4.4 Using Corresponding Parts of Congruent Triangles
CPCTC – Corresponding Parts of Congruent Triangles are Congruent
If you can prove that two triangles are congruent, then you can prove any pair of corresponding
parts is congruent.
Your goal is to prove that two triangles are congruent!
What have we used to prove that triangles are congruent?
State why the triangles are congruent. Then list all other pairs of corresponding parts.
̅̅̅ ∥ ̅̅̅̅
Given: ̅𝐻𝐽
𝐿𝐾 ; ̅̅̅
𝐽𝐾 ∥ ̅̅̅̅
𝐻𝐿
Prove: LHJ  JKL
H
J
L
K
Given: A is the midpoint of MT and SR
Prove: M  T
M
R
A
S
T
Given: 1  2; 3  4
Prove:
D
BCE  DCE
C 2
E
1
4
3
B
A
4.5 Isosceles, Equilateral, and Right Triangles
Isosceles: ______________________________________________________________________
Legs: _________________________________________________________________________
Base: _________________________________________________________________________
Base angles: ___________________________________________________________________
Vertex angle: ___________________________________________________________________
Label the legs, the base angles, and the vertex in the triangle. Use symbols to show which parts
are congruent.
Isosceles Triangle Theorem – If two sides of a triangle are congruent then __________________
______________________________________________________________________________
Converse of the Isosceles Triangle Theorem – If two angles of a triangle are congruent, then ___
______________________________________________________________________________
Given:
XY  XZ
XB
X
bi sec ts YXZ
Prove: Y  Z
1
Y
2
B
Z
Theorem 4-5: If a line bisects the vertex angle of an isosceles triangle, then the line is also
perpendicular bisector of the base.
Diagram:
Corollary: If a triangle is equilateral, then it’s equiangular.
Corollary: If a triangle is equiangular, then it’s equilateral.
Complete each statement. Explain why it is true.
1.) AB = _____
2.) <BDE = _____
3.) <CBE = _____ = <BCE
Solve for x and y.
1.)
Y
50 50
2.)
12
2x + 8
40
3.)
4.)
5.)
y
Yx
50
x
4.6 Congruence in Right Triangles
Right triangle: __________________________________________________________________
Hypotenuse: ___________________________________________________________________
Leg: __________________________________________________________________________
Label the hypotenuse and leg in the right triangle below.
Hypotenuse – Leg Theorem (HL) – If the hypotenuse and leg of one right triangle are congruent
to the hypotenuse and corresponding leg of another right triangle, then the triangles are
congruent.
Conditions for Using HL:
1.) The two triangles must be _________________________________________________.
2.) The hypotenuses must be __________________________________________________.
3.) There is one pair of congruent ______________________________________________.
Complete the proof.
Given: <V and <W are right angles; WZ  VX
Prove: WVZ  VWX
Statements
Reasons
Given
Given
Reflexive Property of Congruence
HL Theorem
For what values of x and y are the triangles congruent by HL?
1.)
2.)
What additional information would prove each pair of triangles congruent by the Hypotenuse –
Leg Theorem?
3.)
4.)