Geometry Proofs

1. Geometry Proofs Math 416

2. Time Frame • Definition • Congruent Triangles • Axiom & Proofs • Propositions

3. Definitions • Geometric Proofs • The essence of pure mathematics • The creative and artistic center of math • The ability to explain in a detailed concise logical manner how a proposition (problem) is either true or false.

4. Definitions (con’t) • Detailed – hard facts • Concise – short to the point • Logical – set of rules based on reason • A proof generally falls back to things that are either known, accepted or already proven. This is how we attain knowledge

5. Gaining Knowledge Proposition Enlightenment Proposition Proposition Proposition Proposition Axiom Definition Thoerem

6. Definitions • Definition: You define something once you identify its essential characteristics • For example, triangle – a two dimensional polygon with three sides Must Not

7. Axiom • Axioms: An obvious statement that is acceptable without proof • For example, the shortest distance between two points is a straight line

8. Propositions • Propositions are statements that require proof • Once proven they are called theorems • For example Proof 1 STATEMENT AUTHORITIES 3 <1 + <3 = 180° DEFINITION 2 DEFINITION <2 + < 3 = 180° <1 = <2 = 180 ALGEBRA

9. Theorums • This proposition now becomes a theorem • Hence, vertically opposite angle theorem • Theorems can be used in a proof as an authority • Definitions must use terms that are already defined Be reversible once you have the characteristics you have the object not give unnecessary information

10. Examples #1 of Definitions Definition: A belingas is a shape with a dot on a vertex are belingas Which of the following is a belingas?

11. Example #2 of a Definition Stencil #1  Are Gatus Which of the following is a Gatu? Definition: A Gatu is a shape with at least one curved side

12. Axioms • A statement not requiring proof • A whole is equal to the sum of its part • Completion C A B D < ABD = <ABD + <CBD • Any quantity can be replaced by another equal quantity

13. Axioms Easiest thing to do is to assign numbers to letters… a=0;b=4;c=4;q=4 • Replacement… • If a + b = c • AND b = q • Then a + q = • The shortest distance between two points is a straight line • Only one line can pass through the same two points • Given a point and a direction, only one line with that direction can pass through the point c

14. Postulates • Theorems we will not prove are called postulates specifically the congruence postulates • Hypothesis: Given two triangles with corresponding sides equal we say • CONC: Two triangles are congruent X A ABC YZX By S S S B C Y Z

15. Postulates • Hypothesis: Given two triangles with two corresponding sides equal and the contained angle equal • Conclusion: The two triangles are congruent X A ABC ZXY ° By SAS Y ° Z B C

16. Postulates • Hypothesis: Given two triangles with two corresponding angles equal and the contained side equal • Conclusion: The two triangles are congruent A X ABC ZXY O By ASA O X C Y B X Z Do #2 

17. Theorems • Once again we will not prove • But you may be required to • You should be able to

18. Theorems • The 90° completion theorem or the complementary angle theorem • The 180° Completion Theorem HYP: Diagram CONC < X + <Y = 90° x y HYP Diagram CONC <x + <y = 180 x y

19. Vertically Opposite Angle Theorem 1 4 3 2 < 1 = < 2 < 3 = <4 Conclusion

20. Triangle Sum Theorem 1 3 2 Conclusion <1 + <2 + <3 = 180°

21. Isosceles Triangle Theorem Given an isosceles triangle, the angles opposite the equal sides are equal 1 2 Conclusion <1 = <2

22. Isosceles Triangle Theorem Converse Given an isosceles triangle, the sides opposite the equal angles are equal A C B Conclusion AB = AC

23. Parallel Line Theorem Note: The converse is true also to prove // lines 1 2 Sometimes called Corresponding angles 3 4 a b c d <1 < a <2 = <b <3 = < c <4 = <d Conclusion <4 = < a < 3 < b <3 + <a = 180° <4 + <b = 180°

24. Parallelogram Theorem and Converse Conclusion: D A AD = BC AB = DC Opposite Sides < BAD = <DCB < ABC = < ADC Opposite Angles x B C BX = XD AX = XC Diagonals Bisected In a parallelogram opposite sides are equal, opposite angles are equal and the diagonals bisect each other

25. Triangle Parallel Similarity Theorem A C B Conc ABC ˜ ADE D E Do #3 

26. Test Question • If ABC ˜ XYZ and then < XYZ is 50°, how much is angle ABC? • 50° • Vertically opposite angles is an example of a a) Theorum b) axiom c) definition d) postulate

27. Pythagoras Theorem A Given a right angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides b c HYP: Diagram CONC: b2 = a2 + c2 B C a

28. Solve for x Solve for x Pythagoras Examples 202 = x2 + x2 400 = 2x2 200 = x2 14.14 = x2 x = 14.14 x2 = 62 + 82 x2= 36 + 64 x = 10 x 6 20 8 x x

29. The 30-60-90 Theorem A The side opposite the 30° angle is half the hypotenuse. 60° b c HYP: Diagram CONC: c = ½ b OR b = 2c 30° C B

30. The 30-60-90 Theorem Converse A If the hypotenuse is twice the length of one of the legs, the angle opposite the leg is 30° 2b b HYP: Diagram CONC: <ACB = 30° C B

31. 30-60-90 Examples (2x)2=x2+196 4x2=x2+196 3x2=196 x2= 65.33 x = 8.08 Opposite the 30° 30° It is half the hypotenuse x = 12 x 6 14 30° x

32. Exam Question D A Hyp: Diagram Conc: < ABC = < ADC B Construction AC C

33. Exam Questions Con’t • Fill in the missing authorities Statement Authorities < DAC = <ACB < DCA = <BAC AC = AC Thus DAC BCA <ABC = < ADC // Line Theorum // Line Theorum Reflex ASA Definition

34. Prove the following Statement Authorities A < BAD=<ACD HYP < ABC = <ABD Reflex ABD˜ CBA AA AB = BD = AD CB BA CA DEFN B D C HYP: diagram CONC: AB2 = BC • BD AB2 = BC • BD Cross Multipln Do #5 & 6 

35. Tips for Success • Always work on what you know • The more facts you put into a question the closer you will get to the answer • Extend the lines

36. Exam Questions & Practice • We will do more examples on the board together… • P262, p266, 267, 268, p272, 274 • Study Guide • Test 