Bisect an angle
Cut an angle into two equal parts with arcs.
Fold or construct?
Fold a paper angle so that one arm lies exactly on the other. The crease cuts the angle into two equal parts. On paper we cannot fold, but we can construct the cut.
What could we use, instead of folding, to make the two halves equal?
What this lesson covers
The idea
To bisect an angle at O, mark A and B on its arms with OA = OB, then cut equal-radius arcs from A and B meeting at C; by SSS the two triangles are congruent, so OC bisects the angle.
Fold or construct?
Fold a paper angle so that one arm lies exactly on the other. The crease cuts the angle into two equal parts. On paper we cannot fold, but we can construct the cut.
What could we use, instead of folding, to make the two halves equal?
- Equal distances, measured with a compass
- A protractor reading, then guessing
- Drawing the line by eye
Build the bisector
Follow the steps. The angle is XOY. First mark A and B on the two arms at the same distance from O.
Two triangles that fit
Draw OA, OB, AC and BC. In the triangles OAC and OBC: OA = OB (one arc from O), AC = BC (equal arcs from A and B), and OC is shared. By SSS they are congruent, so the angles at O are equal.
To bisect an angle at O, mark A and B on its arms with OA = OB. Cut equal-radius arcs from A and B meeting at C. By SSS the two triangles are congruent, so OC bisects the angle.
- Triangles OAC and OBC | Why
- OA = OB | radii of the arc from O
- AC = BC | equal arcs from A and B
- OC = OC | common side
Notes
To bisect an angle at O, mark A and B on its arms with OA = OB. Cut equal-radius arcs from A and B meeting at C. By SSS the two triangles are congruent, so OC bisects the angle.
Check yourself
Why must the arcs from A and from B have the same radius?
∠XOY = 80°. The bisector OC cuts it into two equal angles. How many degrees is each angle?
Answer: 40
The two parts are equal and together make 80°, so each is 80° ÷ 2 = 40°.
Ravi bisects an angle of 64°. Then he bisects one of the two halves again. How many degrees is the smallest part?
Answer: 16
64° ÷ 2 = 32°, and 32° ÷ 2 = 16°.
Meena draws the arc from A with radius 2 cm, but from B with radius 3 cm. They meet at C. Does OC bisect the angle?
- So that AC = BC. Then the triangles OAC and OBC have three pairs of equal sides: SSS. — correct. Yes! Equal arcs give AC = BC.
- So that the arcs look neat.. It is not for looks. With different radii, AC and BC differ and the triangles are not congruent.
- So that OA = OB.. OA = OB comes from the first arc, drawn from O.
- Yes. Any point C where two arcs meet is on the bisector.. Only when the arcs have the same radius, so that AC = BC.
- Yes, but only if the angle is less than 90°.. The size of the angle does not matter. The radii do.
- No. AC and BC are different, so the triangles OAC and OBC are not congruent. — correct. Yes! The arcs from A and B must have the same radius.