4.3-4.6 Proving Triangles Congruent

1. 4.3-4.6 Proving Triangles Congruent Warm up: Are the triangles congruent? If so, write a congruence statement and justify your answer.

2. Proving Triangles Congruent… • How can you prove sides congruent? (things to look for) • How can you prove angles congruent? Given Shared side(reflexive POE) Midpoints Segment Addition Property Segment bisector Transitive POE others? Given Shared angle(reflexive POE) //→Alt. Int <s, . . . Angle Addition Property Angle bisector Vertical Angles Right Angles(┴) Transitive POE

3. Now you try! N GIVEN: R M MN = RS MO = RT PROVE: ΔMNO ΔRST O M S R T what is the first step?

4. Now you try! N GIVEN: R M MN = RS MO = RT PROVE: ΔMNO ΔRST O M S R T STEP 1 – DRAW IT AND MARK IT!

5. Now you try! N GIVEN: R M MN = RS MO = RT PROVE: ΔMNO ΔRST O M S R T STEP 1 – DRAW IT AND MARK IT! STEP 2 – CAN YOU PROVE THE Δs =? HOW?

6. Now you try! N GIVEN: R M MN = RS MO = RT PROVE: ΔMNO ΔRST O M S R T YES, BY SAS FROM THE GIVENS!

7. REAL LIFE EXAMPLES Bridges – Golden Gate, Brooklyn Bridge, New River Bridge . . . .

8. Real Life

9. Real Life

10. Types of Proofs Traditional two-column: This looks like a T-chart and has the statements on the left and reasons on the right.

11. Types of Proofs Flow Chart: Starts from a “base line” and all information flows from the given. Great for visual learners. Paragraph: Write it out! Tell me what you’re doing!

12. Helpful Hints with Proofs… • ALWAYS mark the given in your picture. • Use different colors in your picture to see the parts better. • ALWAYS look for a _______________________ which uses the __________________ property. • ALWAYS look for ______________ lines to prove mostly that _____________________________________. • ALWAYS look for ____________ angles which are always ___________. common side/angle reflexive parallel alternate interior angles are congruent vertical congruent

13. Statements PQ  PS; QR  SR; 1 2 PR  PR ∆QPR  ∆SPR 3 4 Reasons Given Reflexive Property SAS Postulate CPCTC Given: PQ  PS; QR  SR; 1 2Prove:  3  4

14. Statements WO  ZO; XO  YO WOX  ZOY ∆WXO  ∆ZYO Reasons Given Vertical angles are . SAS Postulate Given: WO  ZO;XO  YOProve: ∆WXO  ∆ZYO

15. Proof Practice Given: PSU PTR; SU  TR Prove: SP  TP HINT: draw the trianglesseparately!

16. Proof Practice… 1. PSU PTR; SU  TR 1. given 2. <P  <P 2. Reflexive POE 3. ∆SUP ∆TRP 3. AAS Theorem 4. CPCTC 4. SP  TP

17. Proof Practice… PSU PTR <P  <P SU  TR AAS Theorem ∆SUP  ∆TRP CPCTC SP  TP

18. Practice Name the included side for 1 and 5. Name a pair of angles in which DE is not included. If 6 10, and DC  VC, then ∆ DCA ∆ _______, by _________. DC <8, <9, for example VCE ASA

19. Q P R S More Proofs…Using 2 Column • Given: PQ  RQ; S is midpoint of PR. • Prove: P R QS is an auxiliary line 1. PQ  RQ; S is midpoint of PR 1.given 2. PS  SR 2. Def midpoint 3. QS  QS 3. Reflexive POE 4.∆PQS ∆RQS 4.SSS Postulate 5. P R 5. CPCTC

20. Q P R S More Proofs…Using Flow Chart • Given: PQ  RQ; S is midpoint of PR. • Prove: P R PQ  RQ S is midpoint of PR QS  QS PS  SR SSS Postulate ∆PQS  ∆RQS CPCTC P R

21. Q P R S More Proofs…Using Paragraph • Given: PQ  RQ; S is midpoint of PR. • Prove: P R QS is an __________________. We are given that ____________ and ___________________. Because S is the midpoint, we know that __________because of _____________. We drew in QS so that we can use the reflexive property to prove that _________. We now have enough information to prove that ∆PQS  ∆RQS by ____________. Therefore <P  <R by __________________. auxiliary line PQ  RQ S is midpoint of PR def. of midpt PS  SR QS  QS SSS Post. CPCTC

22. Do the following proofs in whatever way you feel comfortable Given: AB  EB; DEC B Prove: ∆ABE is equilateral

23. Group Work Time: • Group 4.3-4.6 proof practice WS • Group presentations • Next Class • Group presentations • More group practice work