1. Chapter 6 Inequalities in Geometry page 202

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

3. Inequalities for One Triangle

4. 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

5. 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

6. 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

7. 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

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

9. 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

10. 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

11. 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

12. 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

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

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

15. 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.

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

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

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

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

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

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

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

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

24. ∴ 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: & 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.

25. ∴ 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.

26. 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

27. 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