CONSTRUCTION OF ANGLE 75°, 90°, 105°, 120°, 135° AND 150°

Construction of angles 75, 90°, 105, 120,135, and 150

Construction of angle 90°
Geometry construction with the use of a compass and straightedge

In this topic, we show how to construct (draw) a 90 degree angle with compass and straightedge or ruler. There are many ways to do this, but in this construction we make use of a property of Thales Theorem. We produce a circle where the vertex of the desired right angle is a point on a circle. Thales Theorem says that any diameter of a circle subtends a right angle to any point on the circle.

Printable step-by-step instructions that is meant to guide you when studying without the use of a computer

The above animation is obtainable as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Explanation of method

This is actually the same construction as Constructing a perpendicular at the endpoint of a ray. Another way to do it is to

• construct a perpendicular at a point on a line or
• construct a perpendicular to a line from an external point

Proof

This construction functions through the use of the Thales theorem. It produces a circle where the apex of the required right angle is a point on a circle.

	Argument	Reason
1	The line segment AB is a diameter of the circle center D	AB is a straight line through the center.
2	Angle ACB has a measure of 90°.	The diameter of a circle always subtends an angle of 90° to any point (C) on the circle.

– Q.E.D

Construction of angle 90° or (right) angle

This is the step-by-step, printable instruction that can both guide you when on the computer and in the absence of a computer.

After you have done this	Your work should look like this
Begin by drawing a ray with endpoint C. The right angle will have C as its vertex.
1. Mark a point, not on the given line, about 6 cm in from C. Its precise location is not essential. Label it D.
2. Set the compasses on point D and set their width to the endpoint C .
3. Draw an arc that crosses the given line and extends over and above the endpoint C. (If you prefer, draw a complete circle.)
4. Draw a diameter through D from the point where the arc crosses the given line, producing points B and A.
5. Draw a line from C to the endpoint A of the diameter line
6. At this point you are done. The angle ACB is a right angle (90 deg).

Perpendicular at a point on a line

This page illustrates the way to draw a perpendicular at a point on a line with compass and straightedge or ruler. It works by efficiently producing two congruent triangles and then drawing a line between their vertices.

Easy to use step-by-step instructions

Proof

This construction works by efficiently creating two congruent triangles. The image below is the final drawing above with the red lines added.

	The case	Explanation
1	Segment KP is congruent to KQ	They were both drawn with the same compass width
2	Segment PR is congruent to QR	They were both drawn with the same compass width
3	Triangles ∆KRP and ∆KRQ are congruent	Three sides congruent (sss). KR is common to both.
4	Angles PKR, QKR are congruent	CPCTC. Corresponding parts of congruent triangles are congruent
5	Angles PKR QKR are both 90°	They are a linear pair and (so add to 180°) and congruent (therefore each must be 90°)

– Q.E.D

This is the step-by-step, printable instruction.

	After you have done this	Your work ought to appear like this
	Start with a line and point K on that line.
1	Set the compasses’ width to a medium setting. The actual width does not matter.
2	Without altering the compasses’ width, mark a short arc on the line at each side of the point K, creating the points P,Q. These two points are therefore the same distance from K.
3	Increase the compasses to merely double the width (again the exact setting is not essential).
4	From P, mark off a short arc above K
5	Without altering the compasses’ width repeat from the point Q in order that the two arcs cross each other, producing the point R
6	Making use of the straight edge, draw a line from K to where the arcs cross.
7	Done. The line immediately drawn is a perpendicular to the line at K

Perpendicular to a line from an external point
Geometry construction with the use of a compass and straightedge

This page shows how to construct a perpendicular to a line through an external point, making use of just a compass and straightedge or ruler. It works by producing a line segment on the given line, then bisecting it. The bisector will be a right angles to the given line.

Printable step-by-step instructions

This easy to use step by step instruction will serve as a handy learning tool in the absence of a computer

Proof

The image below is the final drawing above with the red lines added.

	The case	Explanation
1	Segment RP is congruent to RQ	They were together drawn with the same compass width
2	Segment SQ is congruent	They were together drawn with the same compass width
3	Triangle RQS is congruent to triangle RPS	Three sides congruent (sss), RS is common to both.
4	Angle JRQ is congruent to JRP	CPCTC. Equivalent parts of congruent triangles are congruent.
5	Triangle RJQ is congruent to triangle RJP	Two sides and included angle congruent (SAS), RJ is common to both.
6	Angle RJP and RJQ are congruent	CPCTC. Equivalent parts of congruent triangles are congruent.
7	Angle RJP and RJQ are 90°	They are congruent and supplementary (add to 180°).

– Q.E.D

Construct a line perpendicular to a line that passes through a point, with compass and straightedge

1	Construct a line perpendicular to the one below that passes through the point P

2	(a)	Construct a line perpendicular to AB through P, and another line perpendicular to CD also through P
	(b)	What is the name of the resulting 4-sided shape? Measure its side lengths with a ruler and calculate its area.

Construction of angles 75° 105° 120° 135° 150° angles plus a lot more

We have previously done how to construct angles 30°, 45°, 60° and 90°. Through combination of those angles you would be able to arrive at other angles.

Adding angles

Angles can be effectively ‘added’ by constructing them so they share a side. This is illustrated in Constructing the sum of angles.

For instance, by first constructing a 30° angle and then a 45° angle, you will obtain a 75° angle. The table below illustrates a few angles that can be obtained by the combination of simpler ones in a lot of ways

To create angle	Combine angles
75°	30° + 45°
105°	45° + 60°
120°	30° + 90° or 60° + 60°
135°	90° + 45°
150°	60° + 90°

In addition, through the combination of three angles a lot more can be constructed.

You can as well subtract them

By constructing an angle “inside” another you can effectively subