1. PROOFS

3. Geometry "measuring the earth“ is the branch of math that has to do with spatial relationships.

4. A BIT OF HISTORY Geometry first became associated with land measurement in Egypt. The Egyptians were obliged to invent it in order to restore the landmarks that were destroyed by the periodic inundation of Nile River. Geometry first became associated with land measurement in Egypt. The Egyptians were obliged to invent it in order to restore the landmarks that were destroyed by the periodic inundation of Nile River.

5. Our textbook is based on Euclidean (or elementary) geometry. "Euclidean" (or "elementary") refers to a book written over 2,000 years ago called "The Elements" by a man named Euclid.

6. Around 300 B.C. Euclid organized the Greek knowledge of geometry into a thirteen –volume work called ELEMENTS. His structure and method influence the way that geometry is taught today.

7. Types of Reasoning 1. INDUCTIVE REASONNG - reaching a conclusion based on previous observation. - CONCLUSION IS PROBABLY TRUE BUT NOT NECESSARILY TRUE.

8. EXAMPLES 1² = 1 1 ≤ 1 2² = 4 2 ≤ 4 CONSIDER THIS EXAMPLE (½) ²= ¼ ½> ¼ The conclusion does not hold true

9. 2. Deductive Reasoning reaching a conclusion by combining known truths to create a new truth reaching a conclusion by combining known truths to create a new truth

10. Deductive Reasoning Deductive Reasoning deductive reasoning is certain, provided that the previously known truths are in fact true themselves. deductive reasoning is certain, provided that the previously known truths are in fact true themselves.

11. Types of Reasoning 1. INDUCTIVE REASONNG - reaching a conclusion based on previous observation. - CONCLUSION IS PROBABLY TRUE BUT NOT NECESSARILY TRUE.

12. Types of Reasoning 1. DEDUCTIVE REASONNG - conclusion based on accepted statements (postulates, past theorems, given information, definition).

13. Examples of Postulates/axioms Linear pair postulate If two // lines are cut by a transversal, then corresponding angles are equal. Properties from algebra - SUBSTITUTION PROPERTY - REFLEXIVE PROPERTY - SYMMETRIC PROPERTY - REFLEXIVE PROPERTY - SUBTRACTION PROPERTY - TRANSITIVE PROPERTY

14. Examples of Definitions Midpoint of a segment Bisector of an angle Congruent angles and Segments Perpendicular lines Supplementary angles Complementary angles OTHERS

15. Types of Reasoning 1. DEDUCTIVE REASONNG - conclusion must be true if hypothesis are true.

16. KINDS OF PROOF DIRECT PROOFINDIRECT PROOF  HAS 5 MAIN PARTS - THEOREM - GIVEN - PROVE STATEMENT - DIAGRAM - PROOF with STATEMENT & REASON  USUALLY IN PARAGRAPH FORM  THE OPPOSITE OF THE STATEMENT TO BE PROVEN IS ASSUMED TRUE, UNTIL THIS ASSUMPTION LEADS TO A CONTRADICTION.  IT IS USED WHEN A DIRECT PROOF IS DIFFICULT OR IMPOSSIBLE.

18. GIVEN:  k // m:  Transversal t cuts k and m. DIAGRAM/FIGURE 1 2 3 k t m PROVE:  2 =  3

19.  k // m: t is a transversal.   1 =  2   1 =  3   2 =  3 REASON  Given  Vertical angles are equal  If two parallel lines are cut by a transversal, then corresponding angles are equal  Transitive property STATEMENT