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In these lessons, we will learn how to construct a 45 degrees angle by bisecting a 90 degrees angle.

We can use a similar angle bisector method to construct some other angles from existing angles.

**Example*** : *

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.

Related Topics: More Geometry Lessons

*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.

**Step 1** : Stretch your compasses until it is more then half the length of *AB*. Put the sharp end at *A* and mark an arc above and another arc below line segment *AB*.

** Step 3** : Join the two points where the arcs intersect with a straight line. This line is the perpendicular bisector of *AB*. *P *is the midpoint of* AB*.

Step 4: Bisect the 90 degree angle to form a 45 degree angle.

The following video shows how to construct a 45 degree angle using compass and straightedge. First, construct perpendicular lines (90 degrees) and then bisect that angle to get a 45 degrees angle.

Constructing a 30 degree angle

This video shows two ways to construct a 30 degree angle. The first way starts by constructing part of an equilateral triangle, then bisecting the 60 degree angle. The second method starts by constructing a rhombus with 60 and 120 degree angles, then joining the opposite vertices to leave the 30 degree angle.