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Proving Triangles are
Congruent: SSS, SAS
Section 5.2
Objectives:
• Show triangles are congruent using SSS and
SAS.
Key Vocabulary
• Included angle
• proof
Postulates
• 12 SSS Congruence Postulate
• 13 SAS Congruence Postulate
Review
• Congruent Triangles
▫ Triangles that are the same shape and size
are congruent.
▫ Each triangle has three sides and three
angles.
▫ If all 6 parts (3 corresponding sides and 3
corresponding angles) are
congruent….then the triangles are
congruent.
Review
• CPCTC –
Corresponding Parts of
Congruent Triangles are
Congruent
• Be sure to label  Δs with
proper markings (i.e. if
D  L, V  P, W  M,
DV  LP, VW  PM, and
WD  ML then we must
write ΔDVW  ΔLPM)
So, to prove Δs  we must prove
ALL sides & ALL s are  ?
Fortunately, NO!
There are some shortcuts…
Two of which follow!!!
Postulate 12: Side-Side-Side
Congruence Postulate
SSS Congruence - If three sides of one triangle are congruent
to three sides of a second triangle, then the two triangles are
congruent.
If MN  QR, NP  RS , PM  SQ then, MNP  QRS
Remember!
Adjacent triangles share a side, so you can apply the Reflexive
Property to get a pair of congruent parts.
∆PMN is adjacent to ∆PON, PN≅PN by the Reflexive Property.
Example 1: Using SSS to Prove
Triangle Congruence
Use SSS to explain why ∆ABC  ∆DBC.
It is given that AC  DC and that AB  DB. By the
Reflexive Property of Congruence, BC  BC.
Therefore ∆ABC  ∆DBC by SSS.
Your Turn
Use SSS to explain why
∆ABC  ∆CDA.
It is given that AB  CD and BC  DA.
By the Reflexive Property of Congruence, AC  CA.
So ∆ABC  ∆CDA by SSS.
Example #2 – SSS – Coordinate Geometry
Use the SSS Postulate to show the two triangles are
congruent. Find the length of each side.
AC = 5
BC = 7
2
2
AB = 5  7  74
MO = 5
NO = 7
MN =
52  72  74
VABC VMNO
Example 3
Does the diagram give enough
information to show that the
triangles are congruent? Explain.
SOLUTION
From the diagram you know that HJ  LJ and HKLK.
By the Reflexive Property, you know that JK  JK.
ANSWER
Yes, enough information is given. Because
corresponding sides are congruent, you can use the
SSS Congruence Postulate to conclude that ∆HJK 
∆LJK.
Definition: Included Angle


The angle between two sides in a
figure.
∠B is included between:
BA & BC
INCLUDED ANGLE
Y
C
A
B
X
Z
Angle in between two consecutive sides
Use the diagram. Name the
included angle between the pair
of sides given.
∠MTR
∠RTQ
∠MRT
∠Q
The 2nd Congruence Short Cut
• It can also be shown that only two pairs of
congruent corresponding sides are needed to
prove the congruence of two triangles if the
included angles are also congruent.
• SAS
Postulate 13: Side-Angle-Side
Congruence Postulate
SAS Postulate – 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.
If PQ  WX , QS  XY ,  Q  X then, PQS  WXY
Caution
The letters SAS are written in that order because the
congruent angles must be between pairs of congruent
corresponding sides.
S
S
S
A
S
A
Example 4: Engineering Application
The diagram shows part of the support
structure for a tower. Use SAS to
explain why ∆XYZ  ∆VWZ.
It is given that XZ  VZ and that YZ  WZ. By
the Vertical s Theorem. XZY  VZW.
Therefore ∆XYZ  ∆VWZ by SAS.
Example 5
Does the diagram give enough information to use the SAS
Congruence Postulate? Explain your reasoning.
a.
b.
SOLUTION
a.
From the diagram you know that AB  CB and DB  DB.
The angle included between AB and DB is ABD.
The angle included between CB and DB is CBD.
