# Construct A 45 Degrees Angle

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

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

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.

How to construct a 45° angle by bisecting a 90° 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.

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 2: Without changing the width of the compasses, put the sharp end at B and mark arcs above and below the line segment AB that will intersect with the arcs drawn in step 1.

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.

In summary the steps to construct a 45° angle are:

1. Construct perpendicular lines to obtain a 90° angle.
2. Bisect the 90° angle to obtain a 45° angle.

How to construct a 45-degree angle using compass and straightedge?
Step 1: Construct perpendicular lines (90 degrees).

Step 2: Bisect that angle to get a 45-degree angle.

How to construct a 30-degree angle without using a protractor?
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° angle.
The second method starts by constructing a rhombus with 60° and 120° angles, then joining the opposite vertices to leave the 30° angle.

How to construct a 75-degree angle with a compass?
Step 1: Construct 60-degree angles by constructing an equilateral triangle.
Step 2: Bisect a 60-degree angle to form a 30-degree angle.
Step 3: Bisect the 30-degree angle to form a 15-degree angle.
Step 4: Combine a 60-degree angle with a 15-degree angle to form a 75-degree angle.

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