5-2 Inequalities and Triangles Students will recognize and apply properties of inequalities to the measure of angles and sides of a triangle. S. Calahan 2008

Definition of Inequality • For any real number a and b , a > b if and only if there is a positive number c such that a = b + c.

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 its corresponding interior angles. m<4 > m<1 m<4 > m<2 B 2 1 3 A 4 C

Exterior Angle Inequality Theorem • Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. All angles that are less than m<4. B m>4 > m<5 m>4 > m>6 m>4 > m>2 m>4 > m>1 +m>5 2 7 6 1 5 3 A 4 C

Angle-Side Relationship • 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. order from least to greatest according to the <s B 10 8 <B, <C, <A A 6 C

Side-Angle Relationship • 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. arrange least to greatest according to the sides B 30 AC, AB, BC 60 C A