1. Inequalities in One Triangle
Lesson 5-3
Inequalities in One Triangle
Lesson 5-3: Inequalities in One Triangle

2. Content Standards G-CO.10 Prove theorems about triangles.
Mathematical Practices
1 Make sense of problems and persevere in solving them
2 Reason abstractly and quantitatively.
6 Attend to precision.

3. You found the relationship between the angle measures of a triangle
You found the relationship between the angle measures of a triangle. (Lesson 4–2)
Recognize and apply properties of inequalities to the measures of the angles of a triangle.
Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.

4. Lesson 5-3: Inequalities in One Triangle
Triangle Inequality
The smallest side is across from the smallest angle.
The largest angle is across from the largest side. 54 37 89 B C A
BC = 3.2 cm
AB = 4.3 cm
AC = 5.3 cm
Lesson 5-3: Inequalities in One Triangle

5. Triangle Inequality – examples…
For the triangle, list the angles in order from least to greatest measure. C A B
4 cm
6 cm
5 cm
Lesson 5-3: Inequalities in One Triangle

6. Triangle Inequality – examples…
For the triangle, list the sides in order from shortest to longest measure.
8x-10
7x+6
7x+8 C A B
(7x + 8) ° + (7x + 6 ) ° + (8x – 10 ) ° = 180°
22 x + 4 = 180 °
22x = 176
X = 8
m<C = 7x + 8 = 64 °
m<A = 7x + 6 = 62 °
m<B = 8x – 10 = 54 °
54 °
62 °
64 °
Lesson 5-3: Inequalities in One Triangle

7. Exterior Angle Inequality
The measure of an exterior angle of a triangle is greater than the measure of wither of its corresponding remote interior angles.
Lesson 5-3: Inequalities in One Triangle

8. Lesson 5-3: Inequalities in One Triangle
Converse Theorem & Corollaries
Converse:
If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle.
Corollary 1:
The perpendicular segment from a point to a line is the shortest segment from the point to the line.
Corollary 2:
The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
Lesson 5-3: Inequalities in One Triangle

9. 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
b + c > a
Example:
Determine if it is possible to draw a triangle with side measures 12, 11, and 17.
> 17  Yes
> 12  Yes
> 11  Yes
Therefore a triangle can be drawn.
Lesson 5-3: Inequalities in One Triangle