WARM-UP

1. WARM-UP Rewrite each of the following statements in “If-then” form as the conditional, and converse. then write a biconditional and determine if it is . is true or false. 2. Vietnamese New Years is on January 3. 3. AB+BC=AC, B is between AC

2. CAHSEE prep

4. Proving Statements About segments What is the difference between congruence and equality?

5. Building a Proof • When writing a proof, you can only use facts that have previously been proved (theorems), facts that are assumed true without proof (postulates), and definitions. • Proofs can be written in paragraph form or in a two-column form. • We will use two-column form most often.

6. Two-Column Proofs: Key Elements • Given: state the “given” facts • Diagram: a figure that shows what is given • Prove: a statement of what you have to prove • Statements: (Left Column) numbered logical statements that lead to your conclusion • Reasons: (Right Column) numbered reasons that justify your statements (definitions, postulates, properties of algebra/congruence, previously proven theorems)

7. Properties of Equality (review) • These two properties are often interchangeable: • Substitution Prop. of Equality: If a=b, then we can substitute (plug in) a for b, or b for a. • If x+a=c AND a=b, then x+b=c (Plug it in, plug it in!) • Transitive Prop. of Equality: If a=b & b=c, then a=c. • If Mr. Madden is the same height as Simon, and Simon is the same height a Bradley, then Mr. Madden and Bradley are the same height. =)

8. Properties of Equality (review cont.) • Mirror, mirror, on the wall… • Reflexive Prop. of Equality: a=a • Anything is equal to itself (Think about your reflection in the mirror!) • Symmetric Property of Equality: If a=b, then b=a. • Think about something symmetrical… if you flip it, it still looks the same. You can always flip an equation, Left to Right. (Flip it good!)

9. Properties of Congruence • These 3 properties work for congruence also: • Reflexive: For any segment AB, AB ≅ AB. • Symmetric: If AB ≅ CD, then CD ≅ AB. • Transitive: If AB ≅ CD & CD ≅ EF, then AB ≅ EF.

10. Given: AB = BC, C is the midpoint of BD Prove: AB = CD • AB = BC, • C is the midpoint of BD • Given • Def. Midpoint • BC = CD • AB = CD • Substitution

11. Given: AB=CD Prove: AC=BD AB=CD Given AB+BC=AC Segment Add. Post. Segment Add. Post. BC+CD=BD BC+AB=BD Substitution AC=BD Substitution

12. Given: AC=BD Prove: AB=CD AC=BD Given AB+BC=AC Segment Add. Post. Segment Add. Post. BC+CD=BD AB+BC = BC+CD Substitution AB=CD Subtr. Prop of =

13. Given: QR = RS Prove: QS = 2 RS

14. Given:LE = RM, EG = AR Prove: LG = MA LE = RM and EG = AR Given AR+RM=AM Segment addition LE+EG=LG Segment addition RM+AR=LG Substitution LG = MA Substitution

15. Example 5: Using Segment Relationships • In the diagram, Q is the midpoint of PR. Show that PQ and QR are equal to ½ PR. • GIVEN: Q is the midpoint of PR. • PROVE: PQ = ½ PR and QR = ½ PR.

16. Q is the midpoint of PR. PQ = QR PQ + QR = PR PQ + PQ = PR 2 ∙ PQ = PR PQ = ½ PR QR = ½ PR Given Definition of a midpoint Segment Addition Postulate Substitution Property Distributive property Division property Substitution Statements: Reasons:

17. (over Lesson 2-2) 1-1a Write using two column proofs! Slide 1 of 1

18. (over Lesson 2-2) 1-1b Slide 1 of 1

19. Example 2: Using Congruence • Use the diagram and the given information to complete the missing steps and reasons in the proof. • GIVEN: LK = 5, JK = 5, JK≅ JL • PROVE: LK ≅ JL

20. ________________ ________________ LK = JK LK ≅ JK JK ≅ JL ________________ Given Given Substitution _________________ Given Substitution Statements: Reasons:

21. LK = 5 JK = 5 LK = JK LK ≅ JK JK ≅ JL LK ≅ JL Given Given Substitution Def. Congruent seg. Given Substitution Statements: Reasons:

22. Def. of congruent segments AB = BC DE = EF Substitution (or Transitive) Substitution (or Transitive) Def. of congruent segments