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Proving Triangles Congruent
Part 2
AAS Theorem
If two angles and one of the non-included sides in one triangle are congruent to two angles and one
of the non-included sides in another triangle, then the triangles
are congruent.
AAS Looks Like…
B C D
FGA
ACB DFG
A: A DA: B GS: AC DF
J
K LM
A: K MA: KJL MJLS: JL JLJKL JML
AAS vs. ASA
ASAAAS
Parts of a Right Triangle
legs
hypotenuse
HL TheoremRIGHT TRIANGLES ONLY!
If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.
HL Looks Like…
XTV
W
WTV WXV
NMP RQS
NM
P
SQ
R
Right : M & QH: PN RSL: MP QS
Right : TVW &
XVW
H: TW XW
L: WV WV
There’s no such thing as AAAAAA Congruence:
These two equiangular triangles have all the same angles… but they are not the same size!
Recap:
There are 5 ways to prove that triangles are congruent:
SSS
SAS
ASA
AAS
HL
Examples
A B C
D
B is the midpoint of AC
SAS ABD CBDK
J
LN
M
H
AAS
MLN HJK
S: AB BCA: ABD CBDS: DB DB
A:L JA: M HS: LN JK
Right Angles: ABD & CBDH: AD CDL: BD BD
Examples C
DA
B
E
B C
D
A
DB AC AD CD
HL
ABD CBDA: A CS: AE CEA: BEA DEC
ASA
BEA DEC
Examples
B C
D
A
B is the midpoint of AC
SSS
DAB DCB
Z
Y
X
V
W
Not Enough!
We cannot conclude whether the triangle are
congruent.
S: AB CBS: BD BDS: AD CD
A: WXV YXZS: WV YZ