Download Identifying congruent triangles 1

4.2
1 PLAN
PACING
Basic: 2 days
Average: 2 days
Advanced: 2 days
Block Schedule: 0.5 block with 4.1
0.5 block with 4.3
What you should learn
GOAL 1 Identify congruent
figures and corresponding
parts.
GOAL 1
IDENTIFYING CONGRUENT FIGURES
Two geometric figures are congruent if they have exactly the same size and
shape. Each of the red figures is congruent to the other red figures. None of the
blue figures is congruent to another blue figure.
GOAL 2 Prove that two
triangles are congruent.
Congruent
Not congruent
Why you should learn it
FE
䉲 To identify and describe
congruent figures in real-life
objects, such as the
sculpture described L
AL I
in Example 1.
RE
LESSON OPENER
VISUAL APPROACH
An alternative way to approach
Lesson 4.2 is to use the Visual
Approach Lesson Opener:
•Blackline Master (Chapter 4
Resource Book, p. 26)
•
Transparency (p. 22)
Congruence and Triangles
MEETING INDIVIDUAL NEEDS
• Chapter 4 Resource Book
Prerequisite Skills Review (p. 5)
Practice Level A (p. 30)
Practice Level B (p. 31)
Practice Level C (p. 32)
Reteaching with Practice (p. 33)
Absent Student Catch-Up (p. 35)
Challenge (p. 38)
• Resources in Spanish
•
Personal Student Tutor
When two figures are congruent, there is a correspondence between their
angles and sides such that corresponding angles are congruent and
corresponding sides are congruent. For the triangles below, you can write
¤ABC £ ¤PQR, which is read “triangle ABC is congruent to triangle PQR.”
The notation shows the congruence and the correspondence.
NEW-TEACHER SUPPORT
See the Tips for New Teachers on
pp. 1–2 of the Chapter 4 Resource
Book for additional notes about
Lesson 4.2.
There is more than one way to write a congruence statement, but it is important
to list the corresponding angles in the same order. For example, you can also write
¤BCA £ ¤QRP.
Corresponding angles
5. right scalene triangle yes
Two Open Triangles Up
Gyratory II by George Rickey
Æ
Æ
Æ
Æ
B
™B £ ™Q
BC £ QR
™C £ ™R
CA £ RP
A
C
q
SOLUTION
STUDENT HELP
Study Tip
Notice that single,
double, and triple arcs
are used to show
congruent angles.
R
Naming Congruent Parts
The congruent triangles represent the triangles in
the photo above. Write a congruence statement.
Identify all pairs of congruent corresponding parts.
202
202
Æ
P
WARM-UP EXERCISES
no
Æ
AB £ PQ
™A £ ™P
EXAMPLE 1
Transparency Available
Tell whether it is possible to draw
each triangle.
1. acute scalene triangle yes
2. obtuse equilateral triangle no
3. right isosceles triangle yes
4. scalene equiangular triangle
Corresponding sides
F
R
S
E
The diagram indicates that ¤DEF £ ¤RST.
The congruent angles and sides are as follows.
Angles:
Sides:
™D £ ™R, ™E £ ™S, ™F £ ™T
Æ
Æ Æ
Æ Æ
Æ
DE £ RS, EF £ ST, FD £ TR
Chapter 4 Congruent Triangles
D
T
xy
Using
Algebra
EXAMPLE 2
Using Properties of Congruent Figures
2 TEACH
In the diagram, NPLM £ EFGH.
F
M
8m
a. Find the value of x.
L
b. Find the value of y.
110ⴗ
(2x ⴚ 3) m
87ⴗ
Æ
(7y ⴙ 9)ⴗ
72ⴗ
10 m
P
SOLUTION
G
N
E
H
EXTRA EXAMPLE 1
Write a congruence statement
for the triangles below. Identify
all pairs of congruent corresponding parts. See below.
H
Æ
a. You know that LM £ GH.
So, LM = GH.
8 = 2x º 3
b. You know that ™N £ ™E.
Y
So, m™N = m™E.
72° = (7y + 9)°
11 = 2x
K
Z
X
63 = 7y
5.5 = x
.........
9=y
J
EXTRA EXAMPLE 2
In the diagram, ABCD ⬵ KJHL.
a. Find the value of x. 3
b. Find the value of y. 25
The Third Angles Theorem below follows from the Triangle Sum Theorem.
You are asked to prove the Third Angles Theorem in Exercise 35.
A
9 cm
(4x ⫺ 3) cm
K
B L
91°
THEOREM
Third Angles Theorem
THEOREM 4.3
B
If two angles of one triangle are congruent to
two angles of another triangle, then the third
angles are also congruent.
