GEOMETRY CONSTRUCTION

Geometry construction of the Difference of two angles With the use of a compass and straightedge

This construction illustrates the way to create an angle which is the difference between two given angles.

This is very similar to Sum of angles, apart from the fact that the second angle is drawn inside the first, efficiently subtracting the angles.

The two types of construction (add , subtract) can be combined in any way you like. For instance, for the four angles P,Q,R,S you could construct P+Q-(R+S).

What to do if the second angle is larger than the first

If the second angle is larger than the first then the points M (and so S) will be below the line PQ. The difference angle will still be ∠SPQ as before.

If for instance you ‘subtracted’ 90° angle from a 30°angle, arithmetically this would be negative: But in geometry, you cannot have negative angles, therefore the difference between the two angles is 60°.

Thus the precise definition of what this construction does is, for two angles A and B, it finds the absolute value of a–b. The two vertical bars mean “absolute value” which is always positive, irrespective of the way a–b comes out. This is why the construction is titled “Difference of two angles” – it is not exactly the same as arithmetic subtraction.

* Negative angles do occur in trigonometry nevertheless.

Proof

	The case	Explanation
1	m∠RPS + m∠SPQ = m∠RPQ	Adjacent angles
2	m∠BAC = m∠RPS	Copied with the use of the procedure in Copying an angle. See that tutorial for proof.
3	m∠SPQ = | m∠RPQ – m∠BAC |	Substitute (2) in (1) and transpose.

– Q.E.D

Difference of two angles

This is the step-by-step, instruction shows you how to to this in the absence of a computer

After doing this	Your work ought to look like this
Begin with two given angles ∠BAC and ∠RPQ
1. Set the compasses on A and any suitable width.
2. Make an arc across the angle producing the points J and K.
3. Move the compasses to point P and make an arc to the right of RP, crossing RP at the new point L.
4. Set the compasses’ width to the distance JK
5. Without changing the compasses, set them on point L and make an arc across the previous arc, creating point M
6. Draw a line from P through M and on to new point S.
The angle ∠SPQ is the difference between the angles ∠RPQ and ∠BAC.

Geometry construction of Sum of angles with a compass and straightedge

This construction takes one given angle and copies it adjacent to another, producing a larger angle whose measure is the sum of the two.

By carrying out this construction more than once, any number of angles can be summed. This can be done by adding each successive angle to the left or the right of the accumulating angle.

In a similar way, angles can be ‘subtracted’.

Printable step-by-step instructions

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.

Proof

	Argument	Reason
1	Angle ∠BAC is congruent to angle ∠SPR	Copied using the procedure described in Copying an angle. See that page for proof.
2	m∠SPQ = m∠SPR + m∠RPQ	Adjacent angles.
3	m∠SPQ = m∠BAC + m∠RPQ	From (1), (2)

– Q.E.D

Adding angles with compass and straightedge

1. Add angle B to angle A

2. Construct an angle which is the sum of the three interior angles of the triangle.
(First copy one angle, then add the other two)

Isosceles triangle given the base and one side

How to construct (draw) an isosceles triangle with compass and straightedge or ruler, given the length of the base and one side. First we copy the base segment. Then we make use of the fact that both sides of an isosceles triangle have the same length to mark the apex (topmost point) of the triangle the same distance from each end of the base.

Printable step-by-step instructions

This printable step-by-step instruction can be used for making handouts or when a computer is not accessible.

Proof

The image below is the last drawing you would obtain after the construction.

	The case	Explanation
1	PR = CD	By construction. PR is a copy of the segment CD. See Copying a line segment for method and proof.
2	QP = QR = AB	QP, QR both drawn with same compass width AB..
3	QPR is an isosceles triangle, with base CD and side AB.	An isosceles triangle has two sides the same length..

– Q.E.D

Construct an isosceles triangle of given base and side length with compass and straightedge
Steps

1	Construct an isosceles triangle whose base is equal in length (congruent to) the segment FG, and whose legs are congruent to AB.

2	(a)	Construct two isosceles triangles with base CD and legs AB, which share a common base,
	(b)	The leg and base lengths are in the ratio of 1 : √2. What are the measures of all 6 interior angles in the finished construction?
	(c)	Confirm this by measuring them with a protractor, and explain this result.

How to draw an isosceles triangle given the base and altitude with compass and straightedge or ruler

The base is the unequal side of the triangle and the altitude is the perpendicular height from the base to the apex. It works by first copying the base segment, then constructing its perpendicular bisector. The apex is then marked up from the base.

