Download SSS ans SAS Postulates

Warm Up
On Desk (5 min)
Do Daily Quiz 5.1 (10 min)
Review
-go over the Daily Quiz
items in 5.1
5.2
ESSENTIAL OBJECTIVE
 Show triangles are
congruent using SSS and
SAS.
In Exercises 1–5, use the triangles below.
Determine whether the given angles or sides represent
corresponding angles, corresponding sides, or neither.
1. B and H
ANSWER
Corresponding angles
2. DB and HK
ANSWER
neither
Complete the statement with the corresponding
congruent part.
?
3. J  _____
ANSWER
D
?
4. CB  _____
ANSWER
KH
5. The triangles are congruent. Identify all pairs of
corresponding congruent parts. Then write a
congruence statement.
ANSWER
B  H, D  J, C  K, BD  HJ, BC  HK,
CD  KJ; ∆BCD  ∆HKJ
5.2
VOCABULARY
 A proof is a convincing argument that
shows why a statement is true.
Side-Side-Side Congruence
Postulate (SSS)
 If three sides of one triangle are congruent
to three sides of a second triangle, then
the two triangles are congruent.
Example 1
Use the SSS Congruence Postulate
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.
Side-Angle-Side Congruence
Postulate (SAS)
 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.
Example 2
Use the SAS Congruence Postulate
Does the diagram give enough information to use the
SAS Congruence Postulate? Explain your reasoning.
a.
OLUTION
a. From the diagram you know that AB  CB and DB  DB.
angle ABD and angle CBD are  (coz both are 90)
Yes, we can use the SAS Congruence Postulate to
conclude that ∆ABD  ∆CBD.
Example 2
Use the SAS Congruence Postulate
b. You know that GF  GH and GE  GE. However, it
does not follow the SAS congruence postulate. So,
No, we cannot use the SAS Congruence Postulate.
Introducing: Two Column
Proof
Example 3
Write a Proof
Write a two-column proof that shows ∆JKL  ∆NML.
JL  NL
L is the midpoint of KM.
∆JKL  ∆NML
SOLUTION
To set up the two column proof, start with the given:
Example 3
Write a Proof
Statements
Reasons
1. JL  NL
1. Given
2. L is the
midpoint of
KM.
2. Given
3. KL  ML
4. JLK  NLM
(Side)
3. Definition of (Side)
midpoint
(An included angle!)
4. Vertical
Angles
Theorem
Example 3
Write a Proof
Statements
Reasons
1. JL  NL
1. Given
2. L is the
midpoint of
KM.
2. Given
3. KL  ML
4. JKL  NML
5. ∆JKL  ∆NML
Side
3. Definition of side
midpoint
An included angle
4. Vertical
Angles
Theorem
5. SAS
Congruence
Postulate
Example 4
From the figure,
DR  AG and RA  RG. Write a
proof to show that ∆DRA  ∆DRG.
D
A
R
G
SOLUTION
1. Make a diagram and label it with
the given information.
Checkpoint
Prove Triangles are Congruent
Example.
CB  CE , AC  DC
∆BCA  ∆ECD
Statements
Reasons
1. CB  CE
1. _____
?
ANSWER
2. _____
?
2. Given
ANSWER
3. BCA  ECD 3. _____
?
ANSWER
4. ∆BCA  ∆ECD 4. _____
?
ANSWER
Example 4
Prove Triangles are Congruent
Statements
1. RA  RG
Reasons
side
1. Given
2. DR  AG
2. Given
3. DRA and DRG are
right angles.
4. DRA  DRG angle
3.  lines form right angles.
5. DR  DR
5. Reflexive Property of
Congruence
6. ∆DRA  ∆DRG
side
4. Right angles are
congruent.
6. SAS Congruence
Postulate
Checkpoint
Prove Triangles are Congruent
Fill in.
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 4. _____
?
ANSWER SAS
Congruence
Postulate
Hw 5.2A