Because the included angles are congruent, you can use the
SAS Congruence Postulate to conclude that ∆ABD  ∆CBD.
Example 5
b.
You know that GF  GH and GE  GE. However, the
congruent angles are not included between the congruent
sides, so you cannot use the SAS Congruence Postulate.
Your Turn
Use SAS to explain why ∆ABC 
∆DBC.
It is given that BA  BD and ABC  DBC. By the
Reflexive Property of , BC  BC. So ∆ABC  ∆DBC
by SAS.
Example 6A: Verifying Triangle
Congruence
Show that the triangles are congruent for the given value of the
variable.
∆MNO  ∆PQR, when x = 5.
PQ = x + 2
=5+2=7
QR = x = 5
PQ  MN, QR  NO, PR  MO
∆MNO  ∆PQR by SSS.
PR = 3x – 9
= 3(5) – 9 = 6
Example 6B: Verifying Triangle
Congruence
Show that the triangles are congruent for the given value of the
variable.
∆STU  ∆VWX, when y = 4.
ST = 2y + 3
= 2(4) + 3 = 11
TU = y + 3
=4+3=7
mT = 20y + 12
ST  VW, TU  WX, and T  W.
∆STU  ∆VWX by SAS.
= 20(4)+12 = 92°
Your Turn
Show that ∆ADB  ∆CDB, t = 4.
DA = 3t + 1
= 3(4) + 1 = 13
DC = 4t – 3
= 4(4) – 3 = 13
mD = 2t2
= 2(16)= 32°
ADB  CDB Def. of .
DB  DB
Reflexive Prop. of .
∆ADB  ∆CDB by SAS.
#1
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they
are congruent.
R
S
T
ΔRST  ΔYZX by SSS
#2
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they
are congruent.
P
R
S
Q
ΔPQS  ΔPRS by SAS
#3
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they
are congruent.
R
T
S
Not congruent.
Not enough Information to Tell
#4
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they
are congruent.
P
S
U
Q
R
T
ΔPQR  ΔSTU by SSS
#5
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they
are congruent.
M
P
R
N
Q
Not congruent.
Not enough Information to Tell
Proof
• A proof is an argument that uses logic,
definitions, properties, and previously
proven statements to show that a
conclusion is true.
• An important part of writing a proof is
giving justifications to show that every
step is valid.
Two-Column Proof
• Two-Column Proof – A proof format used in
geometry in which an argument is presented with two
columns, statements and reasons, to prove
conjectures and theorems are true. Also referred to as a
formal proof.
• Two- Column Proof is a formal proof because it has a
specific format.
• Two-Column Proof
▫ Left column – statements in a logical progression.
▫ Right column – Reason for each statement (definition, postulate,
theorem or property).
Two-Column Proof
Given:
Prove:
Proof:
Statements
Reasons
How to Write A Proof
1.
2.
3.
4.
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
5. List statements in order. If a statement relies
on another statement, list it later than the
statement it relies on
6. End the proof with the statement you are
trying to prove
Geometric Proof
• Since geometry also uses variables, numbers,
and operations, many of the algebraic properties
of equality are true in geometry. For example:
Property
Segments
Angles
Reflexive
AB = AB
m ∠1 = m ∠1
Symmetric
If AB = CD,
then CD = AB.
If m ∠1 = m ∠2,
then m ∠2 = m ∠1.
Transitive
If AB = CD and
CD = EF, then
AB = EF.
If m ∠1 = m ∠2 and
m ∠2 = m ∠3, then
m ∠1 = m ∠3.
• These properties can be used to write geometric
proofs.
Remember!
Numbers are equal (=) and figures are
congruent ().
Example 7
Write a two-column proof that shows ∆JKL  ∆NML.
JL  NL
L is the midpoint of KM.
∆JKL  ∆NML
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.
Example 7
Statements
Reasons
1. JL  NL
1. Given
2. L is the
midpoint of
KM.
2. Given
3. JKL  NML
3. Vertical
Angles
Theorem
These are the given
statements.
This information is from the
diagram.
