESSON F5.3 GEOMETRY III EXPLORE
509
Homework
Concept 1: Triangles and Parallelograms
The Sum of the Angle Measures of a
Triangle
4.
In Figure 61, find the measure of ∠E.
P
For help working these types of problems, go back to Examples
1–3 in the Explain section of this lesson.
1.
40˚
E
In Figure 58, find the measure of ∠W.
?
75˚
H
K
Figure 61
65˚
?
W
5.
In Figure 62, find the measure of ∠Q.
75˚
R
E
30˚
Figure 58
2.
?
Q
100˚
In Figure 59, find the measure of ∠T.
W
T
Figure 62
?
50˚
30˚
O
6.
In Figure 63, find the measure of ∠M.
U
U
Figure 59
3.
50°
In Figure 60, find the measure of ∠N.
Y
50˚
M
?
70° A
Figure 63
7.
?
60˚
N
L
In Figure 64, find the measure of ∠I.
Q
40˚
Figure 60
65˚
?
I
Figure 64
510
TOPIC F5 GEOMETRY
B
8.
13. In Figure 70, find the measure of ∠Z.
In Figure 65, find the measure of ∠S.
S
Z
?
?
100˚
50˚
D
F
Figure 65
9.
O
In Figure 66, find the measure of ∠A.
66 32 ˚
80˚
T
Figure 70
14. In Figure 71, find the measure of ∠W.
A
D
?
42.5˚
P
2
66 3 ˚
1
33 3 ˚
T
Figure 66
E
?
32.5˚
10. In Figure 67, find the measure of ∠R.
W
Figure 71
15. In Figure 72, find the measure of ∠B.
T
B
105˚
?
A
42.5˚
?
R
Figure 67
H
11. In Figure 68, find the measure of ∠D.
22.2˚
129˚
X
Figure 72
16. In Figure 73, find the measure of ∠T.
D
?
A
U
22.2˚
28.8˚
H
Figure 68
30˚
12. In Figure 69, find the measure of ∠T.
R
T
80 21 ˚
?
69 21 ˚
C
69 21 ˚
?
T
Figure 73
Y
Figure 69
LESSON F5.3 GEOMETRY III HOMEWORK
511
17. In Figure 74, find the measure of ∠BAT.
21. In Figure 78, find the measure of ∠N.
N
T
?
?
B
70˚
70˚
Figure 74
U
Figure 78
18. In Figure 75, find the measure of ∠AGS.
G
72˚ ?
40˚
R
A
22. In Figure 79, find the measure of ∠M.
M
S
?
A
Figure 75
C
118˚
19. In Figure 76, find the measure of ∠GLO.
94˚
L
Y
Figure 79
118˚ ?
23. In Figure 80, find the measure of ∠Y.
G
O
Figure 76
20. In Figure 77, find the measure of ∠EVM.
V
96˚
96˚ H
E
?
?
38˚
Y
Figure 80
24. In Figure 81, find the measure of ∠E.
D
M
E
70˚
Figure 77
30˚
K
Figure 81
512
TOPIC F5 GEOMETRY
?
E
Congruent Triangles
For help working these types of problems, go back to Examples
4–8 in the Explain section of this lesson.
28. In Figure 85, ∆DEF is congruent to ∆GHI.
Which angle of ∆DEF is congruent to ∠I?
D
F
25. In Figure 82, ∆ABC is congruent to ∆DEF.
Which angle of ∆DEF is congruent to ∠B?
I
C
H
G
F
A
E
D
Figure 85
29. In Figure 86, ∆ABC is congruent to ∆DEF.
Which side of ∆DEF is congruent to AB?
B
E
Figure 82
C
26. In Figure 83, ∆BCD is congruent to ∆EFG.
Which angle of ∆EFG is congruent to ∠D?
A
F
D
B
G
B
E
E
Figure 86
30. In Figure 87, ∆BCD is congruent to ∆EFG.
Which side of ∆BCD is congruent to EG?
C
D
F
G
Figure 83
B
E
27. In Figure 84, ∆CDE is congruent to ∆FGH.
Which angle of ∆CDE is congruent to ∠F?
C
C
F
D
F
Figure 87
D
H
31. In Figure 88, ∆CDE is congruent to ∆FGH.
Which side of ∆CDE is congruent to GH?
E
F
C
G
Figure 84
H
D
E
G
Figure 88
LESSON F5.3 GEOMETRY III HOMEWORK
513
32. In Figure 89, ∆DEF is congruent to ∆GHI.
Which side of ∆GHI is congruent to DF?
36. In Figure 93, the triangles are congruent.
Write a congruence relation for these triangles.
E
D
F
H
G
I
I
H
G
F
D
E
Figure 93
Figure 89
37. In Figure 94, ∆CDE is congruent to ∆FGH.
Find the measure of ∠G.
33. In Figure 90, the triangles are congruent.
Write a congruence relation for these triangles.
C
A
40˚
D
E
F
C
B
E
D
?
∆CDE ≅ ∆FGH
Figure 90
H
30˚
F
G
34. In Figure 91, the triangles are congruent.
Write a congruence relation for these triangles.
F
Figure 94
38. In Figure 95, ∆DEF is congruent to ∆GHI.
Find the measure of ∠H.
C
D
I
G
E
B
Figure 91
35. In Figure 92, the triangles are congruent.
Write a congruence relation for these triangles.
?
