4.2 Congruence & Triangles

1. 4.2 Congruence & Triangles Geometry Mrs. Kinser Fall 2012

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. Corresponding angles A ≅ P B ≅ Q C ≅ R Corresponding Sides AB ≅ PQ BC ≅ QR CA ≅ RP Triangles 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. The congruent triangles. Write a congruence statement. Identify all parts of congruent corresponding parts. Ex. 1 Naming congruent parts

8. 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 Ex. 1 Naming congruent parts

9. 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 Ex. 2 Using properties of congruent figures 8 m 110° (2x - 3) m (7y+9)° 72° 87° 10 m

10. 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 Ex. 2 Using properties of congruent figures 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. 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 Ex. 3 Using the Third Angles Theorem (2x + 30)° 55° 65°

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

14. 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. Ex. 5 Proving two triangles are congruent

15. 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: Proof: Given: AB║DC, AB≅DC. E is the midpoint of BC and AD. Prove ∆AEB ≅∆DEC

16. 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 Proof:

17. 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 Proof:

18. 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 Proof:

19. 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 Given Proof:

20. 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 Given Definition of a midpoint Proof:

21. 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 Given Definition of a midpoint Definition of congruent triangles Proof:

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.