D
E
If ™A £ ™D and ™B £ ™E,
then ™C £ ™F.
H
(5y ⫺ 12)°
J
EXTRA EXAMPLE 3
Find the value of x. 14
C
A
86° 113°
6 cm C
B
F
D
(4x ⫹ 15)°
87°
22°
A
F
C
THEOREM
D
EXAMPLE 3
Using the Third Angles Theorem
Find the value of x.
CHECKPOINT EXERCISES
M
R
For use after Example 1:
1. Identify all pairs of congruent
corresponding parts.
T
(2x ⴙ 30)ⴗ
SOLUTION
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for extra examples.
In the diagram, ™N £ ™R and ™L £ ™S.
From the Third Angles Theorem, you know
that ™M £ ™T. So, m™M = m™T.
From the Triangle Sum Theorem,
m™M = 180° º 55° º 65° = 60°.
m™M = m™T
60° = (2x + 30)°
åA • åR; åB • åQ; åC • åP;
A
苶B
苶 • R苶Q
苶; B
苶C苶 • Q
苶P苶 ; A
苶C苶 • R苶P苶
55ⴗ
N
65ⴗ
L
R
A
S
B Q
Third Angles Theorem
P
C
Substitute.
30 = 2x
Subtract 30 from each side.
15 = x
Divide each side by 2.
4.2 Congruence and Triangles
For use after Examples 2 and 3:
2. Find the value of x. 11
G
K
203
J
F
Extra Example 1 Sample answer:
†XYZ • †HKJ; åX • åH; åY • åK; åZ • åJ;
X苶Y苶 • H
苶K苶; Y苶Z苶 • K苶J苶; X苶Z苶 • H
苶J苶
E
100°
45°
H
I
(4x ⫺ 9)°
203
GOAL 2
EXTRA EXAMPLE 4
Decide whether the triangles
are congruent. Justify your
reasoning.
Determining Whether Triangles are Congruent
EXAMPLE 4
H
E
PROVING TRIANGLES ARE CONGRUENT
Decide whether the triangles are congruent.
Justify your reasoning.
58°
R
92ⴗ
G
58°
J
F
Proof
From the diagram, you are given
that all three pairs of corresponding sides are congruent: E苶F苶 • J苶H
苶,
E苶G
苶 • J苶G
苶, F苶G
苶• H
苶G
苶. Because åF
and åH have the same measure,
åF • åH. By the Vertical
Angles Theorem, you know that
åEGF • åJGH. By the Third
Angles Theorem, åE • åJ. So
all three sides and all three
angles are congruent. By the
definition of congruent triangles,
†EFG • †JHG.
q
M
P
SOLUTION
Paragraph Proof From the diagram, you are given that all three pairs of
corresponding sides are congruent.
Æ
Æ Æ
Æ
Æ
Æ
RP £ MN, PQ £ NQ, and QR £ QM
Because ™P and ™N have the same measure, ™P £ ™N. By the Vertical Angles
Theorem, you know that ™PQR £ ™NQM. By the Third Angles Theorem,
™R £ ™M.
䉴
So, all three pairs of corresponding sides and all three pairs of corresponding
angles are congruent. By the definition of congruent triangles,
¤PQR £ ¤NQM.
Proving Two Triangles are Congruent
EXAMPLE 5
EXTRA EXAMPLE 5
Given: M
苶N
苶⬵Q
苶P苶, M
苶N
苶 储 P苶Q
苶,
O is the midpoint of 苶M
苶Q
苶 and 苶P苶N
苶.
Prove: 䉭MNO ⬵ 䉭QPO
M
N
92ⴗ
FOCUS ON
APPLICATIONS
The diagram represents the triangular stamps
shown in the photo. Prove that ¤AEB £ ¤DEC.
Æ
Æ Æ
A
Æ
GIVEN 䉴 AB ∞ DC , AB £ DC ,
Æ
B
E
Æ
E is the midpoint of BC and AD.
N
PROVE 䉴 ¤AEB £ ¤DEC
O
C
D
Plan for Proof Use the fact that ™AEB and ™DEC are vertical angles to show
Æ
that those angles are congruent. Use the fact that BC intersects parallel segments
Æ
Æ
AB and DC to identify other pairs of angles that are congruent.
Q
L
AL I
FE
Statements (Reasons)
1. M
苶N
苶• Q
苶P苶, M
苶N
苶 ‚ P苶Q
苶, (Given)
2. åOMN • åOQP,
åMNO • åQPO (Alternate
Interior Angles Theorem)
3. åMON • åQOP (Vertical
Angles Theorem)
4. O is the midpoint of M
苶Q
苶 and P苶N
苶.
