A ray that divides an angle into two equal angles is called an angle bisector. It can be constructed accurately using a compass.
What Is an Angle Bisector?
The ray OC that bisects ∠AOB is called an angle bisector.
∠AOC ＝ ∠COB ＝ (1/2) × ∠AOB
Construction Procedure
① Draw an arc with vertex O as the center and an appropriate radius, and let its intersections with rays OA and OB be P and Q.
② With P as the center, draw an arc with a radius greater than half of PQ.
③ With Q as the center, draw an arc with the same radius as in ②, and let its intersection with the arc in ② be R.
④ Draw ray OR → This is the angle bisector.
State the procedure for constructing the angle bisector of angle AOB. Also explain why this construction divides the angle into two equal angles.
Solution:
The procedure is ①–④ above.
Reason: OP = OQ (on the same arc), PR = QR (same radius), and OR is common. Therefore, △OPR ≡ △OQR (SSS congruence).
Thus, ∠POR = ∠QOR, so OR is the angle bisector of ∠AOB.
Properties of Angle Bisectors
A point on an angle bisector is at an equal distance from the two sides of the angle.
Conversely, a point that is equidistant from the two sides of an angle lies on its angle bisector.
What figure represents all the points that are equidistant from two lines ℓ and m?
Solution:
The angle bisectors of the angles formed by lines ℓ and m.
(Strictly speaking, when the two lines intersect, they form four angles, so there are two angle bisectors, including the bisectors of vertical angles.)
- Angle bisector: a ray that bisects an angle
- Construction: Draw an arc centered at the vertex to intersect both sides, then connect the vertex to the intersection of arcs with the same radius drawn from the two intersection points.
- A point on an angle bisector is equidistant from the two sides.
- A point equidistant from two lines ⇔ a point on an angle bisector
Practice Problems
- Briefly explain the procedure for constructing an angle bisector.
- Construct the angle bisector of ∠A from vertex A of triangle ABC.
- Two walls ℓ and m intersect. You want to walk along a path that is equally distant from both walls. What line should you follow?
Answers and Explanations
- Answer: Draw an arc centered at the vertex and let its intersections with the two sides be P and Q. Draw arcs with the same radius from P and Q so that they intersect at R. OR is the angle bisector.
Explanation: The angle is bisected because congruent triangles are formed. - Answer: Draw an arc centered at A and mark its intersections with sides AB and AC. From those two points, draw arcs with the same radius to find their intersection, then connect it to A.
Explanation: The construction procedure is the same. - Answer: Follow the angle bisector.
Explanation: The set of points equidistant from two lines is an angle bisector.