1. Properties of Triangles
Unit 6

2. 5.5 Inequalities in One Triangle
Lesson Goals
State Standards for Geometry
Use triangle measurements to decide which side is longest or which angle is largest.
Apply the Triangle Inequality Theorem to determine if 3 lengths can form a triangle, and find possible lengths of a 3rd side given two.
Know and use the Triangle Inequality Theorem.
Prove relationships between angles in a polygon.
ESLRs: Becoming Effective Communicators,
Competent Learners and Complex Thinkers

3. theorem Triangle Angle-Side Relationships Theorem
The longest side of a triangle is opposite
the largest angle
The shortest side of a triangle is opposite the smallest angle. A B C
shortest side
largest angle
longest side
smallest angle

4. example
Write the measures for the sides of the triangle in order from least to greatest A B C
111o
46o
23o

5. You Try
Write the measures for the angles of the triangle in order from least to greatest T U 10 V 7 11

6. You Try J L K

7. theorem Exterior Angle Inequality Theorem
The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior angles. A 1 C B

8. theorem Exterior Angle Inequality Theorem
The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior angles. A 1 C B

9. example 6 7 3 5 4 2 1

10. example
Write an equation or inequality to describe the relationship between the measures of all angles. ao do co bo

11. theorem Triangle Inequality Theorem
The sum of the lengths of any two sides
of a triangle is greater than the length of
the third side. A B C A B C A C

12. theorem Triangle Inequality Theorem
The sum of the lengths of any two sides
of a triangle is greater than the length of
the third side. A B C B A C B C

13. theorem Triangle Inequality Theorem
The sum of the lengths of any two sides
of a triangle is greater than the length of
the third side. A B C A C B A B

14. example 5, 7, 8 5 + 7 > 8 5 + 8 > 7 7 + 8 > 5 Yes
Can a triangle be constructed with sides of the following measures?
5, 7, 8
5 + 7 > 8
5 + 8 > 7
7 + 8 > 5
Yes
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.

15. example
Can a triangle be constructed with sides of the following measures?
3, 6, 10
3 + 6 > 10 No

16. example 11, 15, 21 11 + 15 > 21 15 + 21 > 11 11 + 21 > 15 Yes
Can a triangle be constructed with sides of the following measures?
11, 15, 21
> 21
> 11
> 15
Yes
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.

17. example
Can a triangle be constructed with sides of the following measures?
4.2, 4.2,
> 8.4 No

18. A triangle cannot be constructed.
You Try
Can a triangle be constructed with sides of
the following measures?
3 in, 3 in , 8 in
3 + 3 > 8
A triangle cannot be constructed.
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.

19. A triangle cannot be constructed.
You Try
Can a triangle be constructed with sides of
the following measures?
6 in, 6 in , in
6 + 6 > 12
A triangle cannot be constructed.
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.

20. A triangle can be constructed.
You Try
Can a triangle be constructed with sides of
the following measures?
9 in, 5 in , in
9 + 5 > 11
A triangle can be constructed.
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.

21. example
Solve the inequality.

22. example x + 8 > 17 x + 17 > 8 8 + 17 > x x > 9 9 < x
A triangle has one side of 8 cm and another of 17 cm. Describe the possible lengths of the third side.
x + 8 > 17
x + 17 > 8
> x
x > 9
9 < x
x > -9
25 > x
x < 25 17 8 x
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.

23. example x + 11 > 16 x + 16 > 11 11 + 16 > x x > 5 5 < x
A triangle has one side of 11 in and another of 16 in. Describe the possible lengths of the third side.
x + 11 > 16
x + 16 > 11
> x
x > 5
5 < x
x > -5
27 > x
x < 27 16 11 x
By the Triangle Inequality Theorem, the sum of the measures
of any two sides must by greater than the third side.

24. Today’s Assignment
p. 298: 1 – 6, 9, 12, 13, 16, 17, 20, 23 – 25, 27, 28
(worksheet in packet)