Inequalities with addition and subtraction: set 2

Learning Objectives Apply the Triangle Inequality Theorem to determine if three segments can form a triangle. Calculate the range of possible lengths for the third side of a triangle given the lengths of the other two sides. Apply the Exterior Angle Inequality Theorem to compare angle measures and side lengths in a triangle. Use the Segment Addition Postulate in conjunction with inequality properties to prove relationships between segment lengths. Compare the lengths of two sides of a triangle based on the measures of their opposite angles. Construct simple geometric proofs involving segment inequalities. Can you build a triangular garden with fence panels of 3 meters, 4 meters, and 8 meters? 🤔 Let's find out why geometry says no! This tutorial moves beyond basic alge...

TermDefinitionExample Line SegmentA part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. Its length is a positive real number.In a triangle ABC, the sides AB, BC, and AC are all line segments. Segment Addition PostulateIf three points A, B, and C are collinear and B is between A and C, then the length of segment AB plus the length of segment BC is equal to the length of segment AC.If point P is on segment Q_R, then QP + PR = QR. Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle must be greater than the length of the third side.For a triangle with side lengths a, b, and c, it must be true that a + b > c, a + c > b, and b + c > a. Exterior Angle of a PolygonAn angle formed by a side of...

Triangle Inequality Theorem For a triangle with side lengths a, b, and c: \\ a + b > c \\ a + c > b \\ b + c > a Use this to verify if three given lengths can form a triangle or to find the possible range of lengths for a missing side. Range of the Third Side If a triangle has sides of length a and b, the length of the third side, c, must be in the range: \\ |a - b| < c < a + b This is a direct consequence of the Triangle Inequality Theorem. It provides a shortcut to find the lower and upper bounds for the third side of a triangle. Comparison Property of Inequality If a = b + c and c > 0, then a > b. This algebraic property is frequently used in geometric proofs. It allows you to establish an inequality when one quantity is expressed as the sum o...