Chapter 7 Triangle Inequalities Segments, Angles and Inequalities Comparison Property •For any two real numbers, a and b, exactly one of the following statements is true. a<b a=b ab Theorem 7-1 • If point C is between points A and B, and A, C, and B are collinear, then AB AC and ABCB. Theorem 7-2 • If EP is between ED and EF, then mDEF mDEP and mDEF mPEF. Transitive Property •If a<b and b<c, then a<c. •If ab and bc, then ac. Addition and Subtraction Properties •If a<b, then a + c<b + c and a - c<b – c •If ab, then a + cb + c and a - cb – c Multiplication and Division Properties • If c0 and a<b, then ac<bc and a/c<b/c • If c 0 and a b, then ac bc and a/c b/c Exterior Angle Theorem Exterior Angle •An angle that forms a linear pair with one of the angles of a triangle Remote Interior Angles •The two angles in a triangle that do not form a linear pair with the exterior angle Exterior Angle Theorem •The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Exterior Angle Inequality Theorem •The measure of an exterior angle of a triangle is greater than the measure of either of its two remote interior angles. Theorem 7-5 •If a triangle has one right angle, then the other two angles must be acute. Inequalities Within a Triangle Theorem 7-6 •If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides are unequal in the same order. Theorem 7-7 •If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles are unequal in the same order. Theorem 7-8 •In a right triangle, the hypotenuse is the side with the greatest measure. Triangle Inequality Theorem Triangle Inequality Theorem •The sum of the measures of any two sides of a triangle is greater than the measure of the third side.