Download 5.2 Proving Triangles are Congruent: SSS and SAS

Page 1 of 9
5.2
Goal
Show triangles are
congruent using SSS
and SAS.
Key Words
Proving Triangles are
Congruent: SSS and SAS
Geo-Activity
Copying a Triangle
1 Use a straightedge to draw a large
●
C
triangle. Label it TABC.
• proof
A
2 Open your compass to measure
●
3 Open your compass to measure
●
AB
&* of TABC. Use this length to
construct DE
&* so that it is
congruent to AB
&*.
D
B
AC
&*. Use this length to draw an
arc with the point of the
compass at D.
E
D
4 Open your compass to measure
●
E
5 Label the point of intersection F.
●
Then draw TDEF. Measure the
angles of the triangles to
confirm that TABC c TDEF.
BC
&*. Use this length to draw an
arc centered at E that intersects
the arc from Step 3.
F
F
D
D
E
E
The Geo-Activity suggests the following postulate.
POSTULATE 12
Side-Side-Side Congruence Postulate (SSS)
Words
If three sides of one triangle are congruent to three sides
of a second triangle, then the two triangles are congruent.
Symbols
If Side MN
&** c QR
&*, and
Side
NP
&* c RS
&*, and
Side
PM
&** c SQ
&*,
then TMNP c TQRS.
5.2
P
M
N
P
R
S
Proving Triangles are Congruent: SSS and SAS
241
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EXAMPLE
Use the SSS Congruence Postulate
1
Does the diagram give enough
information to show that the
triangles are congruent? Explain.
J
L
H
K
Solution
&* c LJ
&* and HK
&** c LK
&*.
From the diagram you know that HJ
&* c JK
&*.
By the Reflexive Property, you know that JK
ANSWER
© Yes, enough information is given. Because corresponding
sides are congruent, you can use the SSS Congruence
Postulate to conclude that THJK c TLJK.
POSTULATE 13
Visualize It!
Side-Angle-Side Congruence Postulate (SAS)
In the triangle below,
aB is the included
angle between sides
AB
&* and BC
&*.
Words
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of a
second triangle, then the two triangles are congruent.
Symbols
C
PQ
&* c WX
&&, and
If Side
P
X
Angle aQ c aX, and
A
QR
&* c XY
&**,
Side
B
R
P
then TPQR c TWXY.
included angle
EXAMPLE
2
W
Y
W
Use the SAS Congruence Postulate
Does the diagram give enough information to use the SAS Congruence
Postulate? Explain your reasoning.
a.
b.
D
H
G
A
B
C
E
F
Solution
&* c CB
&* and DB
&* c DB
&*.
a. From the diagram, you know that AB
&* and DB
&* is aABD.
The angle included between AB
&* and DB
&* is aCBD.
The angle included between CB
Because the included angles are congruent, you can use the
SAS Congruence Postulate to conclude that TABD c TCBD.
&* c GH
&** and GE
&* c GE
&*. However, the congruent
b. You know that GF
angles are not included between the congruent sides, so you
cannot use the SAS Congruence Postulate.
242
Chapter 5
Congruent Triangles
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Writing Proofs A proof is a convincing argument that shows why a
statement is true. A two-column proof has numbered statements and
reasons that show the logical order of the argument. Each statement
has a reason listed to its right.
HOW TO WRITE A PROOF
• List the given information first.
• Use information from the diagram.
• Give a reason for every statement.
• Use given information, definitions, postulates, and theorems
as reasons.
• List statements in order. If a statement relies on another statement,
list it later than the statement it relies on.
• End the proof with the statement you are trying to prove.
EXAMPLE
3
Write a Proof
Write a two-column proof that shows T JKL c TNML.
Given
© JL
&* c NL
&*
J
&**.
L is the midpoint of KM
Prove © T JKL c TNML
K
M
L
N
Solution
The proof can be set up in two columns. The proof begins with
the given information and ends with the statement you are trying
to prove.
Statements
Reasons
&* c NL
&*
S 1. JL
1. Given
2. L is the midpoint
Student Help
STUDY TIP
You can remind yourself
about side and angle
congruences by writing
letters as shown.
2. Given
These are the given
statements.
&**.
of KM
A 3. aJLK c aNLM
3. Vertical Angles
Theorem
&* c ML
&**
S 4. KL
4. Definition of
midpoint
5. TJKL c TNML
5. SAS Congruence
Postulate
5.2
This information is
from the diagram.
Statement 4 follows from
Statement 2.
Statement 5 follows from
the congruences of
Statements 1, 3, and 4.
