Constructing Angles of 60º, 120º, 30º and 90ºwith the compass

In this section, we will consider the construction of some angles with special sizes. Constructing a 60º Angle We know that the angles in an equilateral triangle are all 60º in size.  This suggests that to construct a 60º angle we need to construct an equilateral triangle as described below. Step 1:  Draw the arm PQ. Step 2:  Place the point of the compass at P and draw an arc that passes through Q. Step 3:  Place the point of the compass at Q and draw an arc that passes through P.  Let this arc cut the arc drawn in Step 2 at R. Constructing a 30º Angle We know that: So, to construct an angle of 30º, first construct a 60º angle and then bisect it. Often, we apply the following steps. Step 1:  Draw the arm PQ. Step 2:  Place the point of the compass at P and draw an arc that passes through Q. Step 3:  Place the point of the compass at Q and draw an arc that cuts the arc drawn in Step 2 at R. Step 4:  With the point of the compass still at Q, draw an arc near T as shown. Step 5:  With the point of the compass at R, draw an arc to cut the arc drawn in Step 4 at T. Step 6:  Join T to P.  The angle QPT is 30º. Constructing a 120º Angle We know that: This means that 120º is the supplement of 60º.  Therefore, to construct a 120º angle, construct a 60º angle and then extend one of its arms as shown below. Constructing a 90º Angle We can construct a 90º angle either by bisecting a straight angle or using the following steps. Step 1:  Draw the arm PA. Step 2:  Place the point of the compass at P and draw an arc that cuts the arm at Q. Step 3:  Place the point of the compass at Q and draw an arc of radius PQ that cuts the arc drawn in Step 2 at R. Step 4:  With the point of the compass at R, draw an arc of radius PQ to cut the arc drawn in Step 2 at S. Step 5:  With the point of the compass still at R, draw another arc of radius PQ near T as shown. Step 6:  With the point of the compass at S, draw an arc of radius PQ to cut the arc drawn in step 5 at T. Step 7:  Join T to P. The angle APT is 90º. Please print this worksheet to practise the angles construction with the compass:

TANGENCIES

DEFINITION

Two figures are tangents when the have only one point in common, called point of tangency. The union of curves with straight lines or with other curves as a result of tangency is called LINKAGE. 

You can find designs based on tangencies in all areas of Art and design. 

The most common tangencies in geometric drawings are those between straight lines and circumferences or between circumferences. 



BASIC PROPERTIES OF TANGENCIES

1. First theorem

A straight line is tangent to a circumference when they have only one point (T) in common. The radius of the circumference at the point of tangency is at right angle to the tangent. 

2. Second theorem

A circumference is tangent to two straight lines from a common point if its centre is on the angle bisector of the angle formed by the straigt lines. 

3. Third theorem
Two circumferences are tangent if they have one point in common, lined up with the centres of the circumferences. The distance between the centres is the addition of the two radius.

RELATIVE POSITIONS

A line and a circumference can have differents positions:

EXTERIOR	TANGENT	SECANT
It doesn't intersect the circumference at any point because it is outside the circumference. The disrance  between the circumference and the line is bigger than the radius.	They have a point in common. It is the tangency point.  The distance between the line and the circumference is equal to the radius	It intersects the circle at 2 points, so they have two points in common.  The distance between the circumference and the line is smaller than the radius.

Two circumferences can also have different positions:

EXTERIOR They do not overlap and the distance between their centers is greater than the sum of their radius. d= d is the distance between centres r1+r2 > d
TANGENT EXTERIOR They have just one common point, the tangency point.  The distance between the centers is the addition of the two radius r1+r2 = d
SECANT They intersect at two different points and the distance between their centers is less than the sum of their radius. r2+r1< d
TANGENT INTERIOR They have a common point and all other points of one of them are interior to the other, exclusively. r1-r2=d
INTERIOR They have no common points and the distance between their centers is greater than 0 and less than the difference of their radius.
CONCENTRIC  They have the same center and a different radius The distance between the centers is 0 d=0

You can practise and find more information in this link:

http://iessierradesantabarbara.juntaextremadura.net/web/cesar/41_relative_positions.html