43
Property of an angle bisector
Textbook: pp. 110–111
GoalKnow and use the property of an angle bisector.
New words
bisector · bissektrisadistance to the sides · tomonlargacha masofaperpendicular · perpendikulyarequidistant · teng uzoqlikda
Explanation
The distance from a point to a line is the length of the perpendicular dropped from the point to the line. Theorem: every point on the bisector of an angle is equidistant from the sides of the angle. Proof: take a point D on the bisector OC of the angle and drop perpendiculars DA and DB to the sides. The right triangles △OAD and △OBD have ∠AOD = ∠BOD (bisector) and a common hypotenuse OD. By hypotenuse-angle △OAD = △OBD, so DA = DB. This property is used to find distances and equal segments.
Worked examples
If D is on the bisector of ∠O and its distance to one side is 4 cm, then its distance to the other side is also 4 cm.
K is on the bisector of ∠EOF = 40°, and KE ⊥ OE; then ∠EOK = 20° and ∠OKE = 90° − 20° = 70°.
Class activity
“Equal distance”: fold a paper angle to find the bisector, mark a point on it and measure its distance to both sides with a set square.
Practice
1
K lies on an angle bisector and its distance to one side is 9 cm. Distance to the other side (cm)?
9
2
OK bisects ∠EOF = 50°, KE ⊥ OE. Find ∠OKE (degrees).
65
3
A point on an angle bisector is at distances 5 cm and x + 2 cm from the sides. Find x.
3
4
Why does the proof use the hypotenuse-angle criterion?
Both triangles are right, the hypotenuse is common and the acute angles are equal.