Find lengths and measures of bisected lines and angles

Learning Objectives Define segment bisectors, midpoints, and angle bisectors. Set up and solve algebraic equations based on the properties of bisected segments. Set up and solve algebraic equations based on the properties of bisected angles. Calculate the total length of a segment or measure of an angle using the value of a variable found from its bisected parts. Apply the concept of bisection to problems involving congruent triangles. Differentiate between a bisector and a perpendicular bisector. Use bisection properties as justifications in geometric proofs. How does a carpenter find the exact center of a wooden beam to ensure perfect balance? 🪚 They use the same geometric principle you're about to master: bisection! This tutorial will teach you how to work with l...

TermDefinitionExample BisectTo divide a geometric figure, such as a line segment or an angle, into two congruent (equal) parts.If you cut a 10 cm rope exactly in the middle, you have bisected it into two 5 cm pieces. Segment BisectorA point, line, ray, or segment that intersects another segment at its midpoint, dividing it into two congruent segments.In segment \overline{AC}, if point B is the midpoint, then line \overleftrightarrow{DB} is a segment bisector of \overline{AC}. MidpointThe point on a line segment that divides it into two segments of equal length. It is the point of bisection.If M is the midpoint of \overline{PQ}, and PQ = 12, then PM = 6 and MQ = 6. Angle BisectorA ray that originates from the vertex of an angle and divides it into two adjacent, congruent angles.If ray \ove...

Definition of a Segment Bisector If a point, line, ray, or segment bisects \overline{AB} at point M, then \overline{AM} \cong \overline{MB}. This implies that AM = MB. Use this rule when you are given that a segment is bisected. Set the lengths of the two smaller, equal parts equal to each other to solve for unknown values. The total length AB = 2 * AM or AB = 2 * MB. Definition of an Angle Bisector If ray \overrightarrow{BX} bisects \angle ABC, then \angle ABX \cong \angle XBC. This implies that m\angle ABX = m\angle XBC. Use this rule when you are given that an angle is bisected. Set the measures of the two smaller, equal angles equal to each other to solve for unknown values. The total angle measure m\angle ABC = 2 * m\angle ABX or m\angle ABC = 2 * m\angle XBC.