Daily Warm-Up Quiz F H, J, and K are midpoints 1. HJ = __ 60 H 65 J FG = __ JK = __ m < HEK = _ E G Reason: _ K 3. Name all 100 parallel segments Geometry1

2 Relationships in Triangles Bisectors in Triangles

Geometry3 Perpendicular Bisectors and Angle Bisectors in Triangles At the end of this lesson, you should be able to use properties of perpendicular bisectors and angle bisectors.

Geometry4 Before we start, recall: A segment bisector is a line or segment which intersects a line segment at its midpoint AM Y

Geometry5 Perpendicular Bisector A perpendicular bisector is a line or segment which intersects a segment at its midpoint to form right angles. I M P

Geometry6 Perpendicular Bisectors ADB C Recall, a perpendicular bisector:  intersects a line segment at its  forms Now, if we draw CA and CB to complete the diagram, we can show that ∆ CAD ≡ ∆ CBD, and that CA ≡ CB. Another way to say this is that C is equidistant from points A and B. midpoint congruent right angles NOTE: If a point lies on the perpendicular bisector, it is equidistant from each of the endpoints.

Geometry7 The distance from a point to a line is the length of the perpendicular segment from the point to the line. P l H PH is the shortest distance!

Geometry8 PERPENDICULAR BISECTOR THEOREM If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. IF THEN

Geometry9 PERPENDICULAR BISECTOR THEOREM (Converse) If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. A IF THEN A

Geometry10 A B C D 5cm 6cm Example 1: Find CA and DB. Explain. CA = 5cm DB = 6cm REASON : C and D are on the perpendicular bisector of AB

Daily Warm-Up Quiz 1.Name and illustrate the three types of bisectors studied, to date, in this class. 2.Completion: a. If a point is on the perpendicular bisector of a segment, then _______. b. If a point is equidistant from the endpoints of a segment, then _______. c. The shortest distance from a point to a line is measured ____________________. Geometry11

Geometry12 Angle Bisector Recall, an angle bisector is a segment or ray which divides an angle into two congruent angles.

Geometry13 The distance from a point to a line is the length of the perpendicular segment from the point to the line. B X C A AX = XC If a point lies on the perpendicular bisector, the it is equidistant from each of the angle’s sides.

Geometry14 ANGLE BISECTOR THEOREM If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. IF THEN

Geometry15 ANGLE BISECTOR THEOREM (Converse) If a point is in the interior of an angle and is equidistant from the sides of the angle, then the point lies on the angle bisector. IF THEN

Geometry16 A B C D E F 2x + 245x Example #2 Find the value of x. Then find FA and FE. FA = FE Why? 5x = 2x x = 24 x = 8 FA = 5(8) = 40 A point on an angle bisector is equidistant from the sides of the angle.

Geometry17 Final Checks for Understanding If a point is on the perpendicular bisector of a segment, then it is ___________________________ ___________________________. If a point is equidistant from the endpoints of a segment, then _____ ___________________________ ___________________________. equidistant from the endpoints of a segment. it is on the perpendicular bisector of the segment.

Geometry18 Final Checks for Understanding Angle Bisector Theorem If a point lies on an angle bisector, Converse of the Angle Bisector Theorem If a point in the interior of an angle it then it is equidistant from each of the endpoints. is equidistant from the sides of the angle, then the point lies on the angle bisector.

Geometry19 Homework: