4.3 Isosceles and Equilateral Triangles 185
Goal
Use properties of isosceles
and equilateral triangles.
Key Words
• legs of an isosceles
triangle
• base of an isosceles
triangle
• base angles
4.3
4.3
Isosceles and Equilateral
Triangles
The Geo-Activity shows that two angles of an
isosceles triangle are always congruent. These
angles are opposite the congruent sides.
The congruent sides of an isosceles triangle
are called .
The other side is called the .
The two angles at the base of the triangle
are called the .base angles
base
legs
●
3 Repeat Steps 1 and 2 for different isosceles triangles. What can
you say about a
H and aK in the different triangles?
●
1 Fold a sheet of paper in half.
●
2 Unfold and label the angles
Use a straightedge to draw a as shown. Use a protractor
line from the fold to the bottom to measure a
H and aK.
edge. Cut along the line to form What do you notice?
an isosceles triangle.
Properties of Isosceles Triangles
Geo-Activity
Base Angles Theorem
Words If two sides of a triangle are
congruent, then the angles opposite
them are congruent.
Symbols If AB
&*
c
AC&*, then aC c aB.
A
B C
THEOREM 4.3
V
OCABULARY
T
IP
Isos- means “equal,”
and -sceles means
“leg.” So, isosceles
means equal legs.
Student Help
H
J
K
Isosceles Triangle
leg leg
base
angles
base
Page 1 of 6
186 Chapter 4 Triangle Relationships
Find the measure of aL.
Solution
Angle L is a base angle of an isosceles
triangle. From the Base Angles
Theorem, aL and aN have the
same measure.
ANSWER
©
The measure of aL is 528.
Find the value of y.
1. 2. 3.
16
y 1 4
9y
508
y8
EXAMPLE
1
Use the Base Angles Theorem
Converse of the Base Angles Theorem
Words If two angles of a triangle are congruent,
then the sides opposite them are
congruent.
Symbols If aB c aC, then AC&* c AB&*.
THEOREM 4.4
Find the value of x.
Solution
By the Converse of the Base Angles Theorem,
the legs have the same length.
DE 5 DF
Converse of the Base Angles Theorem
x 1 3 5 12 Substitute x 1 3 for DE and 12 for DF.
x 5 9 Subtract 3 from each side.
ANSWER
©
The value of x is 9.
EXAMPLE
2
Use the Converse of the Base Angles Theorem
Use Isosceles Triangle Theorems
Rock and Roll Hall of Fame,
Cleveland, Ohio
12
x 1 3
F
E
D
528528??
MM
NN
LL
Base angles don’t have
to be on the bottom of
an isosceles triangle.
A
B
C
Visualize It!
Page 2 of 6
4.3 Isosceles and Equilateral Triangles 187
4.5 Equilateral Theorem
Words If a triangle is equilateral, then it is equiangular.
Symbols If AB&* c AC&* c BC&*, then aA c aB c aC.
4.6 Equiangular Theorem
Words If a triangle is equiangular, then it is equilateral.
Symbols If aB c aC c aA, then AB&* c AC&* c BC&*.
THEOREMS 4.5 and 4.6
Find the length of each side of the equiangular triangle.
Solution
The angle marks show that TQRT is
equiangular. So, TQRT is also equilateral.
3x 5 2x 1 10
x 5 10
Subtract 2x from each side.
3(10) 5 30 Substitute 10 for x.
ANSWER
©
Each side of TQRT is 30.
Sides of an equilateral T
are congruent.
P
2x 1 10 T
R
3x
EXAMPLE
3
Find the Side Length of an Equiangular Triangle
L
OOK
B
ACK
For the definition of
equilateral triangle,
see p. 173.
Student Help
A
B
C
Constructing an Equilateral Triangle
You can construct an equilateral
triangle using a straightedge and compass.
By the Triangle Sum Theorem, the measures of the three congruent
angles in an equilateral triangle must add up to 1808. So, each angle
in an equilateral triangle measures 608.
●
1 Draw AB
&*
. Draw
an arc with center A
that passes through B.
●
2 Draw an arc with
center B that
passes through A.
●
3 The intersection of
the arcs is point C.
TABC is equilateral.
A
B
A
B
A
B
C
60°
60° 60°
Page 3 of 6
1. What is the difference between equilateral and equiangular?
Tell which sides and angles of the triangle are congruent.
