Download 6.4 Prove Triangles Similar by AA

6.4
Prove Triangles Similar by AA
p Use the AA Similarity Postulate.
Goal
Your Notes
POSTULATE 22: ANGLE-ANGLE (AA) SIMILARITY
POSTULATE
K
L
If two angles of one triangle are
congruent to two angles of
another triangle, then the
two triangles are similar.
Z
Y
J
X
nJKL , nXYZ
Use the AA Similarity Postulate
Example 1
Determine whether the triangles
are similar. If they are, write a
similarity statement. Explain
your reasoning.
E
D
228
A
F
Solution
B
Because they are both right angles, ∠
are congruent.
688
C
and ∠
By the Triangle Sum Theorem,
1
1 m∠A 5 1808, so m∠A 5
Therefore, ∠A and ∠
are congruent.
.
So, nABC , nDEF by the
.
Checkpoint Determine whether the triangles are
similar. If they are, write a similarity statement.
1.
P
R
2.
R
718
788
G
538
318
278
P
Q
G
H
1108
278
788
Q
Copyright © Holt McDougal. All rights reserved.
F
F
H
Lesson 6.4 • Geometry Notetaking Guide
159
6.4
Prove Triangles Similar by AA
p Use the AA Similarity Postulate.
Goal
Your Notes
POSTULATE 22: ANGLE-ANGLE (AA) SIMILARITY
POSTULATE
K
L
If two angles of one triangle are
congruent to two angles of
another triangle, then the
two triangles are similar.
Z
Y
J
X
nJKL , nXYZ
Use the AA Similarity Postulate
Example 1
Determine whether the triangles
are similar. If they are, write a
similarity statement. Explain
your reasoning.
E
228
A
D
F
688
Solution
C
B
Because they are both right angles, ∠ B and ∠ E
are congruent.
By the Triangle Sum Theorem,
688 1 908 1 m∠A 5 1808, so m∠A 5 228 .
Therefore, ∠A and ∠ D are congruent.
So, nABC , nDEF by the AA Similarity Postulate .
Checkpoint Determine whether the triangles are
similar. If they are, write a similarity statement.
1.
P
R
2.
R
718
788
G
538
318
278
P
Q
G
H
1108
278
788
Q
F
nFGH , nRQP
Copyright © Holt McDougal. All rights reserved.
F
H
not similar
Lesson 6.4 • Geometry Notetaking Guide
159
Your Notes
Show that triangles are similar
Example 2
Show that the two triangles are similar.
a. nRTV and nRQS
b. nLMN and nNOP
M
R
O
T 498
V
498
Q
L
S
N
P
Solution
a. You may find it helpful to redraw the triangles
separately.
and m∠
both equal 498,
. By the Reflexive Property,
Because m∠
∠
>∠
∠R > ∠
.
So, nRTV , nRQS by the
b. The diagram shows ∠L > ∠
}
MN i
so ∠
>∠
Angles Postulate.
.
. It also shows that
by the Corresponding
So, nLMN , nNOP by the
.
Checkpoint Complete the following exercise.
3. Show that nBCD , nEFD.
F
C
D
B
160
Lesson 6.4 • Geometry Notetaking Guide
E
Copyright © Holt McDougal. All rights reserved.
Your Notes
Show that triangles are similar
Example 2
Show that the two triangles are similar.
a. nRTV and nRQS
b. nLMN and nNOP
M
R
O
T 498
V
498
Q
L
S
N
P
Solution
a. You may find it helpful to redraw the triangles
separately.
Because m∠ RTV and m∠ Q both equal 498,
∠ RTV > ∠ Q . By the Reflexive Property,
∠R > ∠ R .
So, nRTV , nRQS by the AA Similarity Postulate .
b. The diagram shows ∠L > ∠ ONP . It also shows that
}
}
MN i OP so ∠ LNM > ∠ P by the Corresponding
Angles Postulate.
So, nLMN , nNOP by the AA Similarity Postulate .
Checkpoint Complete the following exercise.
3. Show that nBCD , nEFD.
F
C
D
B
E
Because they are both right angles, ∠C > ∠F.
You know that ∠CDB > ∠FDE by the Vertical
Angles Congruence Theorem.
So, nBCD , nEFD by the AA Similarity
Postulate.
160
Lesson 6.4 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 3
Using similar triangles
Height A lifeguard is standing beside the lifeguard chair
on a beach. The lifeguard is 6 feet 4 inches tall and
casts a shadow that is 48 inches long. The chair casts a
shadow that is 6 feet long. How tall is the chair?
Solution
The lifeguard and the chair form sides of two right
triangles with the ground, as shown below. The sun’s rays
hit the lifeguard and the chair at the same angle. You
, so the triangles
have two pairs of congruent
are similar by the
.
6 ft 4 in.
x ft
48 in.
6 ft
You can use a proportion to find the height x. Write 6 feet
4 inches as
inches so you can form two ratios of
feet to inches.
x ft
5
Write proportion of side lengths.
x5
Cross Products Property
x5
Solve for x.
The chair is
feet tall.
Checkpoint Complete the following exercise.
4. In Example 3, how long is the shadow of a person
that is 4 feet 9 inches tall?
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 6.4 • Geometry Notetaking Guide
161
Your Notes
Example 3
Using similar triangles
Height A lifeguard is standing beside the lifeguard chair
on a beach. The lifeguard is 6 feet 4 inches tall and
casts a shadow that is 48 inches long. The chair casts a
shadow that is 6 feet long. How tall is the chair?
Solution
The lifeguard and the chair form sides of two right
triangles with the ground, as shown below. The sun’s rays
hit the lifeguard and the chair at the same angle. You
have two pairs of congruent angles , so the triangles
are similar by the AA Similarity Postulate .
6 ft 4 in.
x ft
48 in.
6 ft
You can use a proportion to find the height x. Write 6 feet
4 inches as 76 inches so you can form two ratios of
feet to inches.
x ft
5
76 in.
6 ft
Write proportion of side lengths.
48 in.
48 x 5 456
x 5 9.5
Cross Products Property
Solve for x.
The chair is 9.5 feet tall.
Checkpoint Complete the following exercise.
4. In Example 3, how long is the shadow of a person
that is 4 feet 9 inches tall?
Homework
3 feet
Copyright © Holt McDougal. All rights reserved.
Lesson 6.4 • Geometry Notetaking Guide
161