1. Topic 3
Congruent Triangles
Unit 2 Topic 4

2. Information
Two triangles are congruent if they have the same size and shape. You can turn, flip and/or slide one so it fits exactly on the other. Congruent angles can be marked with symbols, such as an arc, or a dot. Congruent sides can be marked with small line segments, called hatch marks.

3. Information
If ABC and DEF are congruent, then the corresponding angles and corresponding sides are equal.
The symbol for congruence is Since ABC and DEF are congruent, we write .

4. Information
By definition, if two triangles are congruent, then all corresponding angles are equal and all corresponding sides are equal. However, to prove two triangles are congruent, you do not need to know that all corresponding sides and all corresponding angles are equal. There are three conditions that can be used to prove two triangles are congruent.

5. Information Triangle Congruence Conditions
SSS Congruence Condition
If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. B Q
….. A C P R

6. Information Triangle Congruence Conditions
SAS Congruence Condition
If two sides and the contained angle of one triangle are equal to two sides and the contained angle of another triangle, then the triangles are congruent B Q
….. A C P R

7. Information Triangle Congruence Conditions
ASA Congruence Condition
If two angles and the contained side of one triangle are equal to two angles and the contained side of another triangle, then the triangles are congruent B Q
….. A C P R

8. Example 1
Try this on your own first!!!!
Stating Congruence and Corresponding Angles and Sides.
For each pair of triangles below, state the congruence theorem that proves they are congruent. Then, state the corresponding angles and sides for a). x A B C x X Y Z
a) c)
b) d) A B C
3.6 cm
3.5 cm
2.1 cm X Y Z
3.6 cm
2.1 cm
3.5 cm x A B C X Y Z x A B C
40˚
50˚
2.2 cm
40˚
50˚
2.2 cm X Z Y

9. Example 1a: Solution A B C
3.6 cm
3.5 cm
2.1 cm
We can see that all three sides are equal. Therefore, the triangles are congruent by SSS congruence.
Corresponding sides: Corresponding angles:
AB = XY A = X
BC = YZ B = Y
AC = XZ C = Z X Y Z
3.6 cm
2.1 cm
3.5 cm

10. Example 1b: Solution x A B C
b) Two angles and the contained side of one triangle is equal to two angles and the contained side of another triangle. Therefore, the triangles are congruent by the ASA congruence theorem. X Y Z x
Corresponding sides: Corresponding angles:
AB = XY A = X
BC = YZ B = Y
AC = XZ C = Z

11. Example 1c: Solution x A B C
c) Two sides and the contained angle of one triangle is equal to two sides and the contained angle of another triangle. Therefore, the triangles are congruent by the SAS congruence theorem. x X Y Z
Corresponding sides: Corresponding angles:
AB = XY A = X
BC = YZ B = Y
AC = XZ C = Z

12. Example 1d: Solution d) Since angle B and angle Y are each
90°, we can add to the diagram. A B C
40˚
50˚
2.2 cm
Two angles and the contained side of one triangle is equal to two angles and the contained side of another triangle. Therefore, the triangles are congruent by the ASA congruence theorem.
90°
40˚
50˚
2.2 cm X Z Y
Corresponding sides: Corresponding angles:
AC = XY A = X
AB = XY B = Y
BC = YZ C = Z
90°

13. More Information
Using algebra is one way to present a proof. Another way to present a proof, often used in geometry, is called a two-column proof. A two-column proof is a presentation of a logical argument involving deductive reasoning in which the statements of the argument are in one column and the justification for the statements are written in another column.
To prove two triangles are congruent you need to provide a logical argument that establishes one of the three congruence conditions: SSS, SAS or ASA. Your proof consists of a set of statements and accompanying reasons. There are many reasons that you can use to justify the statements in your proof.

14. More Information
You can end a proof with Q.E.D., a Latin phrase that means "which had to be demonstrated”.

15. Example 2a Given: A = D and AB = DB Prove: ∆ABC ∆DBE
Try this on your own first!!!!
Proving Two Triangles Are Congruent
Given: A = D and AB = DB
Prove: ∆ABC ∆DBE
Statement
Reason

16. Example 2a Given: A = D and AB = DB Prove: ∆ABC ∆DBE
Proving Two Triangles Are Congruent
Given: A = D and AB = DB
Prove: ∆ABC ∆DBE
Statement
Reason
Given
Opposite angles are equal
ASA Congruency condition A A S

17. Example 2b Try this on your own first!!!!
Proving Two Triangles Are Congruent
Statement
Reason

18. Example 2b: Solution Proving Two Triangles Are Congruent Statement
Reason
Given
Common side
SAS Congruency condition A S S

19. Example 3a Given: point E is the midpoint AC,
Completing a Proof Using Definitions
Given: point E is the midpoint AC,
point E is the midpoint of BD
Prove: AB = CD
Statement
Reason

20. Example 3a: Solution Statement Reason BE=DE CE=AE
By definition of a midpoint
Opposite angles are equal
SAS Congruency condition C D A E B S A S

21. Example 3b Given: TP is perpendicular to AC, represented as TP ⊥ AC
Try this on your own first!!!!
Completing a Proof Using Definitions
Given: TP is perpendicular to AC, represented as TP ⊥ AC
TP bisects ATC
Prove: AT = CT
Helpful Hint
To bisect is to divide in exactly half.
Statement
Reason T P A C

22. Example 3b: Solution
Given: TP is perpendicular to AC, represented as TP ⊥ AC
TP bisects ATC
Prove: AT = CT
TP is perpendicular to AC meaning they meet at a right angle.
Statement
Reason
TPA=TPC
TP=TP
ATP=CTP
AT=CT
By definition of a perpendicular
Common side
By definition of a bisect of ATC
ASA Congruency
By congruency T P A C A S A

23. Example 4 Given: and Prove: Try this on your own first!!!!
Completing a Proof with Parallel Lines
Given: and
Prove: T U V W X
Statement
Reason

24. Example 4 Given: and Prove: Completing a Proof with Parallel Lines
Statement
Reason
VTU=VXW
TU=XW
VUT=VWX
Alternate interior angles are equal
Given
ASA Congruency S T U V W X A A

25. Example 5 Given: BC=ED,  OBA=  OEF, and  OCB=  ODE.
Try this on your own first!!!!
Example 5
Completing a Proof Using Supplementary Angles
Given: BC=ED,  OBA=  OEF, and  OCB=  ODE.
Prove:  BOC =  EOD.
Statement
Reason F O E D C B A  

26. Example 5: Solution Given: BC=ED,  OBA=  OEF, and  OCB=  ODE.
Completing a Proof Using Supplementary Angles
Given: BC=ED,  OBA=  OEF, and  OCB=  ODE.
Prove:  BOC =  EOD.
Statement
Reason
ABO=FEO
OBC=OED
BC=ED
BCO=EDO
BOC=EOD
Given
Supplementary of the given angles
ASA Congruency
By Congruency F O E D C B A   A A S

27. Need to Know: Congruent triangles have the same size and shape.
To prove that triangles are congruent, it must be shown that the corresponding sides and corresponding angles in the triangles are equal.
To do this, the following Triangle Congruence Conditions are used:
SSS Congruence Theorem
SAS Congruence Theorem
ASA Congruence Theorem
You’re ready! Try the homework from this section.