# Math Help - Constructing an angle using only a ruler and compasses...?!?!

1. ## Constructing an angle using only a ruler and compasses...?!?!

Hi I needed to construct a 60 degree angle with only a ruler and compasses. I understand how to construct an angle of 90 degrees, and how to do an angle bisector but cannot understand how to do this.

Thankyou

2. Hello,

Originally Posted by Turple
Hi I needed to construct a 60 degree angle with only a ruler and compasses. I understand how to construct an angle of 90 degrees, and how to do an angle bisector but cannot understand how to do this.

Thankyou
In an equilateral triangle, angles are 60 degrees.

So you can try to do this way :

- let A and B be two points of the plane
- get the length AB with your compass
- draw the circle of center A, and length AB. This will make all the points that are at equal distance to A than B.
- draw the same circle, but of center B. This will make all the points that are at equal distance to B than A.

There will be two intersections. See what the properties of the sides of the formed triangle are

3. Hello, Turple!

I needed to construct a 60° angle with only a ruler and compass.
We need to construct an equilateral triangle.

Draw a horizontal line; label the left end $A.$

Place the point on $A$ and swing an arc cutting the line at $B.$

Keep the point on $A$ and swing an arc above the line.
Place the point on $B$ and swing an arc cutting the first arc at $C.$

Triangle ABC is equilateral.

Therefore: . $\angle A \,=\,\angle B\,=\,\angle C \,=\,60^o$

Edit: Too fast for me, Moo!
.

4. Originally Posted by Turple
Hi I needed to construct a 60 degree angle with only a ruler and compasses. I understand how to construct an angle of 90 degrees, and how to do an angle bisector but cannot understand how to do this.

Thankyou
Here comes the result:

5. Originally Posted by Turple
Hi I needed to construct a 60 degree angle with only a ruler and compasses. I understand how to construct an angle of 90 degrees, and how to do an angle bisector but cannot understand how to do this.

Thankyou
1. Draw a circle around midpoint M
2. Draw a diameter. I labeled the endpoints of the diameter by A and B.
3. Construct a perpendicular bisector of MB which cuts the circle line in 2 points.
4. Connect one of the intersection points with M. I've labeled the point in question by C.
5. The angle $\angle(BMC)=60^\circ$

By the way: This method is very convenient to construct a regular hexagon. I've marked the vertices of the hexagon in red.