This page shows how to construct (draw) a 30 degree angle with compass and straightedge or ruler. It works by first creating a rhombus and then a diagonal of that rhombus. Using the properties of a rhombus it can be shown that the angle created has a measure of 30 degrees. See the proof below for more on this.
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
This construction works by creating a rhombus. Its two diagonals form four 30-60-90 triangles.
The image below is the final drawing above with the red items added.
Argument | Reason | |
---|---|---|
1 | Line segments PT, TR, RS, PS, TS are congruent (5 red lines) | All created with the same compass width. |
2 | PTRS is a rhombus. | A rhombus is a quadrilateral with four congruent sides. |
3 | Line segment AS is half the length of TS, and angle PAS is a right angle | Diagonals of a rhombus bisect each other at right angles. See Rhombus definition. |
4 | Line segment AS is half the length of PS | PS is congruent to TS. See (1), (3) |
5 | Triangle ∆PAS is a 30-60-90 triangle. | ∆PAS is a right triangle with two sides in the ratio 1:2. (third side would be √3 by pythagoras). |
6 | Angle APS has a measure of 30°. | In any triangle, smallest angle is opposite shortest side. |