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Construct A 45 Degrees Angle

Related Topics: More Geometry Lessons

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

We can use the 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.

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


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


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.

Perpendicular Bisector

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

construct 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 degrees 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 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. 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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