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Lesson 4-5
Objective – To prove triangles congruent by
using ASA, AAS, and HL.
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
Given: AB  CD,
CD AC  BD
Prove: ADC DAB
Statement
1) AB  CD, AC  BD
2) 1  4, 2  3
3) AD  AD
4) ADC DAB
A
1
C
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
Construction – Copy a Triangle Using ASA
B
B
2
3
Steps
1) Construct  
4
D
Reasons
Given
Alt. Int. s Thm.
Reflexive Prop. of 
ASA  Postulate
A
C
X
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
Construction – Copy a Triangle Using ASA
Construction – Copy a Triangle Using ASA
B
B
Steps
Steps
1) Construct  
1) Construct  
A
A
C
C
X
X
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
Construction – Copy a Triangle Using ASA
Construction – Copy a Triangle Using ASA
B
B
A
Steps
Steps
1) Construct  
1) Construct  
2) Copy adjacent
side length
A
C
X
C
X
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
1
Lesson 4-5
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
Construction – Copy a Triangle Using ASA
Construction – Copy a Triangle Using ASA
B
B
Steps
1) Construct  
2) Copy adjacent
side length
A
Steps
A
C
X
1) Construct  
2) Copy adjacent
side length
3) Construct  
adjacent to side
C
X
Z
Z
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
Construction – Copy a Triangle Using ASA
Construction – Copy a Triangle Using ASA
B
B
Steps
1) Construct  
2) Copy adjacent
side length
3) Construct  
adjacent to side
A
C
X
Steps
1) Construct  
2) Copy adjacent
side length
3) Construct  
adjacent to side
A
C
X
Z
Z
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
ASA Congruence Postulate
If two angles and an included side of one triangle
are congruent to two angles and the included side
of another, then the triangles are congruent.
Construction – Copy a Triangle Using ASA
Construction – Copy a Triangle Using ASA
B
B
Steps
1) Construct  
2) Copy adjacent
side length
3) Construct  
adjacent to side
A
C
X
Z
ABC XYZ by ASA
Y
A
C
X
Steps
1) Construct  
2) Copy adjacent
side length
3) Construct  
adjacent to side
Z
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
2
Lesson 4-5
AAS Congruence Theorem
If two angles and a nonincluded side of one triangle
are congruent to two angles and the nonincluded side
of another, then the triangles are congruent.
Why is this a theorem?
Given: A  D, C  F
BC  EF
P
Prove:
ABC DEF
B
A
Statement
Statement
F
Reasons
Given
1) A  D, C  F
BC  EF
2) B  E
3) ABC DEF
Third Angles Theorem
ASA  Postulate
HL Congruence Theorem
If the hypotenuse and a leg of a right triangle are
congruent to the hypotenuse and the leg of another
right triangle, then the triangles are congruent.
Given: D & E are right s,
B is midpoint of AC & DE
Prove: ABD CBE
D
B
5) ABD CBE
HL  Theorem
2) ABD &CBE are right
s
3) B is midpoint of AC & DE
Reasons
1) BD bisects ABC
2) ABD  CBD
3) A  C
Given
Def. of  bisector
Given
4) BD  BD
Reflexive Prop of 
AAS Theorem
5) ABD CBD
How can the triangles be proved congruent?
1)
3)
HL
E
4) AB  BC, DB  BE
1) D & E are right s
C
A
Reasons
Given
Def. of right
Given
Def. of midpoint
Statement
D
A
E
CD
B
Given: A  C, BD bisects ABC
Prove: ABD CBD
ASA
C

2)
4)
AAS
AAS
How can the triangles be proved congruent?
5)
7)
AAS
ASA
6)
8)
AAS
No Conclusion
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
3