(Given)
5. M
苶O
苶• Q
苶O
苶, P苶O
苶• N
苶O
苶
(Def. of midpoint)
6. †MNO • †QPO (Def. of • †s )
RE
P
TRIANGULAR
STAMP
When these stamps were
issued in 1997, Postmaster
General Marvin Runyon said,
“Since 1847, when the first
U.S. postage stamps were
issued, stamps have been
rectangular in shape. We
want the American public
to know stamps aren’t
‘square.’”
SOLUTION
Statements
Æ
1. Given
2.
2. Alternate Interior Angles Theorem
Æ
3.
4.
6.
204
204
Reasons
1. AB ∞ DC,
5.
Checkpoint Exercises for Examples
4 and 5 on next page.
Æ
Æ
AB £ DC
™EAB £ ™EDC,
™ABE £ ™DCE
™AEB £ ™DEC
Æ
E is the midpoint of AD,
Æ
E is the midpoint of BC.
Æ
Æ Æ
Æ
AE £ DE, BE £ CE
¤AEB £ ¤DEC
Chapter 4 Congruent Triangles
3. Vertical Angles Theorem
4. Given
5. Definition of midpoint
6. Definition of congruent triangles
In this lesson, you have learned to prove that two triangles are congruent by the
definition of congruence—that is, by showing that all pairs of corresponding
angles and corresponding sides are congruent. In upcoming lessons, you will
learn more efficient ways of proving that triangles are congruent. The properties
below will be useful in such proofs.
CHECKPOINT EXERCISES
For use after Examples 4 and 5:
1. Decide whether the triangles
are congruent. Justify your
reasoning.
C
THEOREM
THEOREM 4.4
Properties of Congruent Triangles
20° Q
20°
B
REFLEXIVE PROPERTY OF CONGRUENT TRIANGLES
120°
A
Every triangle is congruent to itself.
C
E
SYMMETRIC PROPERTY OF CONGRUENT TRIANGLES
If ¤ABC £ ¤DEF, then ¤DEF £ ¤ABC.
D
F
K
TRANSITIVE PROPERTY OF CONGRUENT TRIANGLES
If ¤ABC £ ¤DEF and ¤DEF £ ¤JKL, then ¤ABC £ ¤JKL.
J
L
GUIDED PRACTICE
Vocabulary Check
Concept Check
✓
✓
2. Third Angles Theorem
3. No; corresponding sides
are not congruent.
Skill Check
✓
1. Copy the congruent triangles shown at the
right. Then label the vertices of your triangles
so that ¤JKL £ ¤RST. Identify all pairs
of congruent corresponding angles and
corresponding sides. ™J and ™R, ™K and
Æ
Æ Æ
Æ Æ
Æ
D
B
2. How does the student know that the
corresponding angles are congruent?
3. Is ¤ABC £ ¤ADE? Explain your answer.
A
C
E
q
N
P
6. What is the measure of ™R? 30°
7. What is the measure of ™N? 30°
Æ
Æ
Æ
Æ
9. Which side is congruent to LN? PR
CLOSURE QUESTION
If 䉭GHJ ⬵ 䉭MNP, what corresponding angles and sides are congruent?
3 square units
105ⴗ
L
From the diagram, you are
given that all three pairs of
corresponding sides are congruent: C苶E苶 • Q
苶P苶, E苶D
苶 • P苶D
苶,
C苶D
苶• Q
苶D
苶. Because åC and åQ
have the same measure, and
åE and åP have the same
measure, åC • åQ and
åE • åP. By the Third Angles
Theorem, åCDE • åQDP. So
all three sides and all three
angles are congruent. By the
definition of congruent triangles, †CED • †QPD.
DAILY PUZZLER
The design shown is created entirely
from congruent triangles. If the area
of the design is 96 square units,
what is the area of one triangle?
45ⴗ
5. What is the measure of ™M? 45°
8. Which side is congruent to QR? MN
P
åG • åM, åH • åN, åJ • åP,
G
苶H
苶• M
苶N
苶, H
苶J苶 • N
苶P苶, G
苶J苶 • M
苶P苶
Use the diagram at the right, where ¤LMN £ ¤PQR.
4. What is the measure of ™P? 105°
120°
D
FOCUS ON VOCABULARY
Draw two figures that are similar
but not congruent. Point out that
these figures have corresponding
sides. Make sure students understand that the term corresponding
sides does not imply that the sides
are necessarily congruent.
™S, ™L and ™T, JK and RS , KL and ST , JL and RT .
ERROR ANALYSIS Use the information and the diagram below.
On an exam, a student says that ¤ABC £ ¤ADE
because the corresponding angles of the triangles
are congruent.
E
M
4.2 Congruence and Triangles
R
205
205