In these lessons, we will learn

- how to construct an angle bisector of a given angle.
- how to use an angle bisector to construct some angles for example, 90 degrees, 45 degrees, 60 degrees, 30 degrees, 120 degrees, 135 degrees, 15 degrees.

**Related Pages**

Types Of Angles

Loci

More Geometry Lessons

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:

- From the vertex, draw an arc across both rays of the angle.
- From each arc intersection draw another pair of arcs that intersect each other.
- 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?**

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

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

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?**

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

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