G
F
D
F
108°
C
H
H
D
48°
E
E
∆DEF ≅ ∆GHI
Figure 95
G
Figure 92
514
TOPIC F5 GEOMETRY
42. In Figure 99, ∆DEF is congruent to ∆GHI.
Find the measure of ∠I.
39. In Figure 96, ∆EFG is congruent to ∆HIJ.
Find the measure of ∠I.
G
I
?
24˚
∆EFG ≅ ∆HIJ
E
H
G
94˚
I
F
D
F
108°
?
48°
H
J
∆DEF ≅ ∆GHI
E
Figure 99
Figure 96
43. In Figure 100, ∆EFG is congruent to ∆HIJ.
Find the measure of ∠H.
40. In Figure 97, ∆FGH is congruent to ∆IJK.
Find the measure of ∠H.
G
G
24˚
E
F
94˚
F
∆FGH ≅ ∆IJK
?
I
I
H
?
J
H
∆EFG ≅ ∆HIJ
38°
J
58°
Figure 100
K
Figure 97
41. In Figure 98, ∆CDE is congruent to ∆FGH.
Find the measure of ∠F.
C
40˚
E
F
?
30˚
H
D
∆CDE ≅ ∆FGH
G
Figure 98
LESSON F5.3 GEOMETRY III HOMEWORK
515
47. In Figure 104, ∆KLM is congruent to ∆NOP.
Find the length of LM.
44. In Figure 101, ∆FGH is congruent to ∆IJK.
Find the measure of ∠F.
G
M
N
10 cm
L
O
6 cm
K
P
? F
∆KLM ≅ ∆NOP
Figure 104
I
48. In Figure 105, ∆STU is congruent to ∆VWX.
Find the length of SU.
H
∆FGH ≅ ∆IJK
U
S
58°
38°
J
K
∆STU ≅ ∆VWX
7.5 cm
Figure 101
45. In Figure 102, ∆SRK is congruent to ∆FIN.
Find the length of SR.
T
V
W
5 cm
K
N
X
3 ft
5 ft
Figure 105
R
I
4 ft
S
F
∆SRK ≅ ∆FIN
Isosceles Triangles and Equilateral
Triangles
For help working these types of problems, go back to Examples
9–14 in the Explain section of this lesson.
Figure 102
46. In Figure 103, ∆CDE is congruent to ∆FGH.
Find the length of FG.
49. In Figure 106, ∆RIT is an isosceles triangle.
Which pair of angles must have the same measure?
D
T
6 in
C
4 in
6 ft
E
6 ft
∆CDE ≅ ∆FGH
H
R
3 ft
Figure 106
F
G
Figure 103
516
TOPIC F5 GEOMETRY
I
50. In Figure 107, ∆BSU is an isosceles triangle.
Which pair of angles must have the same measure?
54. In Figure 111, ∆JET is an isosceles triangle.
Find the measure of ∠E.
J
S
8 cm
4 cm
B
?
E
35˚
T
Figure 111
8 cm
U
Figure 107
51. In Figure 108, ∆WAY is an isosceles triangle.
Which pair of angles must have the same measure?
55. In Figure 112, ∆WHO is an isosceles triangle.
Find the measure of ∠W.
W
?
Y
4 ft
H
6 ft
W
55˚
O
4 ft
Figure 112
56. In Figure 113, ∆SAM is an isosceles triangle.
Find the measure of ∠S.
A
Figure 108
52. In Figure 109, ∆TOY is an isosceles triangle.
Which pair of angles must have the same measure?
3 cm
T
S
?
Y
2 cm
M
2 cm
O
Figure 109
27˚
53. In Figure 110, ∆BAT is an isosceles triangle.
Find the measure of ∠T.
B
A
Figure 113
57. In Figure 114, find the length of BA.
B
A
70˚
?
?
T
Figure 110
A
70˚
8 cm
70˚
T
Figure 114
LESSON F5.3 GEOMETRY III HOMEWORK
517
58. In Figure 115, find the length of JT.
65. In Figure 122, ∆ASK is an equilateral triangle. Find the
length of AS.
J
6 ft
E
A
?
35˚
35˚
T
6 ft
Figure 115
59. In Figure 116, find the length of HO.
S
K
Figure 122
W
66. In Figure 123, ∆PAL is an equilateral triangle. Find the
length of AL.
55˚
3 ft
H
L
?
P
55˚
O
5 cm
Figure 116
A
Figure 123
60. In Figure 117, find the length of AM.
67. In Figure 124, ∆JAM is an equilateral triangle. Find the
length of AM.
S
4 cm
27˚
M
8 cm
M
J
?
27˚
A
A
Figure 117
Figure 124
61. Sketch an isosceles triangle, ∆PUT, with PU and PT of
equal lengths.
68. In Figure 125, ∆BDU is an equilateral triangle. Find the
length of BU.
62. Sketch an isosceles triangle, ∆CAN, with CA and AN of
different lengths.
63. Sketch an isosceles triangle, ∆RAT, with all sides of
equal lengths.
D
2 in
B
U
64. Sketch an isosceles triangle, ∆SLO, with LO and SO of
equal lengths.
Figure 125
69. In Figure 126, find the measure of angle S.
A
S
?
K
Figure 126
518
TOPIC F5 GEOMETRY
70. In Figure 127, find the measure of angle A.
79. In a certain right triangle, the lengths of the sides are 25 in,
60 in, and 65 in. Find the lengths of the legs.
