Constructing  75°  105°  120°  135°  150° angles and more

On other pages there are instructions for constructing angles of 30°, 45°, 60° and 90°. By combining them you can construct other angles.

Adding angles

Angles can be effectively 'added' by constructing them so they share a side. This is shown in Constructing the sum of angles.

As an example, by first constructing a 30° angle and then a 45° angle, you will get a 75° angle. The table below shows some angles that can be obtained by summing simpler ones in various ways

To make Combine angles
75° 30° + 45°
105° 45° + 60°
120° 30° + 90°  or   60° + 60°
135° 90° + 45°
150° 60° + 90°

Furthermore, by combining three angles many more can be constructed.

You can subtract them too

By constructing an angle "inside" another you can effectively subtract them. So if you started with a 70° angle and constructed a 45° angle inside it sharing a side, the result would be a 25° angle. This is shown in the construction Constructing the difference between two angles

Bisecting an angle 'halves' it

By bisecting an angle you get two angles of half the measure of the first. This gives you some more angles to combine as described above. For example constructing a 30° angle and then bisecting it you get two 15° angles. Bisection is shown in Bisecting an Angle.

Complementary and supplementary angles

By constructing the supplementary angle of a given angle, you get another one to combine as above. For example a 60° angle can be used to create a 120° angle by constructing its supplementary angle. This is shown in Constructing a supplementary angle.

Similarly, you can find the complementary angle. For example the complementary angle for 20° is 70°. Finding the complementary angle is shown in Constructing a complementary angle.

The basic constructions are described on these pages:

Other constructions pages on this site




Right triangles

Triangle Centers

Circles, Arcs and Ellipses


Non-Euclidean constructions