Construct An Angle Bisector

An angle bisector is a straight line that divides the angle into two equal parts.

Example:
Construct an angle bisector for the following angle:

Solution:
Step 1: Put the sharp end of your compasses at point B and make one arc on the line BC (point S) and another arc on line AB (point T).

Step 2: Put the sharp end of the compasses at S and make an arc within the lines AB and BC. Do the same at T and make sure that the second arc intersects the first arc.

Step 3: Draw a line from point B to the points of intersection of the 2 arcs. This line bisects ∠ABC.

The steps to construct an angle bisector can be summarized as follows:

1. From the vertex, draw an arc across both rays of the angle.
2. From each arc intersection draw another pair of arcs that intersect each other.
3. Draw a line from the vertex to the intersection point to form the angle bisector.

How to draw an angle bisector using a compass and a straight edge?

1. Place compass point on the vertex, and draw an arc across each ray.
2. Place the compass on each arc intersection and draw a further pair of arcs which intersect each other.
3. Use a straight edge to connect the intersection point to the vertex.

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How to bisect a given angle using only a compass and straightedge?

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Use Angle Bisector To Construct Angles

We can use the angle bisector to construct some other angles from existing angles.

For examples:
A 30˚ angle can be obtained by bisecting a 60˚ angle.
A 15˚ angle can be obtained by bisecting a 30˚ angle.
A 45˚ angle can be obtained by bisecting a 90˚ angle.
A 22.5˚ angle can be obtained by bisecting a 45˚ angle.

Example:
The figure shows a point A on a straight line. Construct an angle of 45˚ at point A.

Solution:
Construct a 90˚ angle, and then construct an angle bisector to obtain a 45˚ angle.

How to construct 30, 45, 60, 90, and 120 degree angles with a compass by constructing angle bisectors?

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How to construct a 75° angle by constructing a 60° angle and a 15° angle?

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