Understanding Postulates and Theorems in Geometry

1. Geometry Notes Sections 2-5 & 2-7

2. What you’ll learn • How to identify and use basic postulates/axioms and theorems about points, lines, and planes

3. Vocabulary • Postulate • Axiom • Theorem • Proof • Paragraph proof • Informal proof

4. Recall the definition of Postulate • DEFN Postulate (axiom is another word for postulate): • A statement that describes a fundamental relationship between the basic terms of geometry • Always accepted as true • DEFN Theorem: • A statement or conjecture that can be proven true using postulates, definitions, and undefined terms • Must be proven

5. Postulates • In conditional format: • If you have two points there is exactly one line that would go through those two points. • Through any two points there is exactly one line. • Symbolically: • 2 pts →exactly one line • Through any three noncollinear points there is exactly one plane. • In conditional format: • If you have three noncollinear points, there is exactly one plane that would contain them. • Symbolically: • 3 noncollinear pts →exactly one plane

6. More Postulates • If you have a line, then that line has at least two points on it. • If you have a plane, then it has at least 3 noncollinear points. • If 2 points lie in a plane, then the entire line containing those 2 points lies in that plane. • If 2 lines intersect, then their intersection is exactly one point. • If 2 planes intersect, then their intersection is a line.

7. Ruler Postulate: • This postulate guarantees all line segments have length or measure • If you have AB then mAB or AB exists Even More Postulates • Symbolically: • AB →mAB or AB • Segment Addition Postulate (hey we already know this . . .right?) • If B is between A and C, then AB + BC = AC • And if AB + BC = AC then B is between A and C. • Symbolically: • B is between A and C↔ AB + BC = AC

8. Definitions we know. . . • DEFN: right angle • An angle is a right angle iff it measures 90 • Symbolically: • Right angle ↔90 • DEFN: congruent segments • Segments are congruent iff they have the same measure • Symbolically: •  ↔= • DEFN: congruent angles • Angles are congruent iff they have the same measure • Symbolically: •  ↔=

9. Postulates (axioms), definitions, and already proven theorems are the facts and rules we use to justify our argument in deductive reasoning. • Proofs are like puzzles or games.You have to memorize the postulates, definitions and theorems—they are the rules to the game.

10. The 5 Essential Parts of a Good Proof • 1. State the theorem or conjecture to be proven. • Okay, I’m not going to lie, sometimes we skip this one • Now for the parts we really don’t skip--ever • 2. A list of the given information • Usually cleverly hidden by the word “Given” • 3. A diagram of what we’re given (and only what we are given) • This is the part that looks like a picture • 4. State what is to be proved • Again, cleverly hidden by the word “Prove” • 5. A system of deductive reasoning • My favorite is a toss up between the flow chart proof and the two – column proof

11. Things everyone needs to know about writing proofs: • The given and prove statements cannot be written in a general format, they must be specific • Example: • 2 angles are right angles is too general • A and B are right angles is what you want • The statements and reasons must be numbered in any proof • You are only allowed to use the word “given”, postulates, definitions, or previously proven theorems for reasons

12. Let’s try one. . . • Yes a proof. • Prove that all right angles are congruent. • Hint: Rewrite the statement you are proving as a conditional statement (in If-then form) • If two angles are right angles, then they are congruent. • This is the part we usually skim over, but since this is our first time we might want to do all the steps. . .

13. If two angles are right angles then they are congruent. • So, do you think it’s true? • Why? • Now that we believe, let’s move on to step 2. . . What are we given to use? • The given information is always listed in the hypothesis of the conditional statement. • The “If” part

14. We are given two right angles. I would feel so much better if we gave them names. . . It would make the whole thing more personal. • Let’s call them 1 and 2 (see we can use numbers sometimes) • Now what did that if part say. . . • If two angles were right angles. . . . • Given: 1 and 2 are right angles • Two essential parts covered, three to go. • What’s next?

15. Next is a diagram of our given information • Given: 1 and 2 are right angles • We have to draw 2 basic right angles and name them 1 and 2 – never add special circumstances like making the angles adjacent, linear pairs, vertical angles. . . 1 2

16. Now what do we have so far? 1 2 • Given: 1 & 2 are right angles What’s the next step in our list?Step 4 out of 5. . .  4. State what is to be proved. • The information to be proved is found in the conclusion of the conditional statement • The part after the word “then” • If two angles are right angles, then they are congruent.

17. 1 2 • Given: 1 & 2 are right angles • Prove: 1 2 Remember they have names now • And now for the last step • 5. A system of deductive reasoning • My favorite is a toss up between the flow chart proof and the two – column proof

18. Prove: 1 2 1 2 • Given: 1 & 2 are right angles Statements Reasons 1. 1& 2are right angles 1. Given 2. m 1 =90 m2=90 2. right s ↔ 90 3. m1 = m2 3. Substitution 4. 1 2 4. =↔ 

19. Have you learned. . . • How to identify and use basic postulates about points, lines, and planes? • We will build on the process of writing proofs. It takes time. You’ll get there.  • Assignment : Worksheet 2.7