Download Practice A3

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
4-1
PERIOD
NAME
DATE
4-1
Practice
Word Problem Practice
Classifying Triangles
Classifying Triangles
1. MUSEUMS Paul is standing in front of
a museum exhibition. When he turns his
head 60° to the left, he can see a statue
by Donatello. When he turns his head
60° to the right, he can see a statue by
Della Robbia. The two statues and Paul
form the vertices of a triangle. Classify
this triangle as acute, right, or obtuse.
Classify each triangle as acute, equiangular, obtuse, or right.
1.
2.
100°
40°
40°
obtuse
3.
85°
30°
65°
90°
60°
30°
acute
right
4. BOOKENDS Two bookends are shaped
like right triangles.
obtuse
Classify each triangle in the figure at the right by its angles and sides.
5. ABC
equiangular; equilateral
right; scalene
6. EDC
E
A
D
2. PAPER Marsha cuts a rectangular piece
of paper in half along a diagonal. The
result is two triangles. Classify these
triangles as acute, right, or obtuse.
C
7. BDC
right; scalene
equilateral
obtuse; isosceles
9. LMN is an isosceles triangle, with LM = LN, LM = 3x - 2, LN = 2x + 1, and
MN = 5x - 2.
x = 3; LM = 7, LN = 7, MN = 13
Find the measures of the sides of KPL and classify each triangle by its sides.
10. K(-3, 2), P(2, 1), L(-2, -3)
, LK = √26
; isosceles
KP = √
26 , PL = 4 √2
11. K(5, -3), P(3, 4), L(-1, 1)
53 , PL = 5, LK = 2 √
13 ; scalene
KP = √
12. K(-2, -6), P(-4, 0), L(3, -1)
KP = 2 √
10 , PL = 5 √
2 , LK = 5 √
2 ; isosceles
Glencoe Geometry
13. DESIGN Diana entered the design at the right in a logo contest
sponsored by a wildlife environmental group. Use a protractor.
How many right angles are there? 5
Chapter 4
001_018_GEOCRMC04_890513.indd 8
8
Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x = 7; FG = 12, GH = 12, FH = 12
5. DESIGNS Suzanne saw this pattern on
a pentagonal floor tile. She noticed many
different kinds of triangles were created
by the lines on the tile.
They are both right triangles.
8. FGH is an equilateral triangle with FG = x + 5, GH = 3x - 9, and FH = 2x - 2.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A3
ALGEBRA For each triangle, find x and the measure of each side.
3. WATERSKIING Kim and Cassandra
are waterskiing. They are holding on to
ropes that are the same length and tied
to the same point on the back of a speed
boat. The boat is going full speed ahead
and the ropes are fully taut.
B
G
A
H
F
C
I
J
E
D
Kim
a. Identify five triangles that appear to
be acute isosceles triangles.
Sample answers: EBD,
CAE, ADC, BGH,
CHI, DIJ, EJF, AGF
Cassandra
Kim, Cassandra, and the point where
the ropes are tied on the boat form the
vertices of a triangle. The distance
between Kim and Cassandra is never
equal to the length of the ropes. Classify
the triangle as equilateral, isosceles, or
scalene.
b. Identify five triangles that appear to
be obtuse isosceles triangles.
Sample answers: AFE,
EJD, CID, BHC, BGA,
AHD, CGE, ACJ
isosceles
Glencoe Geometry
Chapter 4
4/12/08 001_018_GEOCRMC04_890513.indd
12:34:17 AM
9
9
Glencoe Geometry
4/12/08 12:34:26 AM
Answers (Lesson 4-1)
The bottom side of each triangle is
exactly half as long as the slanted side
of the triangle. If all the books between
the bookends are removed and they are
pushed together, they will form a single
triangle. Classify the triangle that can
be formed as equilateral, isosceles, or
scalene.
B
4. ABD
PERIOD
Lesson 4-1
Chapter 4
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
4-2
PERIOD
Study Guide and Intervention
NAME
DATE
4-2
(continued)
Angles of Triangles
Find the measure of each numbered angle.
At each vertex of a triangle, the angle formed by one side
and an extension of the other side is called an exterior angle of the triangle. For each
exterior angle of a triangle, the remote interior angles are the interior angles that are not
adjacent to that exterior angle. In the diagram below, ∠B and ∠A are the remote interior
angles for exterior ∠DCB.
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two remote interior angles.
m∠1 = m∠A + m∠B
Example 1
Find m∠1.
Example 2
1.
S
TIGER
1
2.
146°
1
2
73°
m∠1 = 27
B
m∠1 = m∠2 = 17
Find each measure.
D
1
A
C
Find x.
P
78° Q
80°
80°
3. m∠1
55
4. m∠2
55
5. m∠3
70
85°
55°
1
2
40°
3
1
60°
T
x°
S
Exterior Angle Theorem
Substitution
Simplify.
55°
R
Find each measure.
m∠PQS = m∠R + m∠S
78 = 55 + x
23 = x
Exterior Angle Theorem
Substitution
side.