L
5 cm
80. In a certain right triangle, the lengths of the sides are 5m,
12m, and 13m. Find the lengths of the legs.
P
5 cm
5 cm
81. In ∆STW, the lengths of the sides are 3 cm, 4 cm, and 5 cm.
Is ∆STW a right triangle?
?
A
Figure 127
71. In Figure 128, find the measure of angle M.
J
?
M
82. In ∆LUG, the lengths of the sides are 5 ft, 11 ft, and 13 ft.
Is ∆LUG a right triangle?
83. In ∆PTA, the lengths of the sides are 9 cm, 10 cm, and
12 cm. Is ∆PTA a right triangle?
84. In ∆TIP, the lengths of the sides are 10 ft, 24 ft, and 26 ft.
Is ∆TIP a right triangle?
85. In ∆BIN, the lengths of the sides are 10 cm, 12 cm, and
24 cm. Is ∆BIN a right triangle?
A
Figure 128
72. In Figure 129, find the measure of angle U.
86. In ∆COW, the lengths of the sides are 15 in, 36 in, and
39 in. Is ∆COW a right triangle?
87. In ∆IRT, the lengths of the sides are 60 cm, 63 cm, and
87 cm. Is ∆IRT a right triangle?
D
88. In ∆MTA, the lengths of the sides in are 5 ft, 7 ft, and 9 ft.
Is ∆MTA a right triangle?
B
?
U
Figure 129
89. The triangle in Figure 130 is a right triangle. Find c, the
length of its hypotenuse.
3 in
Right Triangles and the Pythagorean
Theorem
4 in
c
For help working these types of problems, go back to Examples
15– 19 in the Explain section of this lesson.
73. In a right triangle, the measure of one angle is 28˚. Find the
measures of the other angles.
74. In a right triangle, the measure of one angle is 80˚. Find the
measures of the other angles.
75. In a right triangle, the measure of one angle is 30˚. Find the
measures of the other angles.
76. In a right triangle, the measure of one angle is 45˚. Find the
measures of the other angles.
Figure 130
90. The triangle in Figure 131 is a right triangle. Find c, the
length of its hypotenuse.
c
10 cm
24 cm
Figure 131
77. In a certain right triangle, the lengths of the sides are 60 cm,
63 cm, and 87 cm. Find the length of the hypotenuse.
78. In a certain right triangle, the lengths of the sides are 15 ft,
20 ft, and 25 ft. Find the length of the hypotenuse.
LESSON F5.3 GEOMETRY III HOMEWORK
519
91. The triangle in Figure 132 is a right triangle. Find c, the
length of its hypotenuse.
16 cm
96. The triangle in Figure 137 is a right triangle. Find b, the
length of one of its legs.
12 cm
15 ft
9 ft
c
Figure 132
b
Figure 137
92. The triangle in Figure 133 is a right triangle. Find c, the
length of its hypotenuse.
Parallel Lines and Parallelograms
c
18 ft
For help working these types of problems, go back to Examples
20–23 in the Explain section of this lesson.
97. In Figure 138, k || r. The measure of angle 1 is 75˚.
Find the measure of angle 2.
24 ft
Figure 133
t
93. The triangle in Figure 134 is a right triangle. Find b, the
length of one of its legs.
k
3 in
75˚
1
3
b
5 in
r
5
7
Figure 134
94. The triangle in Figure 135 is a right triangle. Find a, the
length of one of its legs.
52 cm
a
48 cm
6
8
Figure 138
98. In Figure 138, k || r. The measure of angle 1 is 75˚.
Find the measure of angle 3.
99. In Figure 138, k || r. The measure of angle 1 is 75˚.
Find the measure of angle 4.
Figure 135
95. The triangle in Figure 136 is a right triangle. Find a, the
length of one of its legs.
a
87 cm
2
4
100. In Figure 138, k || r. The measure of angle 1 is 75˚.
Find the measure of angle 5.
101. In Figure 138, k || r. The measure of angle 1 is 75˚.
Find the measure of angle 6.
63 cm
102. In Figure 138, k || r. The measure of angle 1 is 75˚.
Find the measure of angle 7.
103. In Figure 138, k || r. The measure of angle 1 is 75˚.
Find the measure of angle 8.
Figure 136
104. In Figure 138, k || r. Are angles 1 and 5 supplementary
angles, corresponding angles, or vertical angles?
520
TOPIC F5 GEOMETRY
105. In Figure 139, line segment AB is parallel to line
segment ED. Find the measure of angle A.
112. In Figure 141, line segment KL is parallel to line
segment MN. Find the measure of angle JLK.
113. In Figure 142, the quadrilateral is a parallelogram.
Find the measure of angle Q.
A
E
U
Q
40˚
B
C
D
Figure 139
106. In Figure 139, line segment AB is parallel to line
segment ED. Find the measure of angle C.
110˚
D
107. In Figure 140, line segment TU is parallel to line
segment VW. Find the measure of angle STU.
Figure 142
114. In Figure 142, the quadrilateral is a parallelogram.
Find the measure of angle U.
V
65˚
T
A
115. In Figure 142, the quadrilateral is a parallelogram.
Find the measure of angle A.
116. In Figure 143, the quadrilateral is a parallelogram.
Find the measure of angle F.
S
45˚
W
U
F
L
Figure 140
108. In Figure 140, line segment TU is parallel to line
segment VW. Find the measure of angle SUT.
