Download 4.3 Isosceles and Equilateral Triangles

Page 1 of 6
4.3
Goal
Use properties of isosceles
and equilateral triangles.
Isosceles and Equilateral
Triangles
Geo-Activity
Properties of Isosceles Triangles
1 Fold a sheet of paper in half.
●
Key Words
• legs of an isosceles
triangle
2 Unfold and label the angles
●
Use a straightedge to draw a
line from the fold to the bottom
edge. Cut along the line to form
an isosceles triangle.
as shown. Use a protractor
to measure aH and aK.
What do you notice?
• base of an isosceles
triangle
• base angles
J
H
K
3 Repeat Steps 1 and 2 for different isosceles triangles. What can
●
you say about aH and aK in the different triangles?
Student Help
VOCABULARY TIP
Isos- means “equal,”
and -sceles means
“leg.” So, isosceles
means equal legs.
The Geo-Activity shows that two angles of an
isosceles triangle are always congruent. These
angles are opposite the congruent sides.
leg
The congruent sides of an isosceles triangle
are called legs .
leg
base
angles
The other side is called the base .
base
The two angles at the base of the triangle
are called the base angles .
Isosceles Triangle
THEOREM 4.3
Base Angles Theorem
Words
A
If two sides of a triangle are
congruent, then the angles opposite
them are congruent.
Symbols
&* c AC
If AB
&*, then aC c aB.
4.3
B
C
Isosceles and Equilateral Triangles
185
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EXAMPLE
1
Use the Base Angles Theorem
Find the measure of aL.
M
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.
528
L
N
Rock and Roll Hall of Fame,
Cleveland, Ohio
THEOREM 4.4
Converse of the Base Angles Theorem
Words
A
If two angles of a triangle are congruent,
then the sides opposite them are
congruent.
Symbols
If aB c aC, then AC
&* c AB
&*.
Visualize It!
EXAMPLE
Base angles don’t have
to be on the bottom of
an isosceles triangle.
Find the value of x.
2
B
Use the Converse of the Base Angles Theorem
F
Solution
C
C
12
By the Converse of the Base Angles Theorem,
the legs have the same length.
D
B
DE 5 DF
A
x 1 3 5 12
x59
ANSWER
x13
Converse of the Base Angles Theorem
Substitute x 1 3 for DE and 12 for DF.
Subtract 3 from each side.
© The value of x is 9.
Use Isosceles Triangle Theorems
Find the value of y.
1.
2.
3.
y
508
186
Chapter 4
Triangle Relationships
y8
16
9
y14
E
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Student Help
THEOREMS 4.5 and 4.6
LOOK BACK
4.5 Equilateral Theorem
For the definition of
equilateral triangle,
see p. 173.
Words
Symbols
If AB
&* c AC
&* c BC
&*, then aA c aB c aC.
4.6 Equiangular Theorem
Words
B
If a triangle is equilateral, then it is equiangular.
A
C
If a triangle is equiangular, then it is equilateral.
Symbols
If aB c aC c aA, then AB
&* c AC
&* c BC
&*.
Constructing an Equilateral Triangle You can construct an equilateral
triangle using a straightedge and compass.
&*. Draw
1 Draw AB
●
2 Draw an arc with
●
an arc with center A
that passes through B.
center B that
passes through A.
3 The intersection of
●
the arcs is point C.
TABC is equilateral.
C
60°
60°
A
B
A
B
60°
A
B
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.
EXAMPLE
3
Find the Side Length of an Equiangular Triangle
Find the length of each side of the equiangular triangle.
R
Solution
The angle marks show that TQRT is
equiangular. So, TQRT is also equilateral.
3x 5 2x 1 10
x 5 10
3(10) 5 30
ANSWER
Sides of an equilateral T
are congruent.
3x
P
2x 1 10
T
Subtract 2x from each side.
Substitute 10 for x.
© Each side of TQRT is 30.
4.3
Isosceles and Equilateral Triangles
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4.3 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. What is the difference between equilateral and equiangular?
Tell which sides and angles of the triangle are congruent.
M
2.
R
3.
L
N
U
4.
T
S
W
V
Find the value of x. Tell what theorem(s) you used.
5.
6.
x cm
8.8 cm
x8
508
Practice and Applications
Extra Practice
Finding Measures Find the value of x. Tell what theorem(s)
you used.
See p. 681.
B
7.
8. H
x8
688
J
9.
E
x8
F
8
A 55
x8
C
G
D
Using Algebra Find the value of x.
10.
12
11.
11
12.
13
6x
x14
x
Homework Help
Example 1: Exs. 7–9, 14,
15, 17–19,
27, 28
Example 2: Exs. 10–13
Example 3: Exs. 20–25
188
Chapter 4
13.
(5x 1 7)8
14.
7x 1 5
Triangle Relationships
15.
3x8
19
528
4x8
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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.
IStudent Help
ICLASSZONE.COM
You be the Judge
17.
C
A
HOMEWORK HELP
Extra help with problem
solving in Exs. 17–19 is
at classzone.com
18.
x8
A
19.
B
2x 8
508
x8
x8
B
B
x8
308
A
C
C
Using Algebra Find the value of y.
20.
2y
21.
11
22.
2y 1 5
10
y15
y
23.
4y 2 3
y
24.
25.
8y 2 10
3y
5
Sports
4y 1 2
5y 2 14
26. Challenge In the diagram at the right, TXYZ is
Y
equilateral and the following pairs of segments
&* and LK
&*; ZY
&* and LJ
&; XZ
&* and JK
&.
are parallel: XY
Describe a plan for showing that TJKL must
be equilateral.
J
X
K
L
Z
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.
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
CLASSZONE.COM
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?
4.3
Isosceles and Equilateral Triangles
189
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Z
Y
X
Tiles In Exercises 29–31, use the diagram at the left. In the diagram,
&** c WX
&*** c YX
&* c ZX
&*.
VX
29. Copy the diagram. Use what you know about side lengths
to mark your diagram.
W
V
30. Explain why aXWV c aXVW.
31. Name four isosceles triangles.
32. Technology Use geometry software
to complete the steps.
B
1 Construct circle A.
●
2 Draw points B and C on the circle.
●
3 Connect the points to form TABC.
●
C
A
Is TABC isosceles? Measure the sides
of the triangle to check your answer.
Standardized Test
Practice
Multiple Choice In Exercises 33 and 34, use the diagram below.
33. What is the measure of aEFD?
A
X
C
X
558
B 658
X
D 1808
X
1258
E
34. What is the measure of aDEF ?
F
X
H
X
Mixed Review
508
G 708
X
J 1808
X
1258
1258
F
G
D
&( is the angle bisector. Find maDBC and maABC.
Angle Bisectors BE
(Lesson 2.2)
35.
36.
E
A
E
A
E
A
C
D
428
37.
C
568
B
B
C
758
B
D
Vertical Angles Find the value of the variable. (Lesson 2.4)
38.
39.
558
40.
(x 2 8)8
(x 1 20)8
818
428
(2x 1 1)8
Algebra Skills
Evaluating Square Roots Evaluate. (Skills Review, p. 668)
41. Ï4
w9
w
190
Chapter 4
Triangle Relationships
42. Ï1
w2
w1
w
43. Ï1
w
44. Ï4
w0
w0
w
D