1. 6.5 & 6.6 Inequalities in One and Two Triangle
Geometry
6.5 & 6.6 Inequalities in One and Two Triangle

2. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
6.5 & 6.6 Essential Question
How are the sides related to the angles of a triangle?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

3. Geometry 5.1 Perpendiculars and Bisectors
January 2, 2019
Geometry 5.1 Perpendiculars and Bisectors

4. Geometry 5.1 Perpendiculars and Bisectors
January 2, 2019
Geometry 5.1 Perpendiculars and Bisectors

5. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Example 1
List the angles of the triangle in order from smallest to largest. ∠T
∠ R
∠ S T R S 15 14 8
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

6. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Example 2
List the sides of the triangle in order from smallest to largest. 𝐷𝐹 𝐸𝐹 𝐷𝐸
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

7. Geometry 5.1 Perpendiculars and Bisectors
January 2, 2019
Geometry 5.1 Perpendiculars and Bisectors

8. Example 3 Is this triangle possible?
Yes
4 + 5 > 7
7 + 4 > 5
5 + 7 > 4 4 7 5
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

9. Example 4 Is this triangle possible?NO
> 7
> 4
But,
7 + 4 > 12 7 4 12 7 1 4 12
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

10. Geometry 5.1 Perpendiculars and Bisectors
January 2, 2019
Geometry 5.1 Perpendiculars and Bisectors

11. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Example 5 8 12 c
Find the range of values for c.
c + 8 > 12 and > c
20 > c
c > 4
c < 20
c is greater than 4 and less than 20.
Or, c is between 4 and 20.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

12. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Your Turn
What are the possible values of x? 10 75 x
x is between 65 and 85.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

13. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem c d a a 1 2 b b
Begin with two congruent triangles.
(SAS)
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

14. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem c d a a 1 2 b b
Rotate 1 to make it smaller.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

15. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d c a a 1 2 b b
Rotate 1 to make it smaller.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

16. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d c a a 1 2 b b
Rotate 1 to make it smaller.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

17. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a c a 1 2 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

18. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a c a 1 2 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

19. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a c a 1 2 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

20. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a c a 2 1 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

21. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a c a 2 1 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

22. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a c a 2 1 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

23. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a a c 2 1 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

24. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a a c 2 1 b b
What happens to side c?
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

25. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
The Hinge Theorem d a a c 2 1 b b
It gets smaller.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

26. The Hinge Theorem Therefore, if 1 is smaller than 2,d a a c 2 1 b b
Therefore, if 1 is smaller than 2,
then side c is smaller than side d.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

27. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

28. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
An Easy Memory Aid
The smaller the angle, the smaller the opposite side.
The larger the angle, the larger the opposite side.
As the angle at the hinge increases, the size of the opening increases.
Think of a door.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

29. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Example 6
How does x compare to y? y 5 5 x
30 6 6
Since 30 < 90, x < y.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

30. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Example 7
How does mA compare to mR? C T 16 17 10 10 A B R S 15 15
Since 16 < 17, mA < mR.
January 2, 2019
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

31. Geometry 5.1 Perpendiculars and Bisectors
Your Turn
January 2, 2019
Geometry 5.1 Perpendiculars and Bisectors