ASA
CONGRUENT TRIANGLE Proofs
ASA (angle-side-angle) CONGRUENCE THEOREM
If two angles and the included side of one triangle are congruent to two angles
and an included side of another triangle, then the two triangles are congruent.
If ____________________ (Angle)
E
B
____________________ (Side)
____________________ (Angle)
C
A
D
then, __________________________________
F
“INCLUDED” MEANS THE SIDE BETWEEN THE ANGLES!
1 Given: SQ bisects RQT and RST
R
Prove: QRS ≅ QTS
Q
S
© Gina Wilson (All Things Algebra®, LLC)
T
Statements
Reasons
1. SQ bisects RQT and RST
1.
2. RQS  TQS
2.
3. RSQ  TSQ
3.
4. QS  QS
4.
5. QRS ≅ QTS
5.
2 Given: JK LM , JL KM
J
L
Prove: JKL ≅ MLK
M
K
Statements
Reasons
1. JK LM , JL KM
1.
2. JKL  MLK
2.
3. JLK  MKL
3.
4. KL  LK
4.
5. JKL ≅ MLK
5.
© Gina Wilson (All Things Algebra®, LLC), 2014-2020
3 Given: BAC  DEC, C is the midpoint of AE
E
B
Prove: ABC ≅ EDC
C
D
A
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
AAS
© Gina Wilson (All Things Algebra®, LLC)
CONGRUENT TRIANGLE Proofs
AAS (angle-angle-side) CONGRUENCE THEOREM
If two angles and a non-included side of one triangle are congruent to two angles
and a non-included side of another triangle, then the two triangles are congruent.
If ____________________ (Angle)
E
B
____________________ (Angle)
____________________ (Side)
A
C
D
F
then, ___________________________________
“NON-INCLUDED” MEANS A SIDE OPPOSITE THE ANGLES!
4 Given: YZ bisects WYX , YWZ  YXZ
Y
Prove: WYZ ≅ XYZ
W
Statements
Z
X
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
© Gina Wilson (All Things Algebra®, LLC), 2014-2020
5 Given: ABC  CED, AB CE
E
B
C is the midpoint of AD
Prove: ABC ≅ CED
A
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
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5.
5.
6.
6.
6 Given: PR bisects QRS, PSR  PQR
© Gina Wilson (All Things Algebra®, LLC)
D
C
Q
P
Prove: PSR ≅ PQR
R
S
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
7 Given: LG JM , H is the midpoint of LM
J
L
Prove: LGH ≅ MJH
H
G
Statements
M
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
© Gina Wilson (All Things Algebra®, LLC), 2014-2020
HL
RIGHT TRIANGLE CONGRUENCE Proofs
HL (hypotenuse-leg) CONGRUENCE THEOREM
If the hypotenuse and a leg of one right triangle is congruent to the hypotenuse
and a leg of another right triangle, then the two triangles are congruent.
E
B
If ____________________ (Hypotneuse)
____________________ (Leg)
C
A
D
then, __________________________________
F
The HYPOTENUSE is the side opposite the right angle. A LEG is a side adjacent to the right angle.
1 Given: LMP and MNP are right triangles, ML  MN
M
© Gina Wilson (All Things Algebra®, LLC)
Prove: LMP ≅ NMP
L
Statements
N
P
Reasons
1. LMP and MNP are right triangles
1.
2. ML  MN
2.
3. MP  MP
3.
4. LMP ≅ NMP
4.
2 Given: WVX and YZX are right triangles, WV  YZ
V
Y
X is the midpoint of WY
Prove: WVX ≅ YZX
X
W
Statements
Z
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
© Gina Wilson (All Things Algebra®, LLC), 2014-2020