1. 4.2 Congruence & Triangles

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 A ≅ P B ≅ Q C ≅ R
Corresponding Sides
AB ≅ PQ
BC ≅ QR
CA ≅ RP B Q R A C P

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

7. Ex. 1 Naming congruent parts
The congruent triangles. Write a congruence statement. Identify all parts of congruent corresponding parts.

8. Ex. 1 Naming congruent parts
The diagram indicates that ∆DEF ≅ ∆RST. The congruent angles and sides are as follows:
Angles: D≅ R, E ≅ S, F ≅T
Sides DE ≅ RS, EF ≅ ST, FD ≅ TR

9. Ex. 2 Using properties of congruent figures
In the diagram NPLM ≅ EFGH
A. Find the value of x.
You know that LM ≅ GH. So, LM = GH.
8 = 2x – 3
11 = 2x
11/2 = x
8 m
110°
(2x - 3) m
(7y+9)°
72°
87°
10 m

10. Ex. 2 Using properties of congruent figures
In the diagram NPLM ≅ EFGH
B. Find the value of y
You know that N ≅ E. So, mN = mE.
72°= (7y + 9)°
63 = 7y
9 = y
8 m
110°
(2x - 3) m
(7y+9)°
72°
87°
10 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
Find the value of x.
In the diagram, 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° = 60°
mM = mT
60° = (2x + 30)°
30 = 2x
15 = 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.
RP ≅ MN, PQ ≅ NQ, QR ≅ QM. Because P and N have the same measure, P ≅ N. By vertical angles theorem, you know that PQR ≅ NQM. By the Third Angles Theorem, R ≅ M.
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°

14. Ex. 5 Proving two triangles are congruent
The diagram represents triangular stamps. Prove that ∆AEB≅∆DEC.
Given: AB║DC, AB≅DC. E is the midpoint of BC and AD.
Prove ∆AEB ≅∆DEC
Plan for proof: Use the fact that AEB and DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segment AB and DC to identify other pairs of angles that are congruent.

15. Proof: Statements: AB║DC, AB≅DC EAB ≅ EDC, ABE ≅ DCE AEB ≅ DEC
Given: AB║DC, AB≅DC. E is the midpoint of BC and AD.
Prove ∆AEB ≅∆DEC
Statements:
AB║DC, AB≅DC
EAB ≅ EDC, ABE ≅ DCE
AEB ≅ DEC
E is the midpoint of AD, E is the midpoint of BC.
AE ≅ DE, BE ≅ CE
∆AEB ≅ ∆DEC
Reasons:

16. Proof: Statements: AB║DC, AB≅DC EAB ≅ EDC, ABE ≅ DCE AEB ≅ DEC
E is the midpoint of AD, E is the midpoint of BC.
AE ≅ DE, BE ≅ CE
∆AEB ≅ ∆DEC
Reasons:
Given

17. Proof: Statements: AB║DC, AB≅DC EAB ≅ EDC, ABE ≅ DCE AEB ≅ DEC
E is the midpoint of AD, E is the midpoint of BC.
AE ≅ DE, BE ≅ CE
∆AEB ≅ ∆DEC
Reasons:
Given
Alternate interior angles theorem

18. Proof: Statements: AB║DC, AB≅DC EAB ≅ EDC, ABE ≅ DCE AEB ≅ DEC
E is the midpoint of AD, E is the midpoint of BC.
AE ≅ DE, BE ≅ CE
∆AEB ≅ ∆DEC
Reasons:
Given
Alternate interior angles theorem
Vertical angles theorem

19. Proof: Statements: AB║DC, AB≅DC EAB ≅ EDC, ABE ≅ DCE AEB ≅ DEC
E is the midpoint of AD, E is the midpoint of BC.
AE ≅ DE, BE ≅ CE
∆AEB ≅ ∆DEC
Reasons:
Given
Alternate interior angles theorem
Vertical angles theorem

20. Proof: Statements: AB║DC, AB≅DC EAB ≅ EDC, ABE ≅ DCE AEB ≅ DEC
E is the midpoint of AD, E is the midpoint of BC.
AE ≅ DE, BE ≅ CE
∆AEB ≅ ∆DEC
Reasons:
Given
Alternate interior angles theorem
Vertical angles theorem
Definition of a midpoint

21. Proof: Statements: AB║DC, AB≅DC EAB ≅ EDC, ABE ≅ DCE AEB ≅ DEC
E is the midpoint of AD, E is the midpoint of BC.
AE ≅ DE, BE ≅ CE
∆AEB ≅ ∆DEC
Reasons:
Given
Alternate interior angles theorem
Vertical angles theorem
Definition of a midpoint
Definition of congruent triangles

22. 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.

23. 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.