Download Proving Triangles are Congruent: SSS and SAS

5.2 Proving Triangles are Congruent:
SSS and SAS
Goal
Show triangles are congruent using SSS and SAS.
VOCABULARY
Proof A proof is a convincing argument that shows why a
statement is true.
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
&**
Side NP
&*
Side PM
&*
then TMNP
Example 1
c
c
c
c
QR
&*, and
RS
&*, and
SQ
&*,
T QRS .
Q
M
N
P R
S
Use the SSS Congruence Postulate
Does the diagram give enough
information to use the SSS
Congruence Postulate? Explain.
J
H
L
K
Solution
From the diagram you know that HJ
& c LJ
& and HK
&* c LK
&* .
By the Reflexive Property, you know that JK
& c JK
&.
Because all three pairs of corresponding sides are congruent,
you can use the SSS Congruence Postulate to conclude that
THJK c T LJK .
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Geometry, Concepts and Skills Notetaking Guide • Chapter 5
POSTULATE 13: SIDE-ANGLE-SIDE CONGRUENCE POSTULATE
(SAS)
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 If Side
PQ
&* c WX
&**, and
Angle aQ c aX, and
Side
QR
&* c XY
&*,
then TPQR c T WXY .
Example 2
Q
P
X
R W
Y
Use the SAS Congruence Postulate
Does the diagram give enough information to use the SAS
Congruence Postulate? Explain your reasoning.
a.
b.
D
H
E
G
A
B
C
F
Solution
a. You know that AB
& c CB
&* and DB
&* c DB
&* .
The included angle between AB
& and DB
&* is aABD.
The included angle between CB
&* and DB
&* is a CBD .
Because the included angles are congruent, you can use
the SAS Congruence Postulate to conclude that
TABD c T CBD .
b. You know that GF
&* c GH
&* and GE
&* c GE
&* but the congruent
angles a GEH and a GEF are not included between the
congruent sides, so you cannot use the SAS Congruence
Postulate.
Follow-Up
In Example 2, part (b), what two angles must be congruent in
order to use the SAS Congruence Postulate?
aHGE and aFGE
Lesson 5.2 • Geometry, Concepts and Skills Notetaking Guide
111
Checkpoint Does the diagram give enough information to
determine whether the triangles are congruent?
1.
2.
Yes
3.
No
Yes
STEPS FOR WRITING A PROOF
p List the given information first.
p Decide whether you can use the given information as it is or if
you need to make another statement based on the given
information.
p List your statements in sequential order—in other words, don’t
list a conclusion before you list the statement that allows you to
make that conclusion.
p Give a reason for every statement that you make.
p As reasons in a proof, use given information, labeled diagrams,
postulates, definitions, and theorems.
p End the proof with the statement you are trying to prove .
Example 3
Write a Proof
Write a two-column proof that
shows THJK c TLKJ.
J
Solution
Statements
K
H
Reasons
Side 1. HJ
& c LK
&*
1. Given
Side 2. HK
&* c LJ
&
2. Given
Side 3. JK
&* c JK
&*
3. Reflexive Property of Congruence
4. THJK c TLKJ
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L
Geometry, Concepts and Skills Notetaking Guide • Chapter 5
4. SSS Congruence Postulate
Example 4
Prove Triangles are Congruent
In the diagram at the right, DR
&* ∏ AG
&* and
RA
&* c RG
&*. Write a proof to show that
TDRA c TDRG.
D
Solution
A
1. Label the diagram with the given information.
R
G
2. Write what you are given and what you need to prove.
Given: DR
&* ∏ AG
&* , RA
&* c RG
&*
Prove: TDRA c TDRG
3. Write a two-column proof. List the given statements first.
Statements
Reasons
1. RA
&* c RG
&*
1. Given
2. DR
&* ∏ AG
&*
2. Given
3. aDRG and aDRA
are right angles.
3. Perpendicular lines form right
angles.
4. aDRG c aDRA
4. All right angles are congruent .
5. DR c DR
5. Reflexive Property of
Congruence
6. T DRA c T DRG
6. SAS Congruence Postulate
Checkpoint Complete the proof.
4. Given: CB
&* c CE
&, AC
& c DC
&*
Prove: TBCA c TECD
Statements
B
D
C
A
E
Reasons
1. CB
&* c CE
&
1. Given
2. AC
&* c DC
&*
2. Given
3. aBCA c aECD
3. Vertical Angles Theorem
4. TBCA c TECD
4. SAS Congruence Postulate
Lesson 5.2 • Geometry, Concepts and Skills Notetaking Guide
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