1. Inequalities and Triangles
Geometry ~ Chapter 5.2 and 5.4
Inequalities and Triangles

2. Inequalities What are they?
Angle measures can be compared using inequalities:
m < a = m < b
m < a < m < b
m < a > m < b

3. Exterior Angle Inequality Theorem:
If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of the remote interior angles.
Example:
<1 is an exterior angle
m < 1 > m <PQR
m < 1 > m<RPQ 1

4. The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.
If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

5. The angles from smallest to largest are F, H and G.
Ex. 1 - Write the angles in order from smallest to largest.
The shortest side is GH, so the smallest angle is opposite GH…. F
The longest side is FH, so the largest angle is G
The angles from smallest to largest are
F, H and G.

6. Ex. 2 - Write the sides in order from shortest to longest.
mR = 180° – (60° + 72°) = 48°
The smallest angle is R, so the shortest side is PQ.
The largest angle is Q, so the longest side is PR.
The sides from shortest to longest are PQ, QR, and PR.

7. Draw and label triangle ABC!!!
Example 2A ~ If m  A = 9x – 7, m  B = 7x – 9 and m  C = 28 – 2x, list the sides of ABC in order from shortest to longest.
Draw and label triangle ABC!!! A
B C
101°
(9x – 7) + (7x – 9) + (28 – 2x) = 180
14x + 12 = 180
14x = 168
x = 12
(9x – 7)°
(7x – 9)°
(28 – 2x)° 4°
75°
The sides from shortest to longest are AB, AC, BC

8. A triangle is formed by three segments, but not every set of three segments can form a triangle.

9. If you take 3 straws of lengths 8 inches, 5 inches and 1 inch and try to make a triangle with them, you will find that it is not possible. This illustrates the Triangle Inequality Theorem.
A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

10. Ex. 3 – Applying the Triangle Inequality Thm
Ex. 3 – Applying the Triangle Inequality Thm. Tell whether a triangle can have sides with the given lengths. Explain.
7, 10, 19
No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

11. Yes—the sum of each pair of lengths is greater than the third length.
Ex. 4 - Tell whether a triangle can have sides with the given lengths. Explain.
2.3, 3.1, 4.6   
Yes—the sum of each pair of lengths is greater than the third length.

12. Ex. 5 - Tell whether a triangle can have sides with the given lengths
Ex. 5 - Tell whether a triangle can have sides with the given lengths. Explain.
8, 13, 21
No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

13. Finding the RANGE of side lengths
The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.
Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.
x + 8 > 13
x + 13 > 8
> x
x > 5
x > –5
21 > x
Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.

14. Ex. 6 - The lengths of two sides of a triangle are 23 inches and 17 inches. Find the range of possible lengths for the third side.
Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.
x + 23 > 17
x + 17 > 23
> x
x > –6
x > 6
40 > x
Combine the inequalities. So 6 < x < 40. The length of the third side is greater than 6 inches and less than 40 inches.

15. Lesson Wrap Up 1. Write the angles in order from smallest to largest.
2. Write the sides in order from shortest to longest.
C, B, A

16. Lesson Wrap Up
3. The lengths of two sides of a triangle are 17 cm and 11 cm. Find the range of possible lengths for the third side.
4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain.
6 cm < x < 28 cm
No; is not greater than 9.8.
5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distances
shown be 8 ft and 6 ft? Explain.
Yes; the sum of any two lengths is greater than the third length.