Download Proving Triangles are Congruent

Proving Triangles
are Congruent
EQ: What can you conclude about two
triangles when you know that two pairs of
corresponding sides and corresponding
included angles are congruent?
Today, you will learn to…
* prove that triangles are congruent
* use congruence postulates to solve
problems
SSS Experiment
Using 3 segments, can you ONLY
create 2 triangles that are
congruent?
Side-Side-Side
Congruence Postulate
B
If Side AB  XY
C
A
Y
Side AC  XZ
Side BC  YZ,
then ΔABC  ΔXYZ
by SSS
X
Z
If 3 pairs of sides are congruent, then
the two triangles are congruent.
1. Does the diagram give enough
info to use SSS Congruence?
K
L
C
A
no
J
B
2. Given: LN  NP and
M is the midpoint of LP
Prove: ΔNLM  ΔNPM
N
L
P
M
Def of midpoint
Reflexive Property
LM  MP
NM  NM
NLM  NPM SSS Congruence
SAS Experiment
Using 2 congruent segments and
1 included angle, can you ONLY
create 2 triangles that are
congruent?
B
Side-Angle-Side
Congruence Postulate
If Side AB  XY
C
A
Y
Angle B  Y
Side BC  YZ,
then ΔABC  ΔXYZ
by SAS
X
Z
If 2 pairs of sides and their included
angle are congruent, then the two
triangles are congruent.
SAS?
5.
4.
NO!
SAS
6.
7.
NO!
SAS
8. Does the diagram give enough
info to use SAS Congruence?
A
B
D
C
ABD  AC
_ _D
_ by SAS
9. Does the diagram give enough
info to use SAS Congruence?
W
V
no
Y
X
Z
10. Does the diagram give enough
info to use SAS Congruence?
E
A
no
B
C
D
11. Given:
W is the midpoint of VY
and the midpoint of ZX
Prove: ΔVWZ  ΔYWX
VW  WY and ZW  WX
Def. of midpoint
VWZ YWX Vertical Angles Th
VWZ  YWX SAS Congruence
12. Given: AB  PB , MB  AP
Prove: ΔMBA  ΔMBP
M
A
B
P
ABM & PBM are right s Def of  lines
ABM  PBM
MB  MB
MBA  MBP
All right s are 
Reflexive Property
SAS Congruence
What is the best way to
get better at proofs?
ASA Experiment
Using 2 angles connected by 1
segment, can you ONLY create
two triangles that are congruent?
Angle-Side-Angle
Congruence Postulate
B
If Angle B  Y,
C
A Y
Side BC  YZ,
Angle C  Z
then ΔABC  ΔXYZ
X
Z
by ASA
If 2 pairs of angles and the included
sides are congruent, then the two
triangles are congruent.
Included side?
A
B
C
The included side between
AB
 A and  B is _____
Included side?
A
B
C
The included side between
CB
 B and  C is _____
Included side?
A
B
C
The included side between
 A and  C is _____
AC
ASA?
2.
1.
NO!
ASA
3.
4.
NO!
ASA
5. Does the diagram give enough
info to use ASA Congruence?
B
A
C
Reflexive Property
Third Angles Theorem
D
Δ ABD  Δ ACD by ASA
6. Does the diagram give enough
info to use ASA Congruence?
A
D
Reflexive
Property
B
Alt. Int.
Angles
Theorem
C
yes, ΔACB  ______
ΔC A D by ASA
7. Does the diagram give enough
info to use ASA Congruence?
A
D
Reflexive
Property
B
C
no
8. Does the diagram give enough
info to use ASA Congruence?
K
L
J
C
B
A
KLJ  ACB
_ _ _ by ASA
9. Determine whether the triangles
are congruent by ASA.
L
K
Vertical Angles
Theorem
H
J
Alt. Int. Angles
Theorem
G
HJG   K
_ _J L_ by ASA
Angle-Angle-Side
Congruence Theorem
B
If Angle B  Y
C
A
Angle C  Z
Y
Side AB  XY
then ΔABC  ΔXYZ
by AAS
Z
X
If 2 pairs of angles and a pair of
nonincluded sides are congruent, then
the two triangles are congruent.
AAS?
11.
10.
AAS
NO!
12.
13.
AAS
NO!
14. Does the diagram give enough
info to use AAS Congruence?
B
A
C
Reflexive
Property
D
ABD   ACD
_ _ _ by AAS
15. Does the diagram give enough
info to use AAS Congruence?
K
L
J
C
A
B
KLJ  ACB
_ _ _ by AAS
16. Determine whether the triangles
are congruent by AAS.
L
K
Vertical Angles
Theorem
H
J
Alt. Int. Angles
Theorem
G
HJG  K
_ _JL_ by AAS
SSA Experiment
Using 2 sides and 1 angle that is NOT
included, can you ONLY create two
triangles that are congruent?
NO
AAA Experiment
Using 3 angles, can you ONLY create
two triangles that are congruent?
NO
All of the angles are ,
but the s are NOT 
Mark the given information on the
triangles. What additional congruence
would you need to show ABC  XYZ?
17. CB  ZY , AC  XZ
SAS Congruence
X
A
C
B
Z
C  Z
Y
Mark the given information on the
triangles. What additional congruence
would you need to show ABC  XYZ?
18. CB  ZY , AC  XZ
SSS Congruence
X
A
C
B
Z
AB  XY
Y
Mark the given information on the
triangles. What additional congruence
would you need to show ABC  XYZ?
19. CB  ZY , C  Z
SAA Congruence
X
A
C
B
Z
A  X
Y
What is the best way to
get better at proofs?