Proving Triangles Congruent
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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 7 2 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
T
vertex MAT bisected by AH.
Prove: MH HT
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
mQMN 90o
mPON 90o
QMN PON
MOON
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).