Report

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. 1. 2 = 64; = −8 2. Vietnamese New Years is on January 3. 3. AB+BC=AC, B is between AC CAHSEE prep GEOMETRY GAME PLAN Date 9/30/13 Monday Section / Topic Notes: 2.5 Proving Statements about segments Lesson Goal Students will be able to write proofs with reasons about congruent segments. Geometry California Standard 2.0 Students write geometric proofs. Homework P. 104-107 (#1-11, 16, 31-34) Announcements Math tutoring is available every Mon-Thurs in Room 307, 3-4PM! Test next Tuesday. PROVING STATEMENTS ABOUT SEGMENTS What is the difference between congruence and equality? 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. Two-Column Proofs: Key Elements ① ② ③ ④ ⑤ 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) 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. =) 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!) 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. Given: AB = BC, C is the midpoint of BD Prove: AB = CD Statement Reason 1. 1. AB = BC, C is the midpoint of BD 2. 2. BC = CD 3. AB = CD 3. Given Def. Midpoint Substitution Given: AB=CD Prove: AC=BD Statement Reason 1. AB=CD 1. 2. 2. AB+BC=AC 3. BC+CD=BD 4. 5. BC+AB=BD AC=BD Given Segment Add. Post. 3. Segment Add. Post. 4. 5. Substitution Substitution Given: AC=BD Prove: AB=CD Statement Reason 1. AC=BD 1. 2. 2. AB+BC=AC 3. BC+CD=BD 4. 5. AB+BC = BC+CD AB=CD Given Segment Add. Post. 3. Segment Add. Post. 4. 5. Substitution Subtr. Prop of = Given: QR = RS Prove: QS = 2 RS Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. Given: LE = RM, EG = AR Prove: LG = MA Statement 1. LE = RM and EG = AR 2. AR+RM=AM Reason 1. Given 2. Segment addition 3. LE+EG=LG 3. Segment addition 4. RM+AR=LG 4.Substitution 5. LG = MA 5. Substitution 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. R Q P Statements: 1. 2. 3. 4. 5. 6. 7. Q is the midpoint of PR. PQ = QR PQ + QR = PR PQ + PQ = PR 2 ∙ PQ = PR PQ = ½ PR QR = ½ PR Reasons: 1. 2. Given Definition of a midpoint 3. Segment Addition Postulate 4. 5. 6. 7. Substitution Property Distributive property Division property Substitution (over Lesson 2-2) Write using two column proofs! Slide 1 of 1 (over Lesson 2-2) Slide 1 of 1 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 K • PROVE: LK ≅ JL J L Statements: Reasons: 1. 2. 3. 4. 5. 6. 1. 2. 3. 4. 5. 6. ________________ ________________ LK = JK LK ≅ JK JK ≅ JL ________________ Given Given Substitution _________________ Given Substitution Statements: Reasons: 1. 2. 3. 4. 5. 6. 1. 2. 3. 4. 5. 6. LK = 5 JK = 5 LK = JK LK ≅ JK JK ≅ JL LK ≅ JL Given Given Substitution Def. Congruent seg. Given Substitution Def. of congruent segments AB = BC DE = EF Substitution (or Transitive) Substitution (or Transitive) Def. of congruent segments