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Constructing a 30° angle
Geometry construction using a compass and straightedge
| After doing this |
Your work should look like this |
| 1. Draw a line segment which will become one side of the angle.
(Skip this step if you are given this line.) The exact length is not important. Label it PQ. P will be the angle's vertex. |
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| 2. Set the compass on P, and set its width to any convenient setting. |
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| 3. Draw an arc across PQ and up over above the point P. Label the point where it crosses PQ as point S. |
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| 4. Without changing the compass width, move the compass to the point S.
Draw a broad arc that crosses the first one and goes well to the right.
Label the point where the two arcs cross as point T. |
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| 5. Without changing the compass width, move the compass to the point T, and draw an arc across the previous arc,
creating point R. |
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| 6. Draw a line from P to R. |
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| Done. The angle QPR has a measure of 30° |
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Proof
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.
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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 PS, and angle PAS is a right angle |
Diagonals of a rhombus
bisect each other at right angles. See Rhombus definition. |
| 4 |
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). |
| 5 |
Angle APS has a measure of 30°. |
In any triangle, smallest angle is opposite shortest side. |
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two 30° angle exercises.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Constructions pages on this site
Lines
Angles
Triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
Non-Euclidean constructions
(C) 2009 Copyright John Page
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