Chapter 6: inequalities in Geometry ≤≥

Chapter 6: inequalities in Geometry ≤≥ By: Sean Bonner and Tyler Martin

6-1 Inequalities • Properties of Inequality • If a > b and c ≥ d, then a + c > b + d • If a > b and c > c then ac > bc and a/c > b/c • If a > b and c < 0, then ac < bc and a/c < b/c • Theorem 6-1 The Exterior Angle Inequality Theorem (EAT) • The measure of an exterior angle of a triangle is greater than the measure of either its remote interior angle

Property of Inequality Examples • If x < y, then x + z < y + zExample: Suppose Sylvia's weight < Jennifer's weight, then Sylvia's weight + 4 < Jennifer's weight + 4 Or suppose 1 < 4, then 1 + 6 < 4 + 6If x > y, then x + z > y + zExample: Suppose Sylvia's weight > Jennifer's weight, then Sylvia's weight + 9 > Jennifer's weight + 9 Or suppose 4 > 2, then 4 + 5 > 2 + 5

6-2 Inverses and Contrapositives • Statement: If p, then q • Inverse: If not p, then not q • Contrapositive: if not q, then not p • Example: Given- if p then q • All Runners are athletes 1) Leroy is a runner then Leroy is an Athlete 2) Lucia is not an athlete, then she is not a runner 3) Linda is an atheletelinda might be a runner (No conclusion) 4) All runners are athletes, larry is not a runner, he might be an athlete

6-3 Indirect Proof • Used if it is impossible to find a direct proof • How to write an indirect proof • Assume temporally that the conclusion is not true • Reason logically until you reach a contradiction of a known fact • Point out that the temporarily assumption must be false, and that the conclusion must the be true

Indirect Proof Example Given: Triangle ABC is an isosceles triangle with vertex angle AProve: Angles 1 and 2 are congruent

Example Answer

6-4 Inequalities in One Triangle Larger Angle Longer Side(LALS) -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 Longer Side Larger Angle (LSLA) -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

LALS/LSLA Longest Side Meium Angle Smallest Side Largest Angle Smallest Angle

The Triangle Inequality Theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. • In the figure, the following inequalities hold. • a + b > c • a + c > b • b + c > a A B C

Example: Check whether it is possible to have a triangle with the given side lengths: 7, 9, 13 Steps: Make Sure that any two sides added is greater than the third side The sum of 7+9 = 16 and 16 >13. The sum of 9 + 13 = 21 and 21 > 7. The sum of 7 + 13 = 20 and 20 > 9. These lengths form a triangle because it follows the Triangle Inequality Theorem

BASIC PRACTICE • Example: • Check whether the given side lengths form a triangle. • 4, 8, 15

6-5 Inequalities for Two Triangles 1. 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. 2. 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, then the included angle of the first triangle is larger than the included angle of the second.

SAS Inequality Theorem The two triangles have two congruent sides <A > <D Therefore: Line CB > Line FE D 40 A 105 DEF ABC B E C F

SSS Inequality Theorem The two triangles have two congruent sides Line CB > Line FE Therefore: <A > <D D A DEF ABC B E C F 8 20

BASIC PROOF A D B C E F Given: BA = ED; BC = EF; AC > DF Prove: m<B > m<E

Triangle Inequality Proof Statements Reasons 1. Given 2. Def. Isosceles Triangle 3. 4. 6. Substitution. 7.

Triangle Inequality Proof Statements Reasons 1. Given 2. Def. Isosceles Triangle 3. ITT 5. EAT 6. Substitution. 7. LALS

Chapter 6: inequalities in Geometry ≤≥