Lesson 4.4 Triangle
Congruencies
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
of congruent triangles
Triangles that are the same size and
shape are congruent triangles.
Each triangle has three angles and
three sides. If all six corresponding
parts are congruent, then the
triangles are congruent.
Corresponding parts
of congruent triangles
Y
B
C
A
Z
X
If ΔABC is congruent to ΔXYZ , then vertices
of the two triangles correspond in the same
order as the letter naming the triangles.
~
ΔABC = ΔXYZ
Corresponding parts
of congruent triangles
Y
B
C
A
Z
X
~
ΔABC = ΔXYZ
This correspondence of vertices can be used
to name the corresponding congruent sides
and angles of the two triangles.
Corresponding Parts
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
5.  B   E
6.  C   F
ABC   DEF
Corresponding Sides
Corresponding Angles
AB  PQ
BC  QR
∠B ≅ ∠Q
AC  PR
∠C ≅ ∠R
∠A ≅ ∠P
∆ABC ≅ ∆PQR
Proving Two Triangles are
Congruent
Do we need to use all six
pairs to prove two
triangles are congruent?
Do you need all six ?
NO !
WHY?
That’s why…
That’s why there are some shortcuts
(but applicable to TRIANGLES
ONLY).
These shortcuts, if used correctly, will
help you prove triangle congruency.
Remember that congruency means
EXACT size and shape… don’t
confuse it with “similar”.
Shortcuts in Triangle Congruency
We learned that two triangles
were congruent if EACH
corresponding pair of angles were
congruent AND each pair of
corresponding sides were
congruent.
Remember that you had to
LIST EACH PAIR?
Yikes! That could get very long
and tedious (tiring).
Properties of Triangle Congruence
Congruence of triangles is reflexive,
symmetric, and transitive.
REFLEXIVE
K
L
J
~
ΔJKL = ΔJKL
K
L
J
Properties of Triangle Congruence
Congruence of triangles is reflexive,
symmetric, and transitive.
SYMMETRIC
~
If ΔJKL = ΔPQR,
~
then ΔPQR = ΔJKL.
K
Q
L
J
R
P
Properties of Triangle Congruence
Congruence of triangles is reflexive,
symmetric, and transitive.
TRANSITIVE
~
If ΔJKL = ΔPQR, and
~
ΔPQR = ΔXYZ, then
~
ΔJKL = ΔXYZ.
K
Q
L
R
J
Y
P
Z
X