Angle bisectors

Learning Objectives Define an angle bisector and identify its properties. Construct an angle bisector using a compass and straightedge. State, prove, and apply the Angle Bisector Theorem and its converse. Solve algebraic problems involving angle measures and the Angle Bisector Theorem. Prove that any point on an angle bisector is equidistant from the sides of the angle. Identify the incenter of a triangle as the point of concurrency of the angle bisectors. Use the properties of angle bisectors in geometric proofs. How could you perfectly aim a billiard ball to hit a cushion and then rebound to hit another ball at a specific spot? 🤔 The secret lies in splitting the angle of its path perfectly in half! This tutorial explores angle bisectors, the rays that cut angles into t...

TermDefinitionExample Angle BisectorA ray, line, or segment that divides an angle into two congruent adjacent angles.If ray BD bisects ∠ABC, then m∠ABD = m∠DBC. If m∠ABC = 70°, then m∠ABD = 35° and m∠DBC = 35°. Congruent AnglesAngles that have the exact same measure in degrees or radians.If m∠P = 45° and m∠Q = 45°, then ∠P ≅ ∠Q. EquidistantBeing at an equal distance from two or more objects. The distance from a point to a line is measured along the perpendicular segment from the point to the line.A point P on the bisector of ∠XYZ is equidistant from rays YX and YZ. This means the perpendicular distance from P to ray YX is equal to the perpendicular distance from P to ray YZ. IncenterThe point of concurrency (intersection) of the three angle bisectors of a triangle. The incenter is always...

Angle Bisector Theorem If a ray bisects an angle of a triangle, it divides the opposite side into two segments that are proportional to the other two sides of the triangle. For ΔABC with angle bisector AD, the rule is: \frac{AB}{AC} = \frac{BD}{CD} Use this theorem when you have a triangle with a known angle bisector and you need to find a missing side length. It creates a proportion relating the four side/segment lengths. Converse of the Angle Bisector Theorem If a point D on side BC of ΔABC divides BC such that \frac{AB}{AC} = \frac{BD}{CD}, then the ray AD bisects ∠BAC. Use this to prove that a ray is an angle bisector if you know the side lengths of the triangle and the segments it creates on the opposite side. Equidistance Property of Angle Bisectors A point lies...