Download GETE0405

4-5
4-5
Isosceles and Equilateral
Triangles
1. Plan
Objectives
1
To use and apply properties of
isosceles triangles
Examples
1
2
3
Using the Isosceles Triangle
Theorems
Using Algebra
Real-World Connection
What You’ll Learn
• To use and apply properties
of isosceles triangles
. . . And Why
Lesson 3-4
B x⬚
3. Name the side opposite &A. BC
To find the angles of a garden
path, as in Example 3
4. Name the side opposite &C. BA
A
x 2 5. Algebra Find the value of x. 105
75⬚
30⬚
C
New Vocabulary • legs of an isosceles triangle
• base of an isosceles triangle
• vertex angle of an isosceles triangle
• base angles of an isosceles triangle • corollary
Math Background
Understanding the vocabulary,
including legs, vertex angle, and
base angles, is necessary for
solving many exercises in this
section. Help students remember
the meanings of terms by having
them describe everyday objects
with the same words. Mention
that isosceles derives from the
Greek iso (same) and skelos (leg).
Trapezoids, which will be studied
in Chapter 4, can also be isosceles,
and have two bases.
GO for Help
Check Skills You’ll Need
1. Name the angle opposite AB. lC
2. Name the angle opposite BC. lA
1
The Isosceles Triangle Theorems
Vocabulary Tip
Isosceles is derived from
the Greek isos for equal
and skelos for leg.
Vertex Angle
Isosceles triangles are common in the real world.
You can find them in structures such as bridges and
buildings. The congruent sides of an isosceles triangle
are its legs. The third side is the base. The two
congruent sides form the vertex angle. The other
two angles are the base angles.
Legs
Base
Base Angles
An isosceles triangle has a certain type of symmetry about a line through its
vertex angle. The theorems below suggest this symmetry, which you will study in
greater detail in Lesson 9-4.
More Math Background: p. 196D
Key Concepts
Lesson Planning and
Resources
Theorem 4-3
Isosceles Triangle Theorem
C
If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.
See p. 196E for a list of the
resources that support this lesson.
&A > &B
Theorem 4-4
A
Converse of Isosceles Triangle Theorem
B
C
PowerPoint
If two angles of a triangle are congruent, then the
sides opposite the angles are congruent.
Bell Ringer Practice
AC > BC
Check Skills You’ll Need
A
B
For intervention, direct students to:
Theorem 4-5
Using Exterior Angles of Triangles
CD ' AB and CD bisects AB.
228
A
D
B
Chapter 4 Congruent Triangles
Special Needs
Below Level
L1
Have students collect other landscape designs from
gardening and landscaping books and magazines.
Have them select one design and describe it using
geometric terms.
228
C
The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the base.
Lesson 3-4: Example 3
Extra Skills, Word Problems, Proof
Practice, Ch. 3
learning style: tactile
L2
Discuss whether the equilateral-equiangular
relationship holds for polygons with more than
3 sides. Ask students to support their reasoning
with examples.
learning style: verbal
Proof
In the following proof of the Isosceles Triangle Theorem, you use a special
segment, the bisector of the vertex angle. You will prove Theorems 4-4 and 4-5
in the Exercises.
2. Teach
Proof of the Isosceles Triangle Theorem
Guided Instruction
Begin with isosceles #XYZ with XY > XZ.
Draw XB, the bisector of the vertex angle &YXZ.
X
Connection to Chemistry
1 2
Given: XY > XZ, XB bisects &YXZ.
Ask: What are different forms of
a chemical element with the same
atomic number called? isotopes
Point out that the prefix iso-,
meaning “equal,” is a variation
of the prefix isos- in isosceles.
Prove: &Y > &Z
Real-World
B
Y
Proof: You are given that XY > XZ. By the definition
of angle bisector, &1 > &2. By the Reflexive Property
of Congruence, XB > XB. Therefore, by the SAS
Postulate, #XYB > #XZB, and &Y > &Z by CPCTC.
Z
Visual Learners
Students may think that the
base of an isosceles triangle is
always at the bottom. Illustrate
by rotating a physical model that
the base can be in any orientation,
as in Exercise 30.
Connection
1
This A-shaped roof
has congruent legs and
congruent base angles.