Example 7
Statements
Reasons
4. KL  ML
4. Definition of
midpoint
5. ∆JKL  ∆NML
5. SAS
Congruence
Postulate
Statement 4 follows
from Statement 2.
Statement 5 follows from the
congruences of Statements 1,
3, and 4.
Example 8
You are making a model of the window shown in the figure. You
know that
DR  AG and RA  RG. Write a
proof to show that ∆DRA  ∆DRG.
D
A
R
SOLUTION
1. Make a diagram and label it with
the given information.
G
Example 8
2. Write the given information and the statement you need to
prove.
DR  AG, RA  RG
∆DRA  ∆DRG
3. Write a two-column proof. List the given statements first.
Example 8
Statements
Reasons
1. RA  RG
1. Given
2. DR  AG
2. Given
3. DRA and DRG are
right angles.
4. DRA  DRG
3.  lines form right angles.
5. DR  DR
5. Reflexive Property of
Congruence
6. ∆DRA  ∆DRG
6. SAS Congruence
Postulate
4. Right angles are
congruent.
Your Turn:
1. Fill in the missing statements and reasons.
CB  CE, AC  DC
∆BCA  ∆ECD
Statements
Reasons
1. CB  CE
1. _____
?
ANSWER
Given
2. _____
?
2. Given
ANSWER
AC  DC
3. BCA  ECD 3. _____
?
ANSWER
Vertical Angles
Theorem
4. ∆BCA  ∆ECD
ANSWER
SAS
Congruence
Postulate
4. _____
?
Example 9:
Given: QR  UT, RS  TS, QS = 10, US = 10
Prove: ΔQRS  ΔUTS
U
Q
10
R
10
S
T
Example 9:
Given: QR  UT, RS  TS, QS = 10, US = 10
Prove: ΔQRS  ΔUTS
Q
U
10
R
10
S
Statements
Reasons________
1. QR  UT, RS  TS, 1. Given
QS=10, US=10
2. QS = US
2. Substitution
3. QS  US
3. Def of  segs.
4. ΔQRS  ΔUTS
4. SSS Postulate
T
Your turn: Proof by SSS
Given: RP  RT
RX bisects PT
Prove: PRX  TRX
Statement
1) RP  RT (s)
2) RX bisects PT
3) X is midpoint of PT
4) PX  XT
(s)
5) RX  RX
(s)
6) PRX  TRX
R
P
48
X
T
Reason
1)
2)
3)
4)
5)
6)
Given
Given
Def. line bisector
Def. midpoint
Reflexive Postulate
SSS Postulate
Example 10: Proving Triangles
Congruent
Given: BC ║ AD, BC  AD
Prove: ∆ABD  ∆CDB
Statements
Reasons
1. BC || AD
1. Given
2. CBD  ABD
2. Alt. Int. s Thm.
3. BC  AD
3. Given
4. BD  BD
4. Reflex. Prop. of 
5. ∆ABD  ∆ CDB
5. SAS
Your Turn
Given: QP bisects RQS. QR  QS
Prove: ∆RQP  ∆SQP
Statements
Reasons
1. QR  QS
1. Given
2. QP bisects RQS
2. Given
3. RQP  SQP
3. Def. of bisector
4. QP  QP
4. Reflex. Prop. of 
5. ∆RQP  ∆SQP
5. SAS
Summary – SSS & SAS
Hints to Proofs!
• If two triangles share a side, in the statement
column state that the side is congruent to itself.
The reason is the Reflexive Property.
• If two triangles share a vertex, in the
statement column state that the angles are
congruent. The reason is the Vertical Angles
Theorem.
Hints to Proofs!
• If a vocabulary word is in the given, you will
need its definition as a reason somewhere in
the proof, otherwise you wouldn’t need to know
that piece of information.
• If two lines are parallel, look for congruent
angles pairs, such as Alternate Interior Angles,
Corresponding Angles, or Alternate Exterior
Angles.
Assignment
• Pg. 245 -249 #1 – 25 odd, 29 – 37 odd