4.4 Isosceles Triangles,
Corollaries, &
CPCTC
Isosceles Triangles
♥ Has at least 2 congruent sides.
♥ The angles opposite the congruent
sides are congruent
♥ Converse is also true. The sides
opposite the congruent angles are
also congruent.
♥ This is a COROLLARY.
A corollary naturally follows a
theorem or postulate. We can
prove it if we need to, but it really
makes a lot of sense.
♥ The bisector of the vertex angle of an
isosceles Δ is the perpendicular bisector of
the base.
Vertex angle
Base
In addition, you just learned
that the angles opposite
congruent sides are
congruent…
Corresponding parts
When you use a shortcut (SSS, AAS, SAS,
ASA, HL) to show that 2 triangles are ,
that means that ALL the corresponding parts
are congruent.
EX: If a triangle is congruent by ASA (for
instance), then all the other corresponding
parts are .
B
F
That means that EG  CB
A
E
C
What is AC congruent to?
G
FE
Corresponding parts of congruent
triangles are congruent.
Corresponding parts of congruent
triangles are congruent.
Corresponding parts of congruent
triangles are congruent.
Corresponding parts of congruent
triangles are congruent.
Corresponding Parts of Congruent
Triangles are Congruent.
CPCTC
If you can prove
congruence using a
shortcut, then you
KNOW that the
remaining
corresponding parts
are congruent.
You can only use
CPCTC in a proof
AFTER you have
proved
congruence.
For example:
A
Prove: AB 
DE
B
C
D
Statements
Reasons
AC  DF
Given
C  F
Given
CB  FE
Given
ΔABC  ΔDEF
AB  DE
F
E
SAS
CPCTC
Your
assignment
2 - Cut and paste proofs
2 – DIY proofs
3 - Constructions