# Proving Triangles Congruent - Po

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```					Proving Triangles
Congruent
The Idea of a Congruence

Two geometric figures with
exactly the same size and
shape.
F

B

A     C    E   D
How much do you
need to know. . .

. . . about two triangles
to prove that they
are congruent?
Corresponding Parts
In Lesson 4.2, you learned that if all
six pairs of corresponding parts (sides
and angles) are congruent, then the
triangles are congruent.
1. AB  DE
2. BC  EF
3. AC  DF
4.  A   D
ABC   DEF
5.  B   E
6.  C   F
Do you need all six ?

NO !

SSS
SAS
ASA
AAS
Side-Side-Side (SSS)

1. AB  DE
2. BC  EF            ABC   DEF
3. AC  DF
Postulate 19: SSS Congruence Postulate
EXAMPLE 3     Solve a real-world problem

Structural Support

Explain why the bench with the diagonal support is
stable, while the one without the support can collapse.
EXAMPLE 3     Solve a real-world problem

SOLUTION

The bench with a diagonal support forms triangles with
fixed side lengths. By the SSS Congruence Postulate,
these triangles cannot change shape, so the bench is
stable. The bench without a diagonal support is not
stable because there are many possible quadrilaterals
with the given side lengths.
EXAMPLE 2        Standardized Test Practice

SOLUTION
By counting, PQ = 4 and QR = 3. Use the Distance Formula
to find PR.
d   =     ( x2 – x1 ) 2 + ( y2 – y1 ) 2
EXAMPLE 2            Standardized Test Practice

PR =       ( – 1 – (– 5 ) )2 + ( 1 – 4 ) 2

=      4 2 + (– 3 ) 2 =      25 = 5

By the SSS Congruence Postulate, any triangle with
side lengths 3, 4, and 5 will be congruent to PQR.
The distance from (–1, 1) to (–1, 5) is 4. The distance from
(–1, 5) to (–4, 5) is 3. The distance from (– 1, 1) to (–4, 5) is
( 5 – 1) 2 + ( (–4) – (–1) ) 2 =      4 2 + (– 3 ) 2 =   25 = 5

EXAMPLE 1         Use the SSS Congruence Postulate

Write a proof.

GIVEN    KL       NL, KM    NM

PROVE         KLM          NLM

Proof    It is given that KL     NL and KM     NM
By the Reflexive Property, LM       LM.
So, by the SSS Congruence
Postulate,   KLM     NLM
GUIDED PRACTICE       for Example 1

Decide whether the congruence statement is true.

1.     DFG        HJK

SOLUTION
Three sides of one triangle are congruent to three
sides of second triangle then the two triangle are
congruent.
Side DG   HK, Side DF   JH,and Side FG JK.
So by the SSS Congruence postulate,     DFG          HJK.
Yes. The statement is true.
GUIDED PRACTICE       for Example 1

Decide whether the congruence statement is true.

SOLUTION

PROOF:    It is given that BC AD By Reflexive property
AC AC, But AB is not congruent CD.
Side-Angle-Side (SAS)
Postulate 20

1. AB  DE
2. A   D    ABC   DEF
3. AC  DF
included
angle
Included Angle
The angle between two sides

G          I           H
Included Angle

Name the included angle:
E

YE and ES     E
ES and YS     S

Y   S      YS and YE     Y
EXAMPLE 2        Prove the AAS Congruence Theorem

Prove the Angle-Angle-Side Congruence Theorem.

Write a proof.

GIVEN       A         D,   C     F, BC   EF

PROVE          ABC        DEF
GUIDED PRACTICE         for Examples 1 and 2

1.    In the diagram at the right, what
postulate or theorem can you use to
prove that     RST       VUT ? Explain.

SOLUTION
STATEMENTS                   REASONS
S      U                Given
RS    UV                 Given
RTS       UTV              The vertical angles
are congruent
Do Now
Eliminate the
possibilities….
• Some of the measurements of ABC
and DEF are given below. Can you
determine if the two triangles are
congruent from this information?
2.5

2.5 cm
30                     30

4 cm                   4cm
Angle-Side-Angle (ASA)
Postulate 21

1. A   D
2. AB  DE     ABC   DEF
3.  B   E
include
d
side
Included Side
The side between two angles

GI          HI               GH
Included Side

Name the included angle:
E

Y and E   YE
E and S   ES
Y   S     S and Y   SY
Angle-Angle-Side (AAS)
Theorem 4.6

1. A   D
2.  B   E   ABC   DEF
3. BC  EF
Non-included
side
Angle-Angle-Side (AAS)
• If two angles and a nonincluded side
of a triangle are congruent to two
angles and a nonincluded side of
another triangle, then the two
triangles are congruent.
O                 A

Y
B                 M           N
Angle-Side-Side
• THERE IS NO SUCH THING
AS ANGLE SIDE SIDE
BECAUSE YOU CAN’T USE
THAT KIND OF LANGUAGE
AT SCHOOL.
Warning: No SSA Postulate

There is no such
thing as an SSA
postulate!

B               E

F
A      C
D

NOT CONGRUENT
Warning: No AAA Postulate
There is no such
thing as an AAA
postulate!

E
B

A     C                  F
D

NOT CONGRUENT
Tell whether you can use the
given information at
determine whether
ABC   DEF

A  D, ABDE, ACDF

AB  EF, BC  FD, AC DE
The Congruence Postulates
 SSS correspondence

 ASA correspondence

 SAS correspondence

 AAS   correspondence
 SSA correspondence

 AAA correspondence
Name That Postulate
(when possible)

SAS
ASA

SSA                 SSS
Name That Postulate
(when possible)

AAA
ASA

SAS                     SSA
Name That Postulate
(when possible)

Vertical
Angles
Reflexive
Property    SAS                    SAS

Vertical           Reflexive
Angles     SAS     Property    SSA
HW: Name That Postulate
(when possible)
HW: Name That Postulate
(when possible)
Let’s Practice
to enable us to apply the specified
congruence postulate.

For ASA:   B  D
For SAS:   AC  FE
For AAS:   A  F
Closure
to enable us to apply the specified
congruence postulate.

For ASA:

For SAS:

For AAS:
Now For The Fun Part…
J
Given: JO  SH;
O is the midpoint of SH
Prove:  SOJ  HOJ
S       0   H
Write a two column Proof
Given: BC bisects AD and A   D
Prove: AB  DC
A             C
E
B             D

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