Chapter 6 Inequalities in Geometry page 202

Lesson 6-4 Inequalities for One Triangle (page 134) Essential Question What are the relationships between sides and angles in a triangle?

Inequalities for One Triangle

Theorem 6-2 If one side of a triangle is longer than a 2nd side, then the angle opposite the 1st side is larger than the angle opposite the 2nd side. R T S

Theorem 6-2 If one side of a triangle is longer than a 2nd side, then the angle opposite the 1st side is larger than the angle opposite the 2nd side. R 1st side opposite angle T S

Theorem 6-2 If one side of a triangle is longer than a 2nd side, then the angle opposite the 1st side is larger than the angle opposite the 2nd side. R 2nd side opposite angle T S

Theorem 6-2 If one side of a triangle is longer than a 2nd side, then the angle opposite the 1st side is larger than the angle opposite the 2nd side. R T S

to theorem 6-2 is theorem 6-3. Conversely to theorem 6-2 is theorem 6-3.

Theorem 6-3 If one angle of a triangle is larger than a 2nd angle, then the side opposite the 1st angle is longer than the side opposite the 2nd angle. R T S

Theorem 6-3 If one angle of a triangle is larger than a 2nd angle, then the side opposite the 1st angle is longer than the side opposite the 2nd angle. R opposite side 1st angle T S

Theorem 6-3 If one angle of a triangle is larger than a 2nd angle, then the side opposite the 1st angle is longer than the side opposite the 2nd angle. R opposite side 2nd angle T S

Theorem 6-3 If one angle of a triangle is larger than a 2nd angle, then the side opposite the 1st angle is longer than the side opposite the 2nd angle. R T S

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

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

Theorem 6-4 If sum of the lengths of any two sides of a triangle is greater than the length of the third side. C A B NOTE: This is used to determine whether 3 lengths will determine a triangle.

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (1) 2, 3, 4 YES NO 2 + 3 ? 4 5 > 4

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (2) 5, 6, 9 YES NO 5 + 6 ? 9 11 > 9

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (3) 12, 15, 18 YES NO 12 + 15 ? 18 27 > 18

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (4) 2, 5, 8 YES NO 2 + 5 ? 8 7 < 8

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (5) 1.2, 4.3, 8.4 YES NO

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (6) 16, 18, 34 YES NO

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (7) 2.7, 3.6, 5.4 YES NO

Examples: State whether or not the given Examples: State whether or not the given lengths are sides of a triangle. (8) 15, 24, 40 YES NO

∴ the side must be greater than 2 and less than 12. Given 2 sides of a triangle, how can you determine the lengths of the 3rd side? Example: 5 & 7 are two sides 5 + 7 > S 5 + S > 7 S + 7 > 5 S > 2 S > -2 12 > S S < 12 ∴ the side must be greater than 2 and less than 12.

∴ the side must be greater than 2 (7-5) and less than 12 (7+5). Given 2 sides of a triangle, how can you determine the lengths of the 3rd side? Example: x & y are two sides (x > y) ∴ the side must be greater than 2 (7-5) and less than 12 (7+5). S > x - y S < x + y ∴ the side must be greater than x - y and less than x + y.

First a Classroom Assignment! Classroom Exercises on page 221 1 to 12 all numbers Assignment Written Exercises on pages 222 & 223 1 to 17 odd numbers

Chapter 5: Quadrilaterals and Chapter 6: Inequalities in Geometry Prepare for Test on Chapter 5: Quadrilaterals and Chapter 6: Inequalities in Geometry Assignments: Chapter 5 Review on pages 197 & 198 1 to 11, 13 to 16, and 19 to 22 all numbers Chapter 5 Test on page 199 1 to 17 all numbers Chapter 6 Review on page 236 11, 12, & 14 Chapter 6 Test on page 236 7 to 10 all numbers