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GeometryGeometry
Triangle Congruence TheoremsTriangle Congruence Theorems
Congruent triangles have three congruent sides and and three congruent angles.
However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles.
Congruent Triangles
The Triangle Congruence The Triangle Congruence Postulates &TheoremsPostulates &Theorems
LAHALLHL
FOR RIGHT TRIANGLES ONLY
AASASASASSSS
FOR ALL TRIANGLES
Theorem
If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent.
Think about it… they have to add up to 180°.
A closer look...A closer look...
If two triangles have two pairs of angles congruent, then their third pair of angles is congruent.
But do the two triangles have to be congruent?
85° 30°
85° 30°
ExampleExample
30°
30°
Why aren’t these triangles congruent?
What do we call these triangles?
So, how do we prove that two triangles really are congruent?
ASA (Angle, Side, ASA (Angle, Side, Angle)Angle)
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . .
then the 2 triangles are
CONGRUENT!
F
E
D
A
C
B
AAS (Angle, Angle, Side)AAS (Angle, Angle, Side)Special case of ASASpecial case of ASA
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . .
then the 2 triangles are
CONGRUENT!
F
E
D
A
C
B
SAS (Side, Angle, Side)SAS (Side, Angle, Side)
If in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, . . .then
the 2 triangles are CONGRUENT!
F
E
D
A
C
B
SSS (Side, Side, Side)SSS (Side, Side, Side)
In two triangles, if 3 sides of one are congruent to three sides of the other, . . .
F
E
D
A
C
B
then the 2 triangles are
CONGRUENT!
HL (Hypotenuse, Leg)HL (Hypotenuse, Leg)
If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .
A
C
B
F
E
D
then the 2 triangles are
CONGRUENT!
HA (Hypotenuse, Angle)HA (Hypotenuse, Angle)
If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .
then the 2 triangles are
CONGRUENT!
F
E
D
A
C
B
LA (Leg, Angle)LA (Leg, Angle)
If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .
then the 2 triangles are
CONGRUENT!
A
C
B
F
E
D
LL (Leg, Leg)LL (Leg, Leg)
If both pair of legs of two RIGHT triangles are congruent, . . .
then the 2 triangles are
CONGRUENT!
A
C
B
F
E
D
Example 1Example 1
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?
F
E
D
A
C
B
Example 2Example 2
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?
A
C
B
F
E
D
Example 3Example 3
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?D
A
C
B
Example 4Example 4Why are the two
triangles congruent?
What are the corresponding vertices?A
B
C
D
EF SAS
A D
C E
B F
Example 5Example 5
Why are the two triangles congruent?
What are the corresponding vertices?
A
B
C
DSSS
A C
ADB CDB
ABD CBD
Example 6Example 6Given:
B C
DA
CDABADBC
CDAB
DABC
CAAC
Are the triangles congruent?
SSS
Why?
Example 7Given: QR PS
RHSRSSR
Are the Triangles Congruent?
QSR PRS = 90°
Q
RS
P
T
mQSR = mPRS = 90°
PS QR
Why?
Summary:Summary:ASA - Pairs of congruent sides contained between two congruent angles
SAS - Pairs of congruent angles contained between two congruent sidesSSS - Three pairs of congruent sides
AAS – Pairs of congruent angles and the side not contained between them.
Summary ---Summary ---for Right Triangles for Right Triangles Only:Only:
HL – Pair of sides including the Hypotenuse and one Leg
HA – Pair of hypotenuses and one acute angle
LL – Both pair of legs
LA – One pair of legs and one pair of acute angles
THE END!!!
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