Congruent Polygons
Using the information we already know about the geometric figure, we
are going to learn to prove that two polygons are congruent, beginning
with triangles.
 To prove that two triangles (or any polygons) are congruent, we
must be able to show that their corresponding sides and their
corresponding angles are congruent.
o Corresponding sides — When dealing with two polygons, their
corresponding sides are the sides that sit in the same position or
occupy the same spot in each polygon.
o Corresponding angles — The angles that sit in the same
position or occupy the same spot in the polygon.
A
B
P
C
Q
R
EX: Given ABC  PQR
Corresponding angles:  A corresponds to  P
Corresponding segments: AB corresponds to PQ
1
M&G 3.4 Instruction
Congruent Polygons
Postulate – A statement about relationships in mathematics that are
accepted as true without proof.
We are going to use two cases/conditions that will allow us to prove
that two triangles are congruent.
o Side-Angle-Side congruence postulate (SAS)—says that two
triangles are congruent if two sides and the angle included
between them are congruent to the corresponding two sides and
angle between them in a second triangle.
o Side-Side-Side congruence postulate
(SSS)—says that two
A
triangles are congruent if all three sides of one triangle are
congruent to all three sides of a second triangle.
A
J
EX 1:
C
B
L
K
Prove that  ABC is congruent to  JKL.
1. m AB = m JK
2. m AC  mJL
3. mBC  mKL
4.  ABC   JKL by SSS
2
M&G 3.4 Instruction
Congruent Polygons
M
P
EX 2:
O
N
Q
R
Prove that  MNO is congruent to  PQR
1. OM  RP
2. MN  PQ
3.  OMN   RPQ (  M   P)
4.  MNO   PQR by SAS
S
EX 3:
R
1.
2.
3.
4.
5.
3
T
G
F
H
RS  FG
ST  GH
R  F
Are the triangles congruent?
This is a non-example since the angles that are congruent in
each triangle are not the included angle and we can see that
M&G 3.4 Instruction
Congruent Polygons
the triangles are not congruent. ASS is not a valid triangle
congruence postulate
CHECK FOR UNDERSTANDING:
On your white board, tell me whether or not each pair of triangles is
congruent. If they are congruent, write a statement of congruency and
tell me how you know that they are congruent.
For example:
G
D
1.
F
H
E
DEF  HGF because of the SAS Postulate.
Z
2.
W
4
X
Y
M&G 3.4 Instruction
Congruent Polygons
3.
I
F
D
G
E
B
H
E
4.
D
C
A
F
5.
P
J
M
N
5
K
L
M&G 3.4 Instruction
Congruent Polygons
A
EX5:
B
C
D
 Given  ABC, AB  AC and AD bisects  A.
 Prove that  ACD   ABD
o AB  AC because it is given.
o  BAD =  CAD because  A is bisected by AD and
bisecting an angle creates two equal angles.
o AD  AD because it is itself (reflexive property).
o  ACD   ABD by the SAS postulate.
A
EX 6:
F
B
C
E
D
This is a regular hexagon with 2 diagonals drawn. Prove that
 FAB   CDE
6
M&G 3.4 Instruction
Congruent Polygons
 AB  DE and AF  DC since all sides are congruent in a
regular hexagon.
  A =  D since all angles are congruent in a regular hexagon.
  FAB   CDE by SAS postulate.
CFU – Are the parallelograms pictured below congruent? How
do you know this?
B
A
7
C
D
P
Q
S
M&G 3.4 Instruction
R