Proving Triangles are Congruent: SSS and SAS
243
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IStudent Help
EXAMPLE
4
Prove Triangles are Congruent
ICLASSZONE.COM
You are making a model of the window shown in the photo.
&* ∏ AG
&* and RA
&* c RG
&*. Write a proof to show
You know that DR
that TDRA c TDRG.
MORE EXAMPLES
More examples at
classzone.com
D
A
R
G
Solution
D
1 Make a diagram and label it with
●
the given information.
2 Write the given information and
●
the statement you need to prove.
&* ∏ AG
&*, RA
&* c RG
&*
Given © DR
Prove © TDRA c TDRG
A
R
G
3 Write a two-column proof. List the given statements first.
●
Student Help
Statements
Reasons
STUDY TIP
&* c RG
&*
1. RA
1. Given
Think about what you
can say given that
DR
&* ∏ AG
&*.
&* ∏ AG
&*
2. DR
2. Given
3. aDRA and aDRG are
3. ∏ lines form right angles.
Then think about what
other information you
can deduce from the
diagram.
right angles.
4. aDRA c aDRG
4. Right angles are congruent.
&* c DR
&*
5. DR
5. Reflexive Property of Congruence
6. TDRA c TDRG
6. SAS Congruence Postulate
Prove Triangles are Congruent
1. Fill in the missing statements and reasons.
D
B
&* c CE
&*, AC
&* c DC
&**
Given © CB
Prove © TBCA c TECD
244
Chapter 5
C
A
Statements
Reasons
&* c CE
&*
1. CB
1. _________?_________
2. _________?_________
2. Given
3. aBCA c aECD
3. _________?_________
4. TBCA c TECD
4. _________?_________
Congruent Triangles
E
Page 5 of 9
5.2 Exercises
Guided Practice
Vocabulary Check
Use TJKL to name the angle included between the two sides.
&* and KM
&**
1. JK
&* and MJ
&*
2. JK
K
&* and JL
&
3. KL
&** and LM
&**
4. KM
J
L
M
&* and KM
&**
5. LK
Skill Check
Decide whether enough information is given to show that the
triangles are congruent. If so, tell which congruence postulate
you would use.
6. TABC, TDEC
A
7. TFGH, TJKH
B
P
P
F
G
C
E
8. TPQR, TSRQ
H
D
K
S
R
J
Practice and Applications
Extra Practice
See p. 683.
Naming Included Angles Use the diagram shown to name the angle
included between the two sides.
&* and BD
&*
9. AB
&* and BC
&*
10. CD
&* and CB
&*
11. AC
&* and AD
&*
12. BA
&** and BD
&*
13. DC
&* and BC
&*
14. BD
B
A
D
C
Using SSS Decide whether enough information is given to use
the SSS Congruence Postulate. Explain your reasoning.
Homework Help
15.
16.
C
E
K
L
D
J
M
Example 1: Exs. 15–17,
21–31
Example 2: Exs. 18–31
Example 3: Exs. 34–36
Example 4: Exs. 34–36
A
B
17.
5.2
P
P
S
Proving Triangles are Congruent: SSS and SAS
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Using SAS Congruence Decide whether enough information is given
to use the SAS Congruence Postulate. Explain your reasoning.
18. P
19. B
C
20. J
D
K
N
P
S
R
A
L
E
M
You be the Judge Decide whether enough information is given
to show that the triangles are congruent. If so, tell which congruence
postulate you would use.
21. B
A
G
22.
H
23. M
N
P
P
F
C
D
E
24.
K
25. R
X
W
J
S
T
D
26.
Y
A
B
C
U
Z
Textiles In Exercises 27 and 28, use the photo of the Navajo rug. In
&* c DE
&* and AC
&* c CE
&*.
the triangles on the rug, BC
27. What additional information is needed to use the SSS Congruence
Postulate to show that TABC c TCDE ?
28. What additional information is needed to use the SAS Congruence
Postulate to show that TABC c TCDE ?
Missing Information Determine what single piece of information
you need to know in order to use either the SSS or SAS Congruence
Postulate to show that the triangles are congruent.
29.
B
A
D
246
Chapter 5
30. J
C
Congruent Triangles
K
F
31.
P
S
E
L
M
P
R
Page 7 of 9
Quilting
Quilting Use the labels on the diagrams to explain how you know
that the triangles are congruent.
&* i CD
&*
33. AB
32. WXYZ is a square.
X
TRIANGLES To create the
W
Y
A
B
C
D
quilt pattern in Ex. 33, squares
are cut out of two rectangular
strips of fabrics sewn together.