P
45˚
I
109. In Figure 140, line segment TU is parallel to line
segment VW. Find the measure of angle W.
110. In Figure 141, line segment KL is parallel to line
segment MN. Find the measure of angle M.
Figure 143
117. In Figure 144, the quadrilateral is a parallelogram.
Find the length of SH.
P 4 cm
M
U
10 cm
K
H
J
45˚
S
Figure 144
100˚
L
Figure 141
N
111. In Figure 141, line segment KL is parallel to line
segment MN. Find the measure of angle JKL.
118. In Figure 144, the quadrilateral is a parallelogram.
Find the length of PH.
LESSON F5.3 GEOMETRY III HOMEWORK
521
123. Are the polygons in Figure 148 similar polygons?
119. In Figure 145, the quadrilateral is a parallelogram.
Find the length of DE.
6 ft
D
E
4 ft
4 ft
6 ft
100˚
80˚ 100˚
3 ft
K
80˚
3 ft
S
8 ft
8 ft
100˚ 80˚
Figure 145
3 ft
100˚
120. In Figure 145, the quadrilateral is a parallelogram.
Find the length of ES.
80˚
6 ft
Figure 148
124. Are the polygons in Figure 149 similar polygons?
Concept 2: Similar
Polygons
3 ft
3 ft
80˚ 100˚
Recognizing Similar Polygons
4 ft
4 ft
4 ft
4 ft
100˚ 80˚
For help working these types of problems, go back to Examples
24–27 in the Explain section of this lesson.
3 ft
3 ft
Figure 149
121. Are the polygons in Figure 146 similar polygons?
125. Are the rectangles in Figure 150 similar rectangles?
6 ft
6 ft
80˚
8 ft 8 ft
8 ft
4 cm
100˚
8 ft
3 cm
3 cm
3 cm
2 cm
2 cm
3 cm
100˚
4 cm
80˚
6 ft
6 ft
Figure 150
Figure 146
126. Are the triangles in Figure 151 similar triangles?
122. Are the rectangles in Figure 147 similar rectangles?
6 ft
3 cm
3 ft
8 ft
8 ft
4 ft
4 ft
60˚
60˚
3 cm
60°
60˚
3 cm
4 cm
60˚
6 ft
Figure 147
522
TOPIC F5 GEOMETRY
60˚
4 cm
Figure 151
3 ft
4 cm
133. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle L.
127. Are the hexagons in Figure 152 similar hexagons?
6 cm
4 cm
120˚ 120˚
4 cm
120˚
4 cm
120˚
4 cm
120˚
G
6 cm
6 cm
120˚
120˚ 120˚
4 cm
120˚
4 cm
K
120˚
6 cm
6 cm
120˚
100˚ I
120˚
L
6 cm
H
Figure 152
128. Are the pentagons in Figure 153 similar pentagons?
5 ft
108˚
108˚
108˚
5 ft
108˚
5 ft
108˚
3 ft
108˚
135. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle H.
108˚
3 ft
3 ft
108˚
∆GHI ~ ∆JKL
Figure 155
134. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle G.
5 ft
3 ft
J
25˚
108˚ 108˚
3 ft
5 ft
Figure 153
129. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF.
Find the measure of angle D.
136. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle K.
137. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF.
Find this ratio:
___
length of BC
length of EF
D
8 cm
B
D
E
10 cm
B
80˚
E
A
A
35˚
C
C
∆ABC ~ ∆DEF
F
∆ABC ~ ∆DEF
Figure 154
130. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF
Find the measure of angle B.
131. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF.
Find the measure of angle C.
132. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF.
Find the measure of angle F.
F
Figure 156
138. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF.
Find this ratio:
___
length of DF
length of AC
139. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF.
Find this ratio:
___
length of EF
length of BC
140. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF.
Find this ratio:
___
length of AC
length of DF
LESSON F5.3 GEOMETRY III HOMEWORK
523
141. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL.
Find this ratio:
__
length of HI
length of KL
147. In Figure 158, the triangles are similar. Complete this
similarity statement: ∆KAF ~ ∆_____
148. In Figure 158, the triangles are similar. Complete this
similarity statement: ∆KFA ~ ∆_____
149. In Figure 159, the triangles are similar. Complete this
similarity statement: ∆USA ~ ∆_____
I
J
3 ft
U
4 ft
L
K
G
A
∆GHI ~ ∆JKL
S
H
W
R
Figure 157
142. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL.
Find this ratio:
__
length of JK
length of GH
Figure 159
143. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL.
Find this ratio:
__
length of KL
length of HI
144. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL.
Find this ratio:
__
length of GH
length of JK
Writing a Similarity Statement
For help working these types of problems, go back to Examples
28–30 in the Explain section of this lesson.
145. In Figure 158, the triangles are similar. Complete this
similarity statement: ∆AFK ~ ∆_____
T
A
F
Y
P
Figure 158
146. In Figure 158, the triangles are similar. Complete this
similarity statement: ∆FAK ~ ∆_____
TOPIC F5 GEOMETRY
150. In Figure 159, the triangles are similar. Complete this
similarity statement: ∆ASU ~ ∆_____
151. In Figure 159, the triangles are similar. Complete this
similarity statement: ∆SUA ~ ∆_____
152. In Figure 159, the triangles are similar. Complete this
similarity statement: ∆UAS ~ ∆_____
153. ∆LAW ~ ∆YER
Find the angle in ∆LAW that has the same measure as
angle Y.
