Inequalities I n Geometry

1. Inequalities In Geometry Chapter 6 Ms. Cuervo

2. Inequalities and Indirect Proof Chapter 6 Lesson 1

3. Example 1 B Given: AC>AB; AB>BC Conclusion: AC BC A C

4. Example 2 Given: m<BAC+ m<CAD= m<BAD Conclusion: m<BAD m<BAC; m<BAD m<CAD B C A D

5. Properties of Inequality • If a>b and c≥d, then a+c>b+d • If a>b and c>0, then ac>bc and a/c>b/c • If a>b and c<0, then ac<bc and a/c<b/c • If a>b and b>c, then a>c • If a=b+c and c>0, then a>b.

6. Example 3 B E • Given: AC>BC; CE>CD • Prove: AE>BD C D Reasons Statements A AC>BC; CE>CD AC+CE>BC+CD AC+CE=AE BC+CD=BD 4. AE>BD Given A Prop. Of Inequality Segment Add Post. Substitution Prop

7. Example 4 • Given: <1 is an exterior angle of DEF. • Prove: m<1>m<D; m<1>m<E D 1 E F Reasons Statements The measure of an ext. < of a triangle equals the sum of the measures of the two remote int <‘s. A Prop. Of Ineq. m<1=m<D+m<E m<1>m<D; m<1>m<E

8. Theorem 6-1 • The Exterior Angle Inequality Theorem- The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

9. Inverses and Contrapositives Chapter 6 Lesson 2

10. Vocabulary • Statement: If p, then q • Inverse: If not q, then not p. • Contrapositive: If not q, then not p.

11. Example 1 Write (a) the inverse and (b) the contrapositive of the true conditional: If two lines are not coplanar, then they do not intersect. Solution: Inverse: If two lines are coplanar, then they intersect (False) Contrapositive: If two lines intersect, then they are coplanar. (True)

12. Venn Diagram If P If Q If P and Q

13. Logically Equivalent • When both the conditional and contrapositive are true. • When both the conditional and contrapositive are false. THEN YOU CAN SAY THAT THE STATEMENTS ARE LOGICALLY EQUIVALENT

14. Summary of Related If-Then Statements • Given statement: If p, then q. • Contrapositive: If not q, then not p. • Converse: If q, then p. • Inverse: If not p, then not q. A STATEMENT AND ITS CONTRAPOSITIVE ARE LOGICALLY EQUIVALENT. A STATEMENT IS NOT LOGICALLY EQUIVALENT TO ITS CONVERSE OR TO ITS INVERSE.

15. Example 1 • Suppose this conditional is true: ALL RUNNERS ARE ATHLETES What can you conclude from each additional statement?

16. All Runners are Athletes: Leroy is a Runner Given: If p, then q; All runners are athletes p Leroy is a runner Conclusion: q Leroy is an athlete Athletes Leroy Runners

17. All Runners Are Athletes: Lucia Is Not An Athlete • Given: If p, then q; All runners are athletes. not q Lucia is not an athlete Conclusion: not p Lucia is not a runner Lucia Athletes Runners

18. All Runners are Athletes: Linda is an Athlete • Given: If p, then q; All runners are athletes q Linda is an athlete No Conclusion Follows Linda might be a runner or she might not be Linda Athletes Runners

19. All Runners are Athletes: Larry is Not a Runner • Given: If p, then q; All Runners are Athletes not p Larry is not a runner. No Conclusion Follows Larry might be an athlete or he might not be. Larry Runners Athletes

20. Indirect Proofs Chapter 6 Lesson 3

21. What Is An Indirect Proof? A proof where you assume temporarily that the desired conclusion is not true. Then you reason logically until you reach a contradiction of the hypothesis or some other known fact.

22. How Do You Write an Indirect Proof? • 1. Assume temporarily that the conclusion is not true. • 2. Reason logically until you reach a contradiction of a known fact. • 3. Point out that the temporary assumption must be false, and that the conclusion must then be true.

23. Example 1 Given: n is an integer and n^2 is even Prove: n is even Proof Assume temporarily that n is not even. Then n is odd and n^2 =n x n = odd x odd x= odd But this contradicts the given information that n^2 is even. Therefore the temporary assumption that n is not even must be false. It follows that n is even.

24. Example 2 • Prove that the bases of a trapezoid have unequal lengths. Given: Trap. PQRS with bases PQ and SR Prove: PQ ≠SR Proof: Assume temporarily the PQ=SR. We know that PQ ||SR by the definition of a trapezoid. Since quadrilateral PQRS has two sides that are both congruent and parallel, it must be a parallelogram, and PS must be parallel to QR. But this contradicts the fact that, by definition, trapezoid PQRS can have only one pair of parallel sides. The temporary assumption that PQ=SR must be false. It follows that PQ ≠ SR.

25. Inequalities Chapter 6 Lesson 4

26. Theorem 6-2 • If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

27. Theorem 6-3 • If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle.

28. Corollary 1 • The perpendicular segment from a point to a line is the shortest segment from the point to the line.

29. Corollary 2 • The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

30. The Triangle Inequality (Theorem 6-4) • The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

31. Inequalities For Two Triangles Chapter 6 Lesson 5

32. SAS Inequality Theorem If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

33. SSS Inequality Theorem • If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, hen the included angle of the first triangle is larger than the included angle of the second.