1. Developing Flow Proofs A Geometric Activity High School Geometry By Mrs. C. Henry

2. TYPES OF PROOFS Three types of proofs: 1)TWO-COLUMN PROOF is the most formal type. Lists numbered statements in the left column and the reason for each statement in the right column. 2)PARAGRAPH PROOF describes the logical argument with sentences. It is more conversational than a two-column proof. 3)FLOW PROOF uses the same statements and reasons as a two- column proof, but the logical flow connecting the statements is indicated by arrows. C OMPARING T YPES OF P ROOFS 3.2 Proof and Perpendicular Lines

3. Flow Proof GIVEN  5 and  6 are a linear pair.  6 and  7 are a linear pair. PROVE 5 7 5 6 7  5 and  6 are a linear air.  6 and  7 are a linear pair.  5   7 Congruent Supplements Theorem Given Linear Pair Postulate 3.2 Proof and Perpendicular Lines

4. FLOW PROOF CARDS Neatly write the following terms on index cards. Use these terms and others to create your Flow Proof. SSS SAS ASA AAS HL 3.1 I DENTIFYING A NGLES Reflexive Property Substitution Δ Sum Theorem  Bisector Theorem  Bisector Theorem Corresponding angles Alternate Exterior angles Alternate Interior angles Consecutive Interior angles Vertical angles Supplementary angles Linear pair

5. Prove It! A B C D GIVEN: AC is  bisector of BD. PROVE: ΔABC  ΔADC

6. Prove It! A B C D GIVEN: AC is  bisector of BD. PROVE: ΔADM  ΔABM M

7. Prove It! F G H E GIVEN: AC is  bisector of BD. PROVE: ΔHGF  ΔHEF

8. Prove It! J G H F G E PROVE: ΔFGE  ΔJGH

9. Prove It! LM PN PROVE: ΔLMN  ΔNPL

10. Prove It! J K L M N GIVEN: Point L bisects KM.  KLJ   MLN. PROVE: Δ JKL  Δ NML

11. Prove It! F G H E J GIVEN: GH ‖ FE. PROVE: ΔGHJ  ΔEFJ

12. Prove It! GIVEN: AD & CD are  bisectors. m ABC = 100 PROVE: m ADC = 140. B AC D

13. Prove It! PROVE: ΔAEB  ΔDEC

14. Example Proving Two Triangles are Congruent StatementsReasons  EAB   EDC,  ABE   DCE  AEB   DEC E is the midpoint of AD, E is the midpoint of BC , DE AE  CEBE Given Alternate Interior Angles Theorem Vertical Angles Theorem Given Definition of congruent triangles Definition of midpoint ||, DCAB DCAB  SOLUTION  AEBDEC Prove that . AEBDEC AB C D E

15. Example Using the Third Angles Theorem Find the value of x. SOLUTION 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 ˚ = (2 x + 30) ˚ 30 = 2 x 15 = x Third Angles Theorem Substitute. Subtract 30 from each side. Divide each side by 2.

16. Goal 2 SOLUTION Paragraph Proof From the diagram, you are given that all three corresponding sides are congruent. , NQPQ , MNRP  QMQR and Because  P and  N have the same measures,  P   N. By the Vertical Angles Theorem, you know that  PQR   NQM. By the Third Angles Theorem,  R   M. Decide whether the triangles are congruent. Justify your reasoning. So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, . PQRNQM Proving Triangles are Congruent

17. Flow Proof GIVEN  5 and  6 are a linear pair.  6 and  7 are a linear pair. PROVE 5 7 5 6 7  5 and  6 are a linear pair.  6 and  7 are a linear pair.  5 and  6 are supplementary.  6 and  7 are supplementary.  5   7 Congruent Supplements Theorem Given Linear Pair Postulate 3.2 Proof and Perpendicular Lines