Download AAS Theorem

Prove: 3  4
Given: AB || CD
AE  BE
C
1
A
E
3
4
D
2
B
Statements
1.
AB || CD
2. 1  3, 2  4
3.
AE  BE
Reasons
1.
Given
2. If lines are parallel, then
corresponding angles are
congruent.
3.
Given
4. If two sides of a triangle
4.
1  2
5.
2  3
5.
Substitution
6.
3  4
6.
Substitution
are congruent, then the angles
opposite those sides are
congruent.
AAS Theorem:
If two angles and a non-included side of one triangle
are congruent to the corresponding parts of another
triangle, then the triangles are congruent.
A
C
D
ABC   DEF
___
B
E
F
C
Example 1)
ABC   DBC
___
A
D
B
Example 2)
A
D
C
B
E
ABC   EDC
___
HL Theorem:
If the hypotenuse and a leg of one right triangle are
congruent to corresponding parts of another right triangle,
then the triangles are congruent.
E
C
A
B
D
ABC   DFE
___
F
B
Example 1)
ABC   DBC
___
A
Example 2)
C
C
D
D
ABC   DCB
___
A
B
PROOF EXAMPLE 1:
Given: 1  2
1
C
CD bisects VCW
Prove: DV  DW
CD bisects VCW
D
3
4
2
W
Statements
1.
V
Reasons
1.
Given
2.
3  4
2. Def. of Angle Bisector
3.
4.
1  2
CD  CD
3.
Given
4. Reflexive Property
5.
6.
ΔCVD  ΔCWD
DV  DW
5.
6.
AAS Theorem
CPCTC
X
PROOF EXAMPLE 2:
Given: W and Y are right angles
WX  YX
Prove: WZ  YZ
Statements
Y
W
Reasons
Z
1. W and Y are right angles
1.
2. ΔXWZ and ΔXYZ are right
triangles.
2. Def. of Right Triangle
3.
4.
3.
Given
4. Reflexive Property
5.
6.
WX  YX
XZ  XZ
ΔXWZ  ΔXYZ
WZ  YZ
5.
6.
Given
HL Theorem
CPCTC
PROOF EXAMPLE 3:
Given: KL  LA; KJ  JA;
AK bisects LAJ
Prove: LK  JK
L
K
1
A
2
J
Statements
1.
KL  LA; KJ  JA
Reasons
1.
Given
2. L and J are right angles 2. Def. of Perpendicular Lines
3.
mL = 90; mJ = 90
3.
Def. of Right Angles
4.
mL = mJ; L  J
4.
Substitution
5.
AK bisects LAJ
5.
Given
6.
1  2
6.
Def. of Angle Bisector
7.
KA  KA
7.
Reflexive Property
8.
9.
ΔLKA  ΔJKA
LK  JK
8.
9.
AAS Theorem
CPCTC