154. ∆LAW ~ ∆YER
Find the angle in ∆LAW that has the same measure as
angle E.
155. ∆LAW ~ ∆YER
Find the angle in ∆LAW that has the same measure as
angle R.
156. ∆LAW ~ ∆YER
Find the angle in ∆YER that has the same measure as
angle A.
157. ∆PUT ~ ∆DWN
Find the angle in ∆DWN that has the same measure as
angle P.
K
524
E
158. ∆PUT ~ ∆DWN
Find the angle in ∆DWN that has the same measure as
angle T.
159. ∆PUT ~ ∆DWN
Find the angle in ∆DWN that has the same measure as
angle U.
170. Are the triangles in Figure 161 similar?
T
160. ∆PUT ~ ∆DWN
Find the angle in ∆PUT that has the same measure as
angle D.
161. ∆LAW ~ ∆YER
The ratio of LA to YE is 7 to 3. Find the ratio of AW to ER.
P
162. ∆LAW ~ ∆YER
The ratio of LA to YE is 7 to 3. Find the ratio of LW to YR.
42˚
S
W
163. ∆LAW ~ ∆YER
The ratio of LA to YE is 7 to 3. Find the ratio of ER to AW.
164. ∆LAW ~ ∆YER
The ratio of LA to YE is 7 to 3. Find the ratio of YR to LW.
165. ∆CAT ~ ∆NIP
The ratio of CT to NP is 9 to 4. Find the ratio of CA to NI.
166. ∆CAT ~ ∆NIP
The ratio of CT to NP is 9 to 4. Find the ratio of AT to IP.
42˚
Figure 161
171. Are the triangles in Figure 162 similar?
A
C
E
167. ∆CAT ~ ∆NIP
The ratio of CT to NP is 9 to 4. Find the ratio of NI to CA.
168. ∆CAT ~ ∆NIP
The ratio of CT to NP is 9 to 4. Find the ratio of IP to AT.
D
F
55˚
55˚
B
F
D
Figure 162
Shortcuts for Recognizing Similar
Triangles
For help working these types of problems, go back to Examples
31– 34 in the Explain section of this lesson.
172. Are the triangles in Figure 163 similar?
A
T
45˚
169. Are the triangles in Figure 160 similar?
45˚
A
B
C
C
E
42˚
35˚
V
35˚
Figure 163
B
F
M
D
Figure 160
LESSON F5.3 GEOMETRY III HOMEWORK
525
173. Are the triangles in Figure 164 similar?
177. Complete this similarity statement for the triangles in
Figure 168: ∆IHG ~ ∆_____
G
G
25˚
K
25°
55˚
J
25˚
K
25°
55˚
J
100˚
100˚ I
100˚
100˚ I
L
L
H
Figure 164
H
Figure 168
174. Are the triangles in Figure 165 similar?
178. Complete this similarity statement for the triangles in
Figure 169: ∆BAC ~ ∆_____
N
52˚
A
B 100˚
C
28˚
E
E
35˚
100˚
H
35˚
C
25˚
B
U
Figure 165
175. Are the triangles in Figure 166 similar?
B
F
D
35˚
80˚
A
65˚
35˚
D
Figure 169
80˚ E
179. Complete this similarity statement for the triangles in
Figure 170: ∆TWP ~ ∆_____
F
C
T
Figure 166
176. Are the triangles in Figure 167 similar?
G
P
48˚
I
J
42˚
94˚
38˚
H
K
38˚
48˚
S
W
L
Figure 167
42˚
F
Figure 170
526
TOPIC F5 GEOMETRY
D
180. Complete this similarity statement for the triangles in
Figure 171: ∆CBA ~ ∆_____
A
184. Complete this similarity statement for the triangles in
Figure 175: ∆FDC ~ ∆_____
D
C
E
F
P
53˚
N
55˚
55˚
36˚
B
53˚
C
F
D
181. Complete this similarity statement for the triangles in
Figure 172: ∆BAC ~ ∆_____
B
185. Are the triangles in Figure 176 similar?
15 cm
D
35˚
80˚
65˚
9 cm
12 cm
80˚
35˚
L
Figure 175
Figure 171
A
36˚
E
F
C
25 cm
20 cm
Figure 172
182. Complete this similarity statement for the triangles in
Figure 173: ∆GIH ~ ∆_____
15 cm
Figure 176
G
I
48˚
186. Are the triangles in Figure 177 similar?
J
Y
24 ft
94˚
W
38˚
38˚
K
H
48˚
20 ft
L
Figure 173
28 ft
183. Complete this similarity statement for the triangles in
Figure 174: ∆ABE ~ ∆_____
12 ft
E
O
D
36˚
F
53˚
B
Figure 174
10 ft
14 ft
U
Figure 177
53˚
A 91˚
A
T
36˚
C
LESSON F5.3 GEOMETRY III HOMEWORK
527
190. Complete this similarity statement for the triangles in
Figure 181: ∆WAY ~ ∆_____
187. Are the triangles in Figure 178 similar?
Y
24 ft
7 in
6 in
W
20 ft
4 in
3 in
28 ft
5 in
5 in
A
T
Figure 178
12 ft
10 ft
188. Are the triangles in Figure 179 similar?
O
U
14 ft
24 cm
Figure 181
191. Complete this similarity statement for the triangles in
Figure 182: ∆CJW ~ ∆_____
10 cm
26 cm
24 cm
C
13 cm
12 cm
W
M
10 cm
26 cm
5 cm
Figure 179
J
189. Complete this similarity statement for the triangles in
Figure 180: ∆AKE ~ ∆_____
15 cm
W
E
C
12 cm
9 cm
T
25 cm
A
15 cm
E
Figure 180
528
G
192. Complete this similarity statement for the triangles in
Figure 183: ∆BCA ~ ∆_____
12 in
K
5 cm
Figure 182
A
20 cm
13 cm
12 cm
TOPIC F5 GEOMETRY
U
7 in
10 in
R
B
14 in
C
Figure 183
6 in
5 in
V
Measurements of Corresponding
Angles of Similar Triangles
For help working these types of problems, go back to Example 35
in the Explain section of this lesson.
201. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure
of angle B.
V
B
193. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV.
Find the measure of angle V.
C
V
R
T
58˚
32˚
U
∆ABC ~ ∆UTV
A
Figure 186
36˚
Y
E
∆RYE ~ ∆CBV
202. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure
of angle U.
C
B
Figure 184
203. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure
of angle C.
194. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV.
Find the measure of angle Y.
204. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure
of angle V.
195. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV.
Find the measure of angle C.
205. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO.
Find the measure of angle B.
196. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV.
Find the measure of angle R.
O
197. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure
of angle B.
P
G
B
C
54˚
R 81˚
A
L
∆BCA ~ ∆RLO
T
Figure 187
H
48˚
54˚
L
∆GHL ~ ∆TBP
B
Figure 185
198. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure
of angle P.
206. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO.
Find the measure of angle A.
207. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO.
Find the measure of angle O.
208. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO.
Find the measure of angle L.
199. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure
of angle G.
200. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure
of angle T.
LESSON F5.3 GEOMETRY III HOMEWORK
529
209. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF.
Find the measure of of angle B.
For help working these types of problems, go back to Examples
36–38 in the Explain section of this lesson.
D
B
E
65˚
A
35˚
C
Lengths of Corresponding Sides of
Similar Triangles
217. The triangles in Figure 190 are similar. ∆WAY ~ ∆OUT.
Find x, the length of UT.
F
Y
12 ft
∆ABC ~ ∆DEF
W
10 ft
Figure 188
14 ft
210. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF.
Find the measure of angle C.
A
T
211. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF.
Find the measure of angle D.
6 ft
O
212. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF.
Find the measure of angle E.
213. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle G.
y
U
Figure 190
218. The triangles in Figure 190 are similar. ∆WAY ~ ∆OUT.
Find y, the length of OU.
219. The triangles in Figure 191 are similar. ∆CJW ~ ∆EGM.
Find x, the length of CJ.
G
K
J
55˚
C
24 cm
W
x
100˚ I
M
26 cm
L
H
∆WAY ~ ∆OUT
x
∆GHI ~ ∆JKL
J
y
13 cm
Figure 189
214. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle H.
215. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle J.
216. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL.
Find the measure of angle L.
530
TOPIC F5 GEOMETRY
∆CJW ~ ∆EGM
E
5 cm
G
Figure 191
220. The triangles in Figure 191 are similar. ∆CJW ~ ∆EGM.
Find y, the length of EM.
221. The triangles in Figure 192 are similar. ∆AKE ~ ∆WTC.
Find x, the length of AK.
15 cm
W
225. The triangles in Figure 194 are similar. ∆ABC ~ ∆ADE.
Find x, the length of BC.
A
C
A
y
12 cm
5 cm
T
B
25 cm
C
x
∆ABC ~ ∆ADE
∆AKE ~ ∆WTC
x
20 cm
K
E
15 cm
Figure 192
222. The triangles in Figure 192 are similar. ∆AKE ~ ∆WTC.
Find y, the length of TC.
223. The triangles in Figure 193 are similar. ∆BKR ~ ∆DPC.
Find x, the length of PC.
5 cm
D
B
y
C
x
D
Figure 194
226. The triangles in Figure 195 are similar. ∆JKL ~ ∆JMN.
Find x, the length of MN.
6 ft
P
10 cm
6 cm
E
12 cm
J
2 ft
L
N
1 ft
K
R
8 cm
∆BKR ~ ∆DPC
Figure 193
224. The triangles in Figure 193 are similar. ∆BKR ~ ∆DPC.
Find y, the length of DP.
K
x
∆JKL ~ ∆JMN
M
Figure 195
LESSON F5.3 GEOMETRY III HOMEWORK
531
227. The triangles in Figure 196 are similar. ∆ABC ~ ∆ADE.
Find x, the length of AB.
229. The triangles in Figure 198 are similar. ∆ABC ~ ∆ADE.
Find x, the length of BC.
A
A
x
∆ABC ~ ∆ADE
∆ABC ~ ∆ADE
15 cm
20 cm
C
B
6 cm
20 cm
C
B
D
E
12 cm
Figure 196
x
D
Figure 198
228. The triangles in Figure 197 are similar. ∆JKL ~ ∆JMN.
Find x, the length of KL.
230. The triangles in Figure 199 are similar. ∆PQR ~ ∆PST.
Find x, the length of QR.
6 ft
J
12 ft
L
4 ft
E
12 cm
N
8 ft
P
x
R
T
x
3 ft
6 ft
∆JKL ~ ∆JMN
∆PQR ~ ∆PST
K
Q
M
S
Figure 197
Figure 199
231. The triangles in Figure 200 are similar. ∆PQR ~ ∆PST.
Find x, the length of PR.