2. 3. 4.
Find the value of x. Tell what theorem(s) you used.
5. 6.
Finding Measures
Find the value of x. Tell what theorem(s)
you used.
7. 8. 9.
Using Algebra
Find the value of x.
10. 11. 12.
13. 14. 15.
3x8
4
x8
(5x 1 7)8
528
7x 1 5 19
x
13
6x
12
x 1 411
D
E
F
x
8
G
H J
688
x
8
A
55
8
B
C
x
8
Practice and Applications
VW
U
ST
R
NL
M
Skill Check
Vocabulary Check
Guided Practice
188 Chapter 4 Triangle Relationships
Exercises
4.3
4.3
Example 1: Exs. 7–9, 14,
15, 17–19,
27, 28
Example 2: Exs. 10–13
Example 3: Exs. 20–25
Homework Help
Extra Practice
See p. 681.
x
8
x
8
508508
8.8 cm8.8 cm
x
cm
x
cm
Page 4 of 6
16. Someone in your class tells you that all
equilateral triangles are isosceles triangles. Do you agree? Use
theorems or definitions to support your answer.
Using Algebra
Find the measure of aA.
17. 18. 19.
Using Algebra
Find the value of y.
20. 21. 22.
23. 24. 25.
26.
Challenge
In the diagram at the right, TXYZ is
equilateral and the following pairs of segments
are parallel: XY
&*
and LK
&*
; ZY
&*
and LJ
&
; XZ
&*
and JK
&
.
Describe a plan for showing that TJKL must
be equilateral.
Rock Climbing
In one type of rock climbing, climbers tie themselves
to a rope that is supported by anchors. The diagram shows a red and
a blue anchor in a horizontal slit in a rock face.
27. If the red anchor is longer than the blue
anchor, are the base angles congruent?
28. If a climber adjusts the anchors so they
are the same length, do you think that
the base angles will be congruent?
Why or why not?
X
J
Y
Z
K
L
4y 1 2
8
y 2 10
5y 2 14
3y
5
y
2y 1 5
4y 2 3
10
2
y
y 1 5
11
y
x 8
A
C
B
2x 8
x 8
CA
B
508
x 8
C
B
A
308
x 8
You be the Judge
4.3 Isosceles and Equilateral Triangles 189
ROCK CLIMBING The
climber is using a method of
rock climbing called top
roping. If the climber slips, the
anchors catch the fall.
Application Links
C LAS S Z O NE. C O M
Sports
H
OMEWORK
H
ELP
Extra help with problem
solving in Exs. 17–19 is
at
classzone
.com
IStudent Help
I CL A S SZO N E.C O M
Page 5 of 6
Tiles
In Exercises 29–31, use the diagram at the left. In the diagram,
VX
&**
c
WX
&***
c
YX
&*
c
ZX
&*
.
29. Copy the diagram. Use what you know about side lengths
to mark your diagram.
30. Explain why aXWV c aXVW.
31. Name four isosceles triangles.
32.
Technology
Use geometry software
to complete the steps.
●
1 Construct circle A.
●
2 Draw points B and C on the circle.
●
3 Connect the points to form TABC.
Is TABC isosceles? Measure the sides
of the triangle to check your answer.
Multiple Choice
In Exercises 33 and 34, use the diagram below.
33. What is the measure of aEFD?
X
A 558
X
B 658
X
C 1258
X
D 1808
34. What is the measure of
aDEF ?
X
F 508
X
G 708
X
H 1258
X
J 1808
Angle Bisectors
BE
&(
is the angle bisector. Find
maDBC and maABC.
(Lesson 2.2)
35. 36. 37.
Vertical Angles
Find the value of the variable. (Lesson 2.4)
38. 39. 40.
Evaluating Square Roots
Evaluate. (Skills Review, p. 668)
41.
Ï
4
w
9
w
42.
Ï
1
w
2
w
1
w
43.
Ï
1
w
44.
Ï
4
w
0
w
0
w
Algebra Skills
818
(2
x 1 1)8
428
(
x 2 8)8
558
(x 1 20)8
C
B D
E
758
A
A
B D
E
568
C
A
B C
E
428
D
Mixed Review
D
E
F G
1258
Standardized Test
Practice
190 Chapter 4 Triangle Relationships
A
B
C
W
X
YZ
V
Page 6 of 6