Find the measures of each numbered angle.
X
1.
A
2.
35°
50°
Y
1
65°
Z
m∠1 = 115
1
3
Q
O
M
R
4.
60°
2
C
D
m∠1 = 60, m∠2 = 120
N
3.
2 1
25°
B
W
80°
V
1
60°
m∠1 = 60, m∠2 = 60, m∠3 = 120
3
2
35°
U
P
S
36°
T
m∠1 = 109, m∠2 = 29, m∠3 = 71
Find each measure.
Glencoe Geometry
5. m∠ABC
6. m∠F
A
B
2x°
x°
Chapter 4
001_018_GEOCRMC04_890513.indd 12
145°
C
D
50
H
12
Answers
58°
125
7. m∠2
55
8. m∠3
1
2
55°
150°
70°
95
Find each measure.
9. m∠1
140
10. m∠2
40
11. m∠3
65
12. m∠4
75
13. m∠5
115
40°
80°
60°
1
4
105°
2
5
3
Find each measure.
B
E
95°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
6. m∠1
3
Subtract 55 from each
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A5
m∠1 = m∠R + m∠S
= 60 + 80
= 140
G
x°
F
29
Glencoe Geometry
14. m∠1
27
15. m∠2
27
Chapter 4
4/12/08 001_018_GEOCRMC04_890513.indd
12:34:41 AM
13
1
A
13
2
D
63° C
Glencoe Geometry
5/22/08 8:04:02 PM
Answers (Lesson 4-2)
S
R
Skills Practice
Angles of Triangles
Exterior Angle Theorem
Exterior Angle
Theorem
PERIOD
Lesson 4-2
Chapter 4
NAME
DATE
4-6
Study Guide and Intervention
PERIOD
NAME
DATE
4-6
(continued)
Isosceles and Equilateral Triangles
Skills Practice
Isosceles and Equilateral Triangles
Properties of Equilateral Triangles
Refer to the figure at the right.
An equilateral triangle has three congruent
sides. The Isosceles Triangle Theorem leads to two corollaries about equilateral triangles.
C
−− −−−
1. If AC AD, name two congruent angles.
1. A triangle is equilateral if and only if it is equiangular.
2. Each angle of an equilateral triangle measures 60°.
P 1
Reasons
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
A
−− −−
EB EA
C
4. If ∠CED ∠CDE, name two congruent segments.
−−
−−
Given
Each ∠ of an equilateral measures 60°.
If lines, then corres. are .
Substitution
If a is equiangular, then it is equilateral.
CE CD
Find each measure.
5. m∠ABC 60
6. m∠EDF 70
#
Exercises
&
ALGEBRA Find the value of each variable.
10
2.
6x - 5
6x°
E
5
3.
20
-
5x
J
H
.
4.
"
#
4x
10
X
5.
12
6
6.
15
3x + 8 60°
40
5
60°
Z
$
4x - 4
Y
Given: ABC is equilateral; ∠1 ∠2.
Prove: ∠ADB ∠CDB
4x°
60°
7
7. PROOF Write a two-column proof.
A
D
1
2
B
Reasons
1. Given
2. An equilateral has sides and angles.
3. Given
4. ABD CBD
5. ∠ADB ∠CDB
4. ASA Postulate
5. CPCTC
C
38
Glencoe Geometry
037_054_GEOCRMC04_890513.indd 38
$
'
%
ALGEBRA Find the value of each variable.
14
7.
2x + 4
3x - 10
Proof:
Statements
−− −−
1. CD CG
2. ∠D ∠G
−− −−
DE GF
CDE CGF
−− −−
CE CF
Chapter 4
4/12/08 037_054_GEOCRMC04_890513.indd
12:35:43 AM
39
21
8.
(2x + 3)°
9. PROOF Write a two-column proof.
−−− −−−
Given: CD CG
−−− −−−
DE GF
−−− −−
Prove: CE CF
3.
4.
5.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Geometry
Proof:
Statements
1. ABC is equilateral.
−−
−−
2. AB CB; ∠A ∠C
3. ∠1 ∠2
Chapter 4
60°
"
,
3x°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F
40°
G
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A18
1.
E
3. If ∠EBA ∠EAB, name two congruent segments.
2 Q
B
D
∠BEC ∠BCE
Answers (Lesson 4-6)
Proof:
Statements
−−− −−−
2. If BE BC, name two congruent angles.
A
Prove that if a line is parallel to one side of an
equilateral triangle, then it forms another equilateral triangle.
D
B
∠ACD ∠CDA
Example
−−− −−−
ABC is equilateral; PQ BC.
m∠A = m∠B = m∠C = 60
∠1 ∠B, ∠2 ∠C
m∠1 = 60, m∠2 = 60
APQ is equilateral.
PERIOD
Lesson 4-6
Chapter 4
NAME
D
E
C
F
G
Reasons
1. Given
2. If 2 sides of a are , then the opposite
those sides are .
3. Given
4. SAS
5. CPCTC
39
Glencoe Geometry
5/22/08 8:06:48 PM