1.5 Segment and Angle Bisectors

Goal1. To be able to use bisectors to find angle measures and segment lengths

DefinitionsThe Midpoint of a segment is the point that

divides or bisects the segments into two congruent segments.

A Segment Bisector is a segment, ray, line, or plane that intersects a segment at its midpoint.

If segment AM is congruent to segment MB, then M is the midpoint of segment AB.

If M is the midpoint of segment AB, then segment AM is congruent to segment MB.

Bisects- Divides into congruent parts.

Examples

Ruler Postulate (Again)Using a number line, we can find the

midpoint of a line segment. But how? Start by drawing a number line with points

C=-4 and D=6. (Just an Example)What is the distance between points C and D?Where is the midpoint? Why? The midpoint is the distance between two

points divided by 2. So the midpoint of the segment CD is 1.

The Midpoint FormulaIf we know the coordinates of the endpoints

of the segments, we can find the midpoint by using the midpoint formula.

If A(x₁, y₁) and E(x₂, y₂) are points in a coordinate plane, then the midpoint of ĀĒ has coordinates

Go to power point example 3 for examples

ExampleThe midpoint of segment RP is M(2,4). One

endpoint is R(-1,7). Find the coordinates of the other endpoint.

(-1 + x)/2 = 2 (7 + y)/2=4-1 + x = 4 7 + y = 8X = 5 y = 1So the other endpoinot is P(5,1)

Class WorkUse the midpoint formula to find the midpoint

of these coordinatesA (-1,7) and B (3,-3)A (0,0) and B (-8,6)

Angle BisectorAn Angle Bisector is a ray that divides an

angle into two adjacent angles that are congruent.

ExampleMeasure of angle ABD is (x + 40)°Measure of angle DBC is (3x – 20)°Solve for x(x + 40)° = (3x - 20)°X + 60 = 3x60 = 2xX = 30