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<ba = ba  b

Theorem 7-1 If point C is between points A and B, and A, C, and B are collinear, then AB  AC and AB  CB.

Theorem 7-2 If EP is between ED and EF, then m  DEF  m  DEP and m  DEF  m  PEF.

Transitive Property If a<b and b<c, then a<c. If a  b and b  c, then a  c.

Addition and Subtraction Properties If a<b, then a + c<b + c and a - c<b – c If a  b, then a + c  b + c and a - c  b – c

Multiplication and Division Properties If c  0 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

The sum of the measures of any two sides of a triangle is greater than the measure of the third side.