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 of a triangle related to the angles of a triangle?
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

3. Triangle Inequality Theorem (6.11)
In any triangle, the sum of any two sides is greater than the third side.
a + b > c
b + c > a
a + c > b
All of these must be true. b a c
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

4. Example 1 Is this triangle possible?
Yes
4 + 5 > 7
7 + 4 > 5
5 + 7 > 4 4 7 5
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

5. Example 2 Is this triangle possible?12 7 4 NO
> 7
> 4
But
7 + 4 > 12 7 1 4 12
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

6. Corollary to Triangle Inequality Theorem
If two sides of a triangle measure a and b, with a being the larger side, then the third side, c, is greater than a – b and less than a + b.
a-b < c < a+b
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

7. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles8 12 c
Example 3
a > b
a-b < c < a+b
c is greater than ________ and less than _________.
12 – 8
12 + 8
12−8<𝑐<12+8
4<c<20
c is greater than 4 and less than 20.
Or, c is a number between 4 and 20.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

8. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Your Turn
What are the possible values of x? 10 75 x
75 > 10
75 – 10 < x <
65 < x < 85
x is between 65 and 85.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

9. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Two Other Theorems
6.9: If one side of a triangle is longer than another side, then the angle opposite the larger side is larger than the angle opposite the shorter side. (Given the side lengths, the largest angle is opposite the longest side.)
6.10: If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. (Given the angle measures, the longest side is opposite the largest angle.)
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

10. T R S 15 14 8
Example 4
List the angles of the triangle in order from smallest to largest.
8  T
14  R
15  S
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

11. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Example 5 b
30
80
70 a c
List the sides of the triangle in order from smallest to largest.
30⁰  b
70⁰  c
80⁰  a
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

12. Exterior Angle Theorem Review
The measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles. 1 2 3
m1 + m2 = m3
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

13. Exterior Angle Inequality
The measure of an exterior angle of a triangle is greater than either of the two remote interior angles. 1 2 3
m3 > m1
m3 > m2
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

14. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Example
m1 > 35
m1 > 40
40
35 1
November 29, 2018
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 c d a a 1 2 b b
Begin with two congruent triangles.
(SAS)
November 29, 2018
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 c d a a 1 2 b b
Rotate 1 to make it smaller.
November 29, 2018
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 c a a 1 2 b b
Rotate 1 to make it smaller.
November 29, 2018
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 c a a 1 2 b b
Rotate 1 to make it smaller.
November 29, 2018
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?
November 29, 2018
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 1 2 b b
What happens to side c?
November 29, 2018
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 1 2 b b
What happens to side c?
November 29, 2018
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?
November 29, 2018
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 c a 2 1 b b
What happens to side c?
November 29, 2018
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 c a 2 1 b b
What happens to side c?
November 29, 2018
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
What happens to side c?
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

26. 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?
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

27. 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.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

28. 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.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

29. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

30. Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

31. 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.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

32. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Think of a door.
As the angle at the hinge increases, the size of the opening increases.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

33. 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.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

34. 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.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles

35. Geometry 6.5 & 6.6 Inequalities in One and Two Triangles
Summary
In a triangle, the sum of any two sides is greater than the third side.
If two sides of a triangle measure a and b, then side c is between a – b and a + b.
In any triangle, the largest side is opposite the largest angle and the smallest side is opposite the smallest angle.
Given two triangles with two congruent sides, the longer side is opposite the larger angle and the larger angle is opposite the longer side.
November 29, 2018
Geometry 6.5 & 6.6 Inequalities in One and Two Triangles