Printable step-by-step instructions

This printable step by step instructions can guide you in the absence of a computer.

Proof

The image below is the final drawing you would obtain after the construction

	The case	Explanation
1	PR = CD	By construction. PR is a copy of the segment CD. See Copying a line segment for method and proof.
2	QS is the perpendicular bisector of PR	By construction. See Constructing the perpendicular bisector of a line segment..
3	Triangles QPS and QRS are congruent.	SAS. See Test for congruence side-angle-side. • QS common to both • ∠QSR = ∠QSP = 90° from (2). • SP = SR from (2).
4	QP = QR	CPCTC – equivalent Parts of Congruent Triangles are Congruent
5	QS = AB	By construction.
6	QPR is an isosceles triangle.	From (4). An isosceles triangle has two sides the same length.
7	QPR is an isosceles triangle with base CD and altitude AB.	From (1) (5) (6)

– Q.E.D

Construction of an isosceles triangle given base and altitude

This is the step-by-step, printable instruction to guide you.

After you have done this	Your work ought to look like this
We begin with two line segments AB and CD that define the altitude and the base length of the triangle.
1. Draw a point P that will become one end of the base of the triangle.
2. Place the point of the compasses on the point C and adjust the compasses’ width to the desired length CD of the base of the finished triangle
3. With the compasses’ point on P, draw an arc.
4. Pick a point R anywhere on the arc. This will become the other end of the base of the triangle.
5. Draw the base line PR.
In the next three steps, we form the perpendicular bisector of the base
6. With the compasses’ width set roughly to the base length (exact width is not essential, draw an arc on each side of the base line from points P and R.
7. Draw a line through the two arc intersections. This is the perpendicular bisector of the base, dividing it into two equal parts.
8. Set the compasses’ width to the distance from A to B. This is the preferred altitude of the triangle.
9. Place the point of the compasses on the midpoint of the base line, and draw an arc across the perpendicular drawn earlier. This is the third vertex of the triangle.
10. Draw the two side lines PQ and RQ
11. At this stage you are done. The triangle PQR is an isosceles triangle.

Constructing an isosceles triangle given a side and apex angle

This construction creates an isosceles triangle given the length of the legs and the apex angle (the one at the top). The construction is done in two main steps:

1. Copy the angle, ensuring each leg of the angle is longer than the preferred side length of the triangle. This is precisely the same as the construction Copying an angle with compass and straightedge. 2. Mark off the length of the two legs of the triangle and draw the base line.

Printable step-by-step instructions

Proof

	The case	Explanation
1	Angles BAC and QPR are congruent	By construction. From Copying an angle with compass and straightedge.
2	Line segments FG, PQ and PR are congruent	All drawn with the same compass width.
3	PQR is an isosceles triangle with the given apex angle and leg length.	An isosceles triangle is a triangle with two congruent legs.

– Q.E.D

This is the step-by-step guide to assist you in the absence of a computer

After doing this	Your work ought to look like this
Start with a angle BAC and line segment FG. The length FG will be the side lengths of the triangle and BAC will be the angle at the top (apex) of the triangle.
In Steps 1-8 we copy the apex angle. This is exactly the same as Copy an angle with compass and straightedge.
1. Make a point P that will be the apex of the new triangle.
2. From P, draw a line. This will become one leg of the new triangle, therefore make it longer that FG. • This line can go off in any direction. • It does not have to be parallel to anything else.
3. Place the compasses on point A, set to any suitable width.
4. Draw an arc across both sides of the angle, producing the points J and K as shown.
5. Without altering the compasses’ width, place the compasses’ point on P and draw a similar arc there, producing point M as shown.
6. Set the compasses on K and alter its width to point J.
7. Without altering the compasses’ width, move the compasses to M and draw an arc across the first one, producing point L where they cross.
8. Draw a line from P through L and onwards further. This will become the second side (leg) of the triangle in order to make it longer than FG.
In the rest of the construction we set the lengths of the two legs and draw the base line.
9. Put the compasses on point F and set its width to point G.
10. Without altering the width, place the compasses on P and make an arc across both lines, creating points Q and R.
11. Draw a line from Q to R.
DONE. The triangle PQR is an isosceles triangle with each leg equal to the given FG in length, and the apex angle is equal in measure to the given angle CAB.