EXAMPLE
Using the Isosceles Triangle Theorems
T
Explain why each statement is true.
a. &WVS > &S
U
WV > WS so &WVS > &S by the Isosceles
Triangle Theorem.
b. TR > TS
V
&R > &WVS and &WVS > &S, so &R > &S
by the Transitive Property of >. TR > TS by the
Converse of the Isosceles Triangle Theorem.
Quick Check
2
A
1
2
A
4
5
A
B
B
D
C
C
Using Algebra
M
Multiple Choice Find the value of y.
17
27
54
90
E
D
C
B
E
D
C
B
A
3
EXAMPLE
E
D
C
B
A
D
E
E
Test-Taking Tip
Remember that the
acute angles of a right
triangle are
complementary.
S
1 Can you deduce that #RUV is isosceles? Explain.
No, point U could be anywhere between R and T.
E
D
C
B
Alternative Method
W
R
By Theorem 4-5, you know that MO ' LN, so
&MON = 90. #MLN is isosceles, so &L > &N
and m&N = 63.
m&N + 90 + y = 180 Triangle Angle-Sum Theorem
63 + 90 + y = 180 Substitute for mlN and x.
y = 27
y⬚
N
63⬚
O
L
1
3
2 Find the value of x. 6
A corollary is a statement that follows immediately from a theorem.
Corollaries to the Isosceles Triangle Theorem and its converse appear on
the next page.
Lesson 4-5 Isosceles and Equilateral Triangles
Advanced Learners
229
English Language Learners ELL
L4
Have students explain why Theorems 4-3, 4-4, and 4-5
do or do not apply to equilateral triangles.
learning style: verbal
EXAMPLE
Math Tip
Point out that every corollary is
a theorem that can be proved.
Subtract 153 from each side.
The correct answer is B.
Quick Check
Draw identical copies of XYZ side
by side. Then label congruent parts,
asking the class to justify each step.
• Use a single tick mark to show
XY on the first copy XZ on
the second copy.
• Use an arc to show &X on the
first copy &X on the second
copy.
• Use double tick marks to show
XZ on the first copy XY on
the second copy.
Ask: Which triangles are
congruent? By what postulate?
kXYZ O kXZY; SAS Which
angles are congruent by CPCTC?
lY and lZ
EXAMPLE
Teaching Tip
Have students calculate the total
measure of the four angles
formed by the x-axis and the
y-axis at the origin. Discuss the
fact that there are 360° about
any point. Ask students to suggest
other ways to show that there
are 360° about any point, such as
drawing a straight angle and
adding the measures on
each side.
Point out that equilateral and equiangular can be
used interchangeably for triangles, but not for other
polygons. For example, squares and rectangles are
both equiangular but only the square is equilateral.
learning style: verbal
229
PowerPoint
Key Concepts
Additional Examples
B
Vocabulary Tip
&X > &Y > &Z
Equilateral:
Congruent sides
Equiangular:
Congruent angles
A
Corollary to Theorem 4-3
Y
If a triangle is equilateral, then the triangle
is equiangular.
1 Explain why ABC is isosceles.
X
Corollary
Corollary
X
Corollary to Theorem 4-4
Z
Y
If a triangle is equiangular, then the triangle
is equilateral.
C
XY > YZ > ZX
X
Z
* ) * )
Because XA n BC ,
lABC O lXAB. By
the angles marked O
and the Transitive Prop.,
lABC O lACB. kABC is
isosceles by Converse of
the Isosceles Triangle Thm.
2 Use the diagram for Example 2.
Suppose that m&L 2 63 and
m&L = y. Find the values of x
and y. x ≠ 90, y ≠ 45
3
x˚
In a rectangle, an angle measure is 90;
in an equilateral triangle, it is 60.
For: Isosceles Triangle Activity
Use: Interactive Textbook, 4-5
Quick Check
x + 90 + 60 + 90 = 360
x = 120
3 What is the measure of the angle at each outside corner of the path? 150
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A
Resources
• Daily Notetaking Guide 4-5 L3
• Daily Notetaking Guide 4-5—
L1
Adapted Instruction
Connection
Landscaping A landscaper uses rectangles and
equilateral triangles for the path around the
hexagonal garden. Find the value of x.
3 In the drawing for Example 3,
suppose that a segment is drawn
between the endpoints of the
segments that determine the
angle marked x°. Find the angle
measures of the triangle that is
formed. 120, 30, 30
Real-World
EXAMPLE
Practice by Example
Example 1
GO for
Help
(page 229)
Complete each statement. Explain why it is true.
