1. 4.2 Congruence & Triangles
Geometry

2. Objectives: Identify congruent figures and corresponding parts
Prove that two triangles are congruent

3. Identifying congruent figures
Two geometric figures are congruent if they have exactly the same size and shape.
NOT CONGRUENT
CONGRUENT

4. Congruency
When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.

5. Triangles
Corresponding angles
Corresponding Sides A B C Q P R

6. How do you write a congruence statement?
There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. Normally you would write ∆ABC ≅ ∆PQR, but you can also write that ∆BCA ≅ ∆QRP or ∆CAB ≅ ∆RPQ.

7. Ex. 1 Naming congruent parts
These are congruent triangles. Write congruence statements to identify all congruent corresponding parts.

8. Ex. 1 Naming congruent parts
The diagram indicates that ∆DEF ≅ ∆RST. The congruent angles and sides are as follows:
Angles:
Sides:

9. Ex. 2 Using properties of congruent figures
In the diagram, NPLM ≅ EFGH
Find the value of x.
8 m
110°
87°
10 m
72°
(7y+9)°
(2x - 3) m

10. Ex. 2 Using properties of congruent figures
In the diagram NPLM ≅ EFGH
Find the value of y.
8 m
110°
87°
10 m
72°
(7y+9)°
(2x - 3) m

11. Third Angles Theorem
If any two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If A ≅ D and B ≅ E, then C ≅ F.

12. Ex. 3 Using the Third Angles Theorem
N ≅ R and L ≅ S. From the Third Angles Theorem, you know that M ≅ T. So mM = mT. From the Triangle Sum Theorem, mM = 180° – (55° + 65°) = 180° – 120° = 60°
Then, mM = mT
60° = (2x + 30)°
30 = 2x
15 = x
Find the value of x.
(2x + 30)°
55°
65°

13. Ex. 4 Proving Triangles are congruent
Decide whether the triangles are congruent. Justify your reasoning.
From the diagram, you are given that all three pairs of corresponding sides are congruent.
92°
92°

14. Ex. 4 Proving Triangles are congruent
92°
92°

15. Ex. 4 Proving Triangles are congruent
So all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles,
∆PQR ≅ ∆NQM.
92°
92°

16. Ex. 5 Proving two triangles are congruent

17. Ex. 5 Proving two triangles are congruent

18. Proof:
Statements:
Reasons:

19. Proof:
Statements:
Reasons:
1. Given

20. Proof: 1. Given 2. Alternate Interior Angles Theorem Statements:
Reasons:
1. Given
2. Alternate Interior
Angles Theorem

21. Proof: 1. Given 2. Alternate Interior Angles Theorem
Statements:
Reasons:
1. Given
2. Alternate Interior
Angles Theorem
3. Vertical Angles Theorem

22. Proof: 1. Given 2. Alternate Interior Angles Theorem
Statements:
Reasons:
1. Given
2. Alternate Interior
Angles Theorem
3. Vertical Angles Theorem
4. Given

23. Proof: 1. Given 2. Alternate Interior Angles Theorem
Statements:
Reasons:
1. Given
2. Alternate Interior
Angles Theorem
3. Vertical Angles Theorem
4. Given
5. Definition of a Midpoint

24. Proof: 1. Given 2. Alternate Interior Angles Theorem
Statements:
Reasons:
1. Given
2. Alternate Interior
Angles Theorem
3. Vertical Angles Theorem
4. Given
5. Definition of a Midpoint
6. Definition of Congruent Triangles

25. What should you have learned?
To prove two triangles congruent by the definition of congruence—that is all pairs of corresponding angles and corresponding sides are congruent.
In upcoming lessons you will learn more efficient ways of proving triangles are congruent. The properties on the next slide will be useful in such proofs.

26. Theorem 4.4 Properties of Congruent Triangles
Reflexive property of congruent triangles: Every triangle is congruent to itself.
Symmetric property of congruent triangles: If ∆ABC ≅ ∆DEF, then ∆DEF ≅ ∆ABC.
Transitive property of congruent triangles: If ∆ABC ≅ ∆DEF and ∆DEF ≅ ∆JKL, then ∆ABC ≅ ∆JKL.