12 ft
P
x
R
T
2 ft
Q
6 ft
∆PQR ~ ∆PST
S
Figure 200
532
TOPIC F5 GEOMETRY
232. The triangles in Figure 201 are similar. ∆TUV ~ ∆TMN.
Find x, the length of UT.
18 ft
8 ft
P
U
3 ft
M
235. The triangles in Figure 204 are similar. ∆PQR ~ ∆PST.
Find x, the length of RT.
x
x
R
T
4 ft
T
Q
V
9 ft
12 ft
∆PQR ~ ∆PST
S
∆TUV ~ ∆TMN
Figure 204
N
Figure 201
233. The triangles in Figure 202 are similar. ∆ABC ~ ∆ADE.
Find x, the length of BD.
236. The triangles in Figure 205 are similar. ∆PQR ~ ∆PST.
Find x, the length of RT.
8 cm
P
R
x
T
A
4 cm
∆PQR ~ ∆PST
6 cm
Q
S
∆ABC ~ ∆ADE
Figure 205
15 cm
237. The triangles in Figure 206 are similar. ∆JKL ~ ∆JMN.
Find x, the length of LJ.
M
B
C
9 cm
∆JKL ~ ∆JMN
x
D
18 ft
E
12 cm
K
Figure 202
234. The triangles in Figure 203 are similar. ∆FGH ~ ∆FIJ.
Find x, the length of GI.
I
x
G
5 cm
6 ft
N
8 ft
L x
J
Figure 206
F
3 cm
6 cm
H
∆FGH ~ ∆FIJ
J
Figure 203
LESSON F5.3 GEOMETRY III HOMEWORK
533
238. The triangles in Figure 207 are similar. ∆PEA ~ ∆POD.
Find x, the length of EO.
O
240. The triangles in Figure 209 are similar. ∆PEA ~ ∆POD.
Find x, the length of PA.
O
∆PEA ~ ∆POD
x
E
E
6 cm
18 ft
P
10 cm
A
5 cm
D
239. The triangles in Figure 208 are similar. ∆JKL ~ ∆JMN.
Find x, the length of NL.
M
∆JKL ~ ∆JMN
K
9 cm
6 cm
x
L
4 cm
J
Figure 208
534
P
x
A
Figure 209
Figure 207
N
∆PEA ~ ∆POD
9 ft
TOPIC F5 GEOMETRY
15 ft
D
Evaluate
Take this Practice Test to
prepare for the final quiz in
the Evaluate module of this
lesson on the computer.
Practice Test
1.
One of the angles in a right triangle measures 10˚. Find the measure of the other acute
angle of the triangle.
2.
In Figure 210, ∆ABC is congruent to ∆DEF.
length AB = length BC
m∠A = 71˚
Find the measure of ∠E.
E
B
F
C
71˚
D
A
Figure 210
3.
In Figure 211:
∆JKL is an equilateral triangle.
length LK = 24 cm
h = 123 cm
Find x, the length of JM.
K
h
J
x
M
Figure 211
L
LESSON F5.3 GEOMETRY III EVALUATE
535
4.
In Figure 212:
line AB is parallel to line CD
line AC is parallel to line BD
length AB = 8.1 cm
length AC = 10.2 cm
m∠ACD = 54˚
Find length BD, m∠ABD, and m∠BDC.
A
C
D
B
Figure 212
5.
The length of each side of a given pentagon measures 2 in. A larger pentagon which is
similar to the given pentagon has a side that measures 2.4 in. Find the perimeter of the
larger pentagon.
6.
Triangle ABC is similar to triangle DEF.
AB = 14 cm
BC = 15 cm
AC = 8 cm
DE = 42 cm
Find x, the length of EF. (Hint: Sketch the two triangles, making sure to match
corresponding sides.)
7.
In Figure 213:
AB is parallel to DE
AB = 130 cm
BC = 117 cm
DE = 390 cm
Find y, the length of DC.
E
B
117 cm
390 cm
130 cm
C
A
y
D
Figure 213
536
TOPIC F5 GEOMETRY
8.
In Figure 214:
∆CED ~ ∆AEB
EC = 21 cm
CD = 18.9 cm
AB = 37.8 cm
Find x, the length of EA.
37.8 cm
A
18.9 cm
C
B
D
21 cm
E
Figure 214
LESSON F5.3 GEOMETRY III EVALUATE
537
538
TOPIC F5 GEOMETRY
Topic F5 Cumulative Review
These problems cover the
material from this and
previous topics. You may
wish to do these problems
to check your understanding
of the material before you
move on to the next topic,
or to review for a test.
1
2
1
4
1.
Find: 180° – (48 ° + 22 °)
2.
In Figure F5.1, find the measure of F.
S
30°
U 100°
?
F
Figure F5.1
3.
In Figure F5.2, find the measure of
angle A.
(Hint: ∆PLA is an equilateral triangle.)
L
2 ft
P
2 ft
2 ft
?
A
4.
5.
Figure F5.2
Find: 22.5 cm + 10.2 cm + 22.5 cm + 10.2 cm
In Figure F5.3, name a point, a line
segment, a ray, and a line.
P
R
Q
Figure F5.3
1
3
6.
Use the order of operations to do this calculation: (4.4 + 4.6) + 4
7.