Application Links
Z
CLASSZONE.COM
Reasoning In Exercises 34 and 35, fill in the missing statements
and reasons.
H
E
&* c GH
&*
34. Given © EF
&* c HE
&**
FG
Prove © TEFG c TGHE
G
IStudent Help
ICLASSZONE.COM
HOMEWORK HELP
Extra help with problem
solving in Exs. 34–35 is
at classzone.com
35.
F
Statements
Reasons
&* c GH
&*
1. EF
1. Given
&* c HE
&**
2. FG
2. _________?_________
&* c GE
&*
3. GE
3. _________?_________
4. TEFG c TGHE
4. _________?_________
Given
© &*
&*
SP c TP
P
&* bisects aSPT.
PQ
S
T
Prove © TSPQ c TTPQ
P
Statements
Reasons
&*
1. &*
SP c TP
1. Given
&* bisects aSPT.
2. PQ
2. _________?_________
3. aSPQ c aTPQ
3. _________?_________
4. _________?_________
4. Reflexive Prop. of Cong.
5. TSPQ c TTPQ
5. _________?_________
5.2
Proving Triangles are Congruent: SSS and SAS
247
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36. Reasoning Fill in the missing statements and reasons.
Given
© AC
&* c BC
&*
C
&*.
M is the midpoint of AB
Prove © TACM c TBCM
A
Student Help
STUDY TIP
It is helpful to label
congruent sides and
congruent angles in
steps of a proof.
Statements
Reasons
&* c BC
&*
S 1. AC
1. Given
&*.
2. M is the midpoint of AB
B
M
2. _________?_________
S 3. _________?_________
3. Definition of midpoint
&** c CM
&**
S 4. CM
4. _________?_________
5. TACM c TBCM
5. _________?_________
37. Error Analysis Using the diagram below, Maria was asked whether
it can be shown that TABC c TDEF. Explain her error.
C
Maria
A
B
E
D
TABC c TDEF by
the SAS Congruence
Postulate.
F
Challenge Write a proof to show that the triangles are congruent.
38. TABD and TCBD
&* c YZ
&*, and ZM
&**
39. XZ
are equilateral.
bisects aYZX.
Y
B
A
M
C
X
D
Standardized Test
Practice
Z
&* c AB
&*, ST
&* c BC
&*, and
40. Multiple Choice In TRST and TABC, RS
&* c CA
&*. Which angle is congruent to aT ?
TR
A
X
aA
B aR
X
C aC
X
D aB
X
41. Multiple Choice In the diagram below, TDEF is equilateral
&*. Which of the statements
and G is the midpoint of DE
is not true?
F
X
H
X
&* c EF
&*
DF
&** c EG
&*
DG
&** c DF
&*
G DG
X
J TDFG c TEFG
X
D
248
Chapter 5
Congruent Triangles
F
G
E
Page 9 of 9
Mixed Review
Classifying Angles Use the diagram to determine whether the
angles are corresponding, alternate interior, alternate exterior,
or same-side interior angles. (Lesson 3.3)
42. a1 and a5
43. a2 and a6
44. a7 and a2
45. a5 and a8
46. a9 and a4
47. a5 and a9
1
3 4
5
2
6
7
8 9
Using the Triangle Inequality Determine whether it is possible to
draw a triangle with the given side lengths. (Lesson 4.7)
48. 14, 8, 25
Algebra Skills
49. 20, 10, 28
50. 16, 14, 30
Evaluating Square Roots Evaluate. Give the exact value if possible. If
not, approximate to the nearest tenth. (Skills Review, p. 669)
51. Ï3
w
52. Ï1
w2
w
53. Ï4
w0
w
54. Ï1
w5
w9
w
55. Ï1
w4
w.7
w6
w
56. Ï0
w.8
w7
w
57. Ï1
w.1
w2
w
58. Ï4
w0
w.8
w5
w
Quiz 1
In the diagram, TABC c TQPR. Complete the statement with the
corresponding congruent part. (Lesson 5.1)
C R
&*
1. aR c __?__
2. AB c __?__
3. TBAC c __?__
4. TRPQ c __?__
A
5. Write a congruence statement for
B
Z
the congruent triangles shown at
the right. (Lesson 5.1)
P
P
Y
F
X
E
G
Decide whether enough information is given to show that the
triangles are congruent. If so, tell which congruence postulate
you would use. Explain your reasoning. (Lesson 5.2)
7. D
6. A
B
C
Y
X
G
E
8.
F
F
J
Z
5.2
K
M
Proving Triangles are Congruent: SSS and SAS
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249