Definitions/Postulates REVIEW!

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1 Do NOW! This is on a worksheet I gave you last week called Practice with Proofs. #1 1) A, B, & C are collinear. B is btwn A & C. AC=32, AB = 2x & BC= 4x+2 2) 1) Given 2) Always start with the givens.. A B C Oct 11 7:23 AM Definitions/Postulates REVIEW! 1. Def of midpoint: a point that divides a segment into 2 congruent segments or 2 equal segments. 2. Def of angle bisector: a segment, ray, or line that divides an angle into 2 congruent angles or 2 angles equal in measure. 3. Angle Addition postulate: (2 versions) If point B is on the interior of <AOC a)...then m<aob + m<boc = m<aoc b)...the whole = the sum of its parts 4. Segment Addition postulate: a) If B is a point between endpoints A and C, then AB + BC = AC b)...the whole = the sum of its parts 5. def. of Perpendicular Lines: 2 lines that meet to form right angles. Sep 29 8:58 AM 1

2 Unit 1 Lesson 13 Proving Angle Theorems Remember... A postulate is a statement that is accepted without proof. A theorem is a statement proved by deductive reasoning. Below is a theorem about vertical angles: Lets prove it! "If 2 angles are vertical angles, then they are congruent." In an if...then statement, the "if" is the given information the "then" is what you have to prove. Always start with what you have been given. THIS IS WHAT WE KNOW!!! Sep 30 8:58 AM "If 2 angles are vertical angles, then they are congruent." There are many forms of a proof. A paragraph proof is written as sentences in a paragraph. Like a story. Given: Prove: are vertical angles what you know what you show (prove) Write a story to prove the theorem! Here's my story!! 1 Add a 3 to the diagram 2 3 I was given that angle 1 and 2 are vertical angles. I know that <1 & <3 form a linear pair and so do <2 & < 3. Since a linear pair of angles are supplementary, I know that the m<1+m<3=180 and m<2+m<3=180. If they are both equal to 180, then I know by the substitution property, that they are both equal to each other. *m<1+m<3 = m<2+m<3* If I subtract the m<3 from both equations, then I get that m<1=m<2. If 2 angles are = in measure, then they are, therefore, <1 <2 Sep 29 8:58 AM 2

3 My story was called a paragraph proof. Let's do this in a 2 column proof format. 1) <1 & <2 are vertical angles 1) Given 2) <1 & <3 are supplementary 2) If 2 angles for a linear pair, then they are supplementary. Oct 10 7:51 AM Prove this theorem using a 2-column proof. = implies Now that we've proved this, we can use this rule in a proof!! Sep 29 12:52 PM 3

4 * prove this one! In an if...then statement, the "if" is the given information the "then" is what you have to prove. Lets prove 1 or more of these theorems. Remember, once a theorem is proved, it is yours to keep...meaning, you can simply use the theorem in a proof instead of going through all of the steps required in proving the theorem. Sep 29 12:52 PM Prove: All right angles are congruent 2 1 1) <1 and <2 are right angles 2) 1) Given Sep 29 12:52 PM 4

5 1. <1 <2 2. <1 & <2 are supp 3. m<1 + m<2 = m<1 = m<2 5. m<1 + m<1 = (m<1)= m<1 = m<2 = <1 & <2 are right angles Sep 29 12:53 PM Given: GH = JK Prove: GJ = HK G H J K Sep 29 12:53 PM 5

6 Homework Unit 1 Lesson 13 Sep 29 1:00 PM 6