Triangles and bisectors

Learning Objectives Define angle bisectors and perpendicular bisectors and identify them in diagrams. State and apply the Angle Bisector Theorem and its converse to solve for unknown lengths. State and apply the Perpendicular Bisector Theorem and its converse to find missing segment lengths. Differentiate between the properties of angle bisectors and perpendicular bisectors in problem-solving contexts. Identify the incenter and circumcenter as points of concurrency for a triangle's bisectors. Construct logical arguments and proofs using the properties of bisectors. Imagine you need to place a sprinkler in a triangular garden so it's the exact same distance from two of the garden's edges. How would you find that perfect spot? 🤔 This tutorial explores two fund...

TermDefinitionExample Angle BisectorA ray that divides an angle into two adjacent angles that are congruent (have the same measure).If ray BD bisects ∠ABC, and m∠ABC = 70°, then m∠ABD = m∠DBC = 35°. Perpendicular BisectorA line, ray, or segment that intersects a given segment at a 90° angle and divides it into two congruent segments (passes through its midpoint).If line 'l' is the perpendicular bisector of segment XY, it passes through the midpoint M of XY, and 'l' ⊥ XY. EquidistantBeing at an equal distance from two or more objects. The distance from a point to a line is the length of the perpendicular segment from the point to the line.If point P is 5 cm from line A and 5 cm from line B, it is equidistant from both lines. Concurrent LinesThree or more lines, rays, or...

Angle Bisector Theorem If a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle. Use this theorem when you know a point is on an angle bisector and you need to relate the perpendicular distances from that point to the sides of the angle. These distances will be equal. Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the two sides of the angle, then it lies on the bisector of the angle. Use this when you know the perpendicular distances from a point to the sides of an angle are equal. You can conclude that the point lies on the angle bisector. Perpendicular Bisector Theorem If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of th...