Download does sss establish triangle congruence? yes!!!

G.CO.8 STUDENT NOTES WS #1 – geometrycommoncore.com
1
DOES SSS ESTABLISH TRIANGLE CONGRUENCE? YES!!!
ABC  WER (SSS)
ABD  CBD (SSS)
E
B
THJ  RJH (SSS)
A
H
B
R
C
R
W
A
T
C
J
D
DOES SAS ESTABLISH TRIANGLE CONGRUENCE? YES!!!
GHK  GFK (SAS) or (SSS)
ABC  UYM (SAS)
A
H
o
ABC  HBM (SAS)
Y
o
A
G
K
M
B
B
o
o
C
H
C
F o
M
o
U
DOES ASA ESTABLISH TRIANGLE CONGRUENCE? YES!!!
BTG  ETG (ASA)
E
o
x
T
GTE  GYD (ASA)
E
o
x G
x
T
G
x
TEG  ULK (ASA)
E
o
Y
T
x
U
G
o
o
B
o
x
L
K
D
DOES AAS/SAA ESTABLISH TRIANGLE CONGRUENCE? YES!!!
Yes, this is a special form of ASA. In any triangle, if you have two angles you are able to determine the third
angle because they sum to 180. So by knowing two angles, you actually know three angles. So ASA and
AAS/SAA both work as congruence relationships.
ANGLE – SIDE – ANGLE
E
U
o
T
x
G
x
K
Third Angles are Congruent.
E
U
o
ANGLE – ANGLE - SIDE
E
U
*
o
L
T
*
x
G
x
K
o
*
L
T
*
x
L
G
x
K
G.CO.8 STUDENT NOTES WS #1 – geometrycommoncore.com
2
DOES ASS/SSA ESTABLISH TRIANGLE CONGRUENCE? SOMETIMES…..
This is a difficult case…. it sometimes produces congruence and other times it doesn’t. We will look at each
case to see when it works and when it doesn’t.
CASE #1 – AS1S2, when S2 is greater than S1.
S1
S2
S1
S2
This forms a triangle congruence relationship. There is only one way for S2 to
be placed to complete the triangle.
CASE #2 – AS1S2, when S2 is less than S1. (Too Short)
This is a weird case but when establishing all things that could happen we need
to include this. It is possible that S2 is not long enough to close the triangle.
This of course is not a congruence relationship, it doesn’t even form a triangle.
S1
S2
S1
S2
S1
S2
CASE #3 – AS1S2, when S2 is less than S1 (1 Intersection)
This case is typically known as HL, which stands for Hypotenuse –
Leg. It gets this special name because it is the right triangle that locks this
shape. A way to understand why this forms a congruence relationship is
because in a right triangle if you know two sides, you can use the Pythagorean
Theorem to calculate the third side. Now you have SSS or SAS, which we have
already established to be congruence criteria. HL forms a triangle congruence
relationship.
CASE #4 – AS1S2, when S2 is less than S1. (2 Intersections)
S1
This is known as the AMBIGIOUS CASE because two different triangles can be
formed by this information. Because S2 is shorter it can swing to form two
possible locations. This does NOT form a triangle congruence relationship.
S1
S2
S2