constructing geometric angles (math)

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Introduction to constructions

Constructions: The drawing of various shapes using only a compass and straightedge or ruler. No

measurement of lengths or angles is allowed.The word construction in geometry has a very specific meaning: the drawing of geometric items

such as lines and circles using only a compass and straightedge or ruler. Very importantly, you

are not allowed to measure angles with aprotractor, or measure lengths with a ruler.Compass

The compass is a drawing instrument used for drawing circles andarcs. It has twolegs, one with a point and the other with a pencil or lead. You can adjust the distance between

the point and the pencil and that setting will remain until you change it. (This kind of compass

has nothing to do with the kind used find the North direction when you are lost).

Straightedge

A straightedge is simply a guide for the pencil when drawing straight lines. In mostcases you will use a ruler for this, since it is the most likely to be available, but you must not use

the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see

them.Why we learn about constructions

The ancient Greek mathematicianEuclidis the acknowledged inventor of geometry.

He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate

geometry reference. In that work, he uses these construction techniques extensively, and so theyhave become a part of the geometry field of study. They also provide a greater insight into

geometric concepts and give us tools to draw things when direct measurement is not appropriate.

Why did Euclid do it this way?

Why didn't Euclid just measure things with a ruler and calculate lengths? For example, one of thebasic constructions isbisecting a line(dividing it into two equal parts). Why not just measure itwith a ruler and divide by two?

The answer is surprising. The Greeks could not do arithmetic. They had only whole numbers, no

zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5,

because 2.5 is not a whole number - the only kind they had. Also, their numbers did not use apositional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In

short, they could perform very little useful arithmetic.

So, faced with the problem of finding the midpoint of a line, they could not do the obvious -

measure it and divide by two. They had to have other ways, and this lead to the constructionsusing compass and straightedge or ruler. It is also why the straightedge has no markings. It is

definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and
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the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.

Constructing Angles of 60, 120, 30 and 90In this section, we will consider the construction of some angles with special sizes.

Constructing a 60 Angle

We know that the angles in anequilateral triangleare all 60 in size. This suggests that to

construct a 60 angle we need to construct an equilateral triangle as described below.

Step 1: Draw the armPQ.

Step 2: Place the point of thecompassatPand draw anarcthat passes through Q.

Step 3: Place the point of the compass at Q and draw an arc that passes throughP. Let this arc

cut the arc drawn in Step 2 atR.


We know that:

So, to construct an angle of 30, first construct a 60 angle and thenbisectit. Often, we apply thefollowing steps.

Step 1: Draw the armPQ.

Step 2: Place the point of thecompassatPand draw anarcthat 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 nearTas shown.
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Step 5: With the point of the compass atR, draw an arc to cut the arc drawn in Step 4 at T.

Step 6: Join TtoP. The angle QPTis 30.


We know that:

This means that 120 is the supplement of 60. Therefore, to construct a 120 angle, construct a

60 angle and then extend one of its arms as shown below.


We can construct a 90 angle either by bisecting a straight angle or using the following steps.

Step 1: Draw the armPA.

Step 2: Place the point of thecompassatPand draw anarcthat cuts the arm at Q.

Step 3: Place the point of the compass at Q and draw an arc ofradiusPQ that cuts the arc drawnin Step 2 atR.

Step 4: With the point of the compass atR, draw an arc of radiusPQ to cut the arc drawn in

Step 2 at S.
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Step 5: With the point of the compass still atR, draw another arc of radiusPQ nearTas shown.

Step 6: With the point of the compass at S, draw an arc of radiusPQ to cut the arc drawn in step

5 at T.

Step 7: Join TtoP. The angleAPTis 90.

Example 12

Solution:


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constructing geometric angles (math)

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