Transcript Proving Triangles Congruent

Proving Triangles
Congruent
Geometry D – Chapter 4.4
SSS - Postulate
If all the sides of one triangle are congruent to all
of the sides of a second triangle, then the triangles
are congruent. (SSS)
Example #1 – SSS – Postulate
Use the SSS Postulate to show the two triangles
are congruent. Find the length of each side.
AC = 5
BC = 7
2
2
AB = 5  7  74
MO = 5
NO = 7
MN =
52  72  74
VABC VMNO
Definition – Included Angle
J
 K is the angle between
JK and KL. It is called the
included angle of sides JK
and KL.
K
L
J
What is the included angle
for sides KL and JL?
L
K
L
SAS - Postulate
If two sides and the included angle of one triangle
are congruent to two sides and the included angle
of a second triangle, then the triangles are
congruent. (SAS)
S
L
Q
P
A
S
A
J
S
S
K
VJKL VPQR by SAS
R
Example #2 – SAS – Postulate
K
L
Given: N is the midpoint of LW
N is the midpoint of SK
Prove:
N
VLNS VWNK
W
S
N is the midpoint of LW
N is the midpoint of SK
Given
LN  NW , SN  NK
Definition of Midpoint
LNS WNK
Vertical Angles are congruent
VLNS VWNK
SAS Postulate
Definition – Included Side
J
JK is the side between
 J and  K. It is called the
included side of angles J
and K.
K
L
J
What is the included side
for angles K and L?
KL
K
L
ASA - Postulate
If two angles and the included side of one triangle
are congruent to two angles and the included side
of a second triangle, then the triangles are
congruent. (ASA)
J
X
Y
K
L
VJKL VZXY
by ASA
Z
H
A
Example #3 – ASA – Postulate
W
Given: HA || KS
AW WK
Prove: VHAW VSKW
K
S
HA || KS, AW WK
Given
HAW SKW
Alt. Int. Angles are congruent
HWA SWK
Vertical Angles are congruent
VHAW VSKW
ASA Postulate
Identify the Congruent Triangles.
Identify the congruent triangles (if any). State the
postulate by which the triangles are congruent.
A
J
R
B
C
H
I
S
K
M
O
L
P
VABC VSTR by SSS
VPNO VVUW by SAS
N V
T
U
W
Note: VJHI is not
SSS, SAS, or ASA.
A
Example #4 – Paragraph Proof
Given: VMAT is isosceles with
vertex MAT bisected by AH.
Prove: MH  HT
T
H
M
• Sides MA and AT are congruent by the definition of an
isosceles triangle.
• Angle MAH is congruent to angle TAH by the definition
of an angle bisector.
• Side AH is congruent to side AH by the reflexive property.
• Triangle MAH is congruent to triangle TAH by SAS.
• Side MH is congruent to side HT by CPCTC.
Example #5 – Column Proof
Q
P
QM  MO
QM  PO, MO has midpoint N
Given: QM || PO,
Prove: QN  PN
QM || PO,
QM  PO
QM  MO
PO  MO
mQMN  90o
mPON  90o
QMN PON
MO ON
VQMN VPON
QN  PN
M
N
O
Given
A line  to one of two || lines is  to the other line.
Perpendicular lines intersect at 4 right
angles.
Substitution, Def of Congruent Angles
Definition of Midpoint
SAS
CPCTC
Summary

Triangles may be proved congruent by
Side – Side – Side (SSS) Postulate
Side – Angle – Side (SAS) Postulate, and
Angle – Side – Angle (ASA) Postulate.

Parts of triangles may be shown to be
congruent by Congruent Parts of
Congruent Triangles are Congruent
(CPCTC).