V
1. VT > 9
2. UT > 9 > YX
3. VU > 9
U
4. &VYU > 9
T
Closure
W
Example 2 x 2 Algebra Find the values of x and y.
An angle exterior to the vertex of
an isosceles triangle measures 80.
Find the angle measures of the
triangle. 100, 40, 40
(page 229)
5.
6.
7.
x⬚
x⬚
Example 3
(page 230)
x⬚
100⬚
50⬚ y⬚
1. VX ; Converse of the
Isosc. kThm.
2. UW ; Converse of the
Isosc. kThm.
3. VY ; VT ≠ VX (Ex. 1)
and UT ≠ YX (Ex. 2), so
Y
VU ≠ VY by the Subtr.
Prop. of ≠.
X 4. Answers may vary.
Sample: lVUY; '
opposite O sides are O.
110⬚
y⬚
4
52⬚ y
x ≠ 38; y ≠ 4
x ≠ 40; y ≠ 70
x ≠ 80; y ≠ 40
8. A square and a regular hexagon are placed so that they have
a common side. Find m&SHA and m&HAS. 150; 15
H
9. Five fences meet at a point to form angles with measures x, 2x, S
3x, 4x, and 5x around the point. Find the measure of each angle.
24, 48, 72, 96, 120
230
230
A
Chapter 4 Congruent Triangles
18. Answers may vary.
Sample: Corollary to
Thm. 4-3: Since
XY O YZ , lX O lZ by
Thm. 4-3. YZ O ZX , so
lY O lX by Thm. 4-3
also. By the Trans.
Prop., lY O lZ, so
lX O lY O lZ.
Corollary to Thm. 4-4:
Since lX O lZ, XY O
YZ by Thm. 4-4. lY O
lX, so YZ O ZX by Thm.
4.4 also. By the Trans
Prop., XY O ZX , so
XY O YZ O ZX .
26. No; the k can be
positioned in ways such
that the base is not on
the bottom.
B
Apply Your Skills
Find each value.
3. Practice
K
10. If m&L = 58, then m&LKJ = 7. 64
Assignment Guide
11. If JL = 5, then ML = 7. 2 12
12. If m&JKM = 48, then m&J = 7. 42
13. If m&J = 55, then m&JKM = 7. 35
1 A B 1-32
J
M
L
14. Architecture Seventeen spires, pictured at the left, cover the Cadet Chapel
at the Air Force Academy in Colorado Springs, Colorado. Each spire is an
isosceles triangle with a 40° vertex angle. Find the measure of each base angle.
70
15. Developing Proof Here is another way to prove the
Isosceles Triangle Theorem. Supply the missing parts.
K
Begin with isosceles #HKJ with KH > KJ.
Draw a. 9, a bisector of the base HJ. KM
H
Statements
1.
2.
3.
4.
5.
6.
Exercise 14
Proof
KM bisects HJ.
HM > JM
KH > KJ
KM > KM
#KHM > #KJM
&H > &J
M
Test Prep
Mixed Review
41-44
45-48
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 6, 8, 17, 18, 28.
Exercise 14 The world’s tallest
J
cathedral spire, measuring 528 ft,
is on the Protestant Cathedral of
Ulm in Germany. Construction
began in 1377, but the spire was
not finished until 1890.
Reasons
c.
d.
3.
e.
f.
g.
33-40
Connection to Architecture
Given: KH > KJ, b. 9 bisects HJ. KM
Prove: &H > &J
C Challenge
9 By construction
9 Def. of segment bisector
Given
9 Reflexive Prop. of O
9 SSS
9 CPCTC
16. Supply the missing parts in this statement of the
Converse of the Isosceles Triangle Theorem.
Then write a proof.
Exercise 19 To avoid careless
errors, encourage students to
show the solution steps for
solving their algebraic equation.
R
Begin with #PRQ with &P > &Q.
Draw a. 9, the bisector of &PRQ. RS
Given: &P > &Q, b. 9 bisects &PRQ. RS
Prove: PR > QR See back of book.
17a.
30, 30, 120
S
P
Q
17. Graphic Arts A former logo for
GPS the National Council of Teachers
of Mathematics is shown at the
right. Trace the logo onto paper.
a. Highlight an obtuse isosceles
The triangles in the logo
have these congruent
triangle in the design. Then
sides and angles.
find its angle measures. See left.
b. How many different sizes of angles
can you find in the logo? What are their measures? 5; 30, 60, 90, 120, 150
GPS Guided Problem Solving
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-0405
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
L3
Date
Practice 4-5
Isosceles and Equilateral Triangles
Find the values of the variables.