Melissa is building a fence around her rectangular garden. The garden is 16 yards
long and 10 yards wide. How many yards of fence must she build in order to
completely enclose the garden?
8.
Find: 180° – (32.4° + 27.8°)
9.
Name the polygon that has 8 sides.
1
10. Maurice is painting the ceiling in his living room. The ceiling is 5 yards by
3
6 yards. How many square yards will he have to paint to give the
entire ceiling one coat of paint?
11. Find the area of the parallelogram
shown in Figure F5.4.
5 ft
4 ft
5 ft
3 ft
Figure F5.4
TOPIC F5 CUMULATIVE REVIEW
539
12. Give the degree measure of a right angle.
13. If an angle measures 100.2°, is it an acute, obtuse, right, or straight angle?
14. In Figure F5.6, find the length of ZX.
(Hint: ∆XYZ is an isosceles triangle.)
Z
5 cm
?
Y 35°
Figure F5.6
35°
X
15. The most popular item of a pennant manfacturer is the classic triangular felt pennant.
If this pennant is 16.5 inches long from the center of its base to its tip and its base is 8
inches long, how many square inches of felt are needed to make the pennant?
16. In Figure F5.7, f || g. The measure
of angle 8 is 75°. Find the measure
of angle 2.
t
f
1
3
2
4
g
5
7
6
8
75°
Figure F5.7
17. In Figure F5.8, find the measure
of Y.
H
96°
?
58°
Y
Figure F5.8
18. In a right triangle, the measure of one angle is 62°. Find the measures of the other
angles.
19. The quadrilateral in Figure F5.9
is a parallelogram. Find the
measure of angle L.
F
135°
P
?
I
Figure F5.9
1
3
20. If an angle measures 33 °, is it an acute, obtuse, right, or straight angle?
540
TOPIC F5 GEOMETRY
L
21. Find the area of the trapezoid
shown in Figure F5.10.
10 ft
6 ft
8 ft
Figure F5.10
22. If mU = 82.5° and mV = 97.5°, are angles U and V complementary, supplementary, or neither?
23. The radius of a circle is 18 inches. Find its circumference and its area.
24. In Figure F5.11, what is the measure
of BAC?
C
D
74°
A
B
E
Figure F5.11
25. ∆ABC is congruent to ∆DEF. Which angle of ∆DEF is congruent to F?
26. The lengths of the sides of a certain right triangle are 80 cm, 84 cm, and 116 cm. Find
the length of the hypotenuse.
27. Bud is covering the surface of a gift box with gold foil. The box is a rectangular
prism that is 4 inches tall, 5 inches wide, and 8 inches long. How many square inches
of foil are needed to cover the entire surface of the box with no overlap?
28. Al is building a trunk to store bedding and other items. He figures that a trunk with a
volume of 9 cubic feet should be large enough. If he builds the trunk so that it is 2
feet high and 3 feet long, how wide should he make the trunk so that it has a volume
of 9 cubic feet?
29. ∆ABC is congruent to ∆DEF. Which side of ∆DEF is congruent to BC?
30. The radius of a cylinder is 2 inches. The height of the cylinder is 6 inches. What is
the surface area of the cylinder?
31. The radius of a cylinder is 2 inches. The height of the cylinder is 6 inches. What is
the volume of the cylinder? Use 3.14 to approximate . Round your answer to two
decimal places.
32. Find the value of this expression: –122 – 82
33. The lengths of the sides of ∆USA are 9 ft, 10 ft, and 12 ft. Is ∆USA a right triangle?
34. The diameter of the base of a cone is 3 centimeters. The height of the cone is 12 centimeters. What is the volume of the cone? Use 3.14 to approximate . Round your
answer to two decimal places.
TOPIC F5 CUMULATIVE REVIEW
541
35. Are the polygons in Figure F5.12
12 ft
similar polygons?
8 ft
16 ft
6 ft
Figure F5.12
36. ∆FAR ~ ∆OUT. (Triangle FAR is similar to triangle OUT.) Find the angle in ∆OUT
that has the same measure as angle A.
37. ∆FAR ~ ∆OUT. The ratio of FA to OU is 8 to 3. Find the ratio of AR to UT.
38. Complete this similarity
statement for the triangles in
Figure F5.13:
∆SRC ~ ∆_____
R
U
10 cm
12 cm
7 cm
R
S
C
14 cm
5 cm
6 cm
V
Figure F5.13
39. The triangles in Figure F5.14 are
similar. ∆GHI ~ ∆JKL. Find the
measure of angle G.
G
K
55°
H
J
100°
I
L
∆GHI ~ ∆ JKL
Figure F5.14
2 12
40. Find the missing number, x, that makes this proportion true: = .
5
x
41. The triangles in Figure F5.15 are
similar. ∆PQR ~ ∆PST. Find x,
the length of PR.
P
x
6 ft
R
Q
∆PQR ~ ∆PST
Figure F5.15
542
TOPIC F5 GEOMETRY
T
1 ft
3 ft
S
42. The triangles in Figure. F5.16 are
similar. ∆JKL ~ ∆JMN Find x, the
length of NL.
M
K
18 cm
∆JKL ~ ∆JMN
12 cm
N x
L 8 cm
J
Figure F5.16
43. The volume of a sphere is 36 cubic feet. What is the radius of the sphere?
TOPIC F5 CUMULATIVE REVIEW
543
544
TOPIC F5 GEOMETRY