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

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

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.

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:

- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)

- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more

- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)

- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)

- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle

- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle

(C) 2011 Copyright Math Open Reference.

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All rights reserved