1.
2.
3.
x
110
t
10
y
x
y
5.
s
X
6. WXYZV is a
regular polygon.
y
b
W
Y
r
125 x
x
19. Multiple Choice The perimeter of the triangle at the
right is 20. Find x. C
3
5
6
7
Lesson 4-5 Isosceles and Equilateral Triangles
z
7.
8.
c
Z
V
a
9.
c
a
30
x
z
60
b
2x 6
2x ⫺ 5
Complete each statement. Explain why it is true.
10. AF 9
11. CA 9
231
© Pearson Education, Inc. All rights reserved.
GO
L4
Enrichment
4.
18. Writing Explain how each corollary on page 230
follows from its theorem. First, write one explanation
and then write the second similar to the first. See margin.
L3
12. KI 9
A
K
13. EC 9
B
J
14. JA 9
I
G
15. HB 9
C
H
F
L
D
E
Given mlD ≠ 25, find the measure of each angle.
16. JAB
17. FAL
18. JKI
19. DLA
Find the values of x and y.
20.
y
x
55
21.
x 110
22.
y
y
x
231
4. Assess & Reteach
x 2 Algebra Find the values of x and y.
20.
D
21.
PowerPoint
60⬚
Lesson Quiz
B
1. If m&BAC = 38, find m&C. 71
2. If m&BAM = m&CAM = 23,
find m&BMA. 90
3. If m&B = 3x and m&BAC = 2x
- 20, find x. 25
4.
Real-World
18
60°
Connection
Careers Radio broadcasters
must respond to opinions
given by “call-in” listeners.
0
27. Reasoning What are the measures
450 ft
550 ft
of the base angles of an isosceles
Tower cables extend to both widths.
right triangle? Explain. 45; they are ≠ and have sum 90.
y
60°
60°
x°
Find the values of x and y.
x ≠ 60; y ≠ 9
28. lA O lD by
the Isos. k Thm.
kABE O kDCE
by SAS.
Proof
28. Given: AE > DE, AB > DC
Proof
29. Prove Theorem 4-5. Use the diagram
next to it on page 228. See margin.
30.
126⬚
C
m⬚
E
A
B
C
D
x 2 Algebra Find the values of m and n.
B
F
E
Prove: #ABE > #DCE
5. ABCDEF is a regular hexagon.
Find m&BAC.
A
30⬚
24. Critical Thinking An exterior angle of an isosceles triangle has measure 100.
Find two possible sets of measures for the angles of the triangle.
80, 80, 20; 80, 50, 50
25. a. Communications In the diagram isosc. >
at the right, what type of triangles
Radio
1000
are formed by the cables of the
Tower
ft
1009 ft tall
same height and the ground?
b. What are the two different base
800
lengths of the triangles? 900 ft; 1100 ft
Cables
c. How is the tower related to
600
each of the triangles? The tower
is the ' bis. of the base of each k.
400
26. Critical Thinking Curtis defines
the base of an isosceles triangle as
its “bottom side.” Is his definition a
200
good one? Explain. See margin.
C
M
y⬚
x⬚
A
B
x ≠ 60; y ≠ 30
x ≠ 30; y ≠ 120
ABCDE is a regular pentagon.
x ≠ 36; y ≠ 36
23. Write the Isosceles Triangle Theorem and its converse as a biconditional.
23. Two sides of a k are O
if and only if the '
opp. those sides are O.
A
C
y⬚
y⬚
Use the diagram for
Exercises 1–3.
x⬚
E
x⬚
22.
D
C
Alternative Assessment
33. (4, 0) and (0, 4)
35. (5, 3); (2, 6); (2, 9);
(8, 3); (–1, 6); (5, 0))
29. AC O CB and lACD O
lDCB are given. CD O
CD by the Refl. Prop. of
232 O, so kACD O kBCD by
232
50⬚
m⬚
m⬚
n⬚
34. (0, 0) and (5, 5)
35. (2, 3) and (5, 6)
x 2 36. Algebra A triangle has angle measures x + 15, 3x - 35, and 4x.
34. (5, 0); (0, 5); (–5, 5);
(5, –5); (0, 10); (10, 0)
The diagram illustrates the
Isosceles Triangle Theorem. Have
the class draw similar diagrams
to illustrate the Converse of the
Isosceles Triangle Theorem and
the two corollaries from this
lesson.
n⬚
m ≠ 60; n ≠ 30
m ≠ 20; n ≠ 45
Coordinate Geometry For each pair of points, there are six points that could be
the third vertex of an isosceles right triangle. Find the coordinates of each point.
Challenge
33. (0, 0), (4, 4), (–4, 0),
(0, –4), (8, 4), (4, 8)
32.
n⬚
m ≠ 36; n ≠ 27
30
Draw the diagram below on the
board.
31.
a. Find the value of x. 25
b. Find the measure of each angle.
40; 40; 100
c. What type of triangle is it? Why?
Obtuse isosc. > : 2 of the ' are O and one l is obtuse.
37. State the converse of Theorem 4-5. If the converse is true, write a paragraph
proof. If the converse is false, give a counterexample. See margin.
Chapter 4 Congruent Triangles
SAS. So AD O DB by
CPCTC, and CD bisects
AB. Also lADC O
lBDC by CPCTC,
mlADC ± mlBDC ≠
180 by l Add. Post., so
mlADC ≠ mlBDC ≠
90 by the Subst. Prop.
So CD is the # bis. of
AB.
37. The # bis. of the base of
an isosc. k is the bis. of
the vertex l; given isosc.
kABC with # bis. CD,
lADC O lCDB and AD
O DB by def. of # bis.
Since CD O CD by Refl.
Prop., kACD O kBCD
by SAS. So lACD O
lBCD by CPCTC, and
CD bisects lACB.
Real-World
Connection
The circle is a basic shape for
many tile designs.
38. Crafts The design in Step 3 is used in Hmong crafts and in Islamic and Mexican
tiles. To create it, the artist
starts by drawing a circle
and four equally spaced
diameters.
a. How many different
sizes of isosceles right
Step 1
Step 2
Step 3
triangles can you find
in Step 2? Trace an example of each onto your paper. 5
b. How many times does a triangle of each size in part (a) appear in the Step 2
diagram? See back of book.
Test Prep
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 253
• Test-Taking Strategies, p. 248
• Test-Taking Strategies with
Transparencies
Reasoning What measures are possible for the base angles of each type of
triangle? Explain.
39. an isosceles obtuse triangle
0 R measure of base l R 45
40. an isosceles acute triangle
45 R measure of base l R 90
Test Prep
Multiple Choice
41. In isosceles #ABC, the vertex angle is &A. What can be proved? C
A. AB = CB
B. &A > &B
C. m&B = m&C
D. BC > AC
1
42. In the diagram at the right, m&1 = 40. What is m&2? G
F. 40
G. 50
H. 80
J. 100
2
43. In an isosceles triangle, the measure of the vertex
angle is 4x. The measure of each base angle is
2x + 10. What is the measure of the vertex angle? D
A. 10
B. 20
C. 50
D. 80
Short Response
44. In the figure at the right, m&APB = 60.
a. What is m&PAB? Explain.
b. &PAB and &QAB are complementary.
What is m&AQB? Show your work.
a–b. See margin.
A
P
Q
B
Mixed Review
45. m&R = 59, m&T = 93 = m&H, m&V = 28,
and RT = GH. What, if anything, can you
for
conclude about RC and GV? Explain.
Help
RC ≠ GV; RC O GV by CPCTC since kRTC O kGHV by ASA. R
Lesson 4-4
GO
Lessons 4-2, 4-3
V
G
C
H
Which congruence statement, SSS, SAS, ASA, or AAS, would you use to conclude
that the two triangles are congruent?
46.
Lesson 3-5
T
AAS
47.
SSS
48. How many sides are in a regular polygon whose exterior angles measure 158?
24 sides
lesson quiz, PHSchool.com, Web Code: aua-0405
[2] a. 60; since mlPAB ≠ mlPBA,
and mlPAB ± mlPBA ≠ 120,
mlPAB ≠ 60.
Lesson 4-5 Isosceles and Equilateral Triangles
233
b. 120; mlAPB ≠ 60 so mlPAB ≠ 60.
Since lPAB and lQAB are compl.,
mlQAB ≠ 30. kQAB is isosc. so
mlAQB ≠ 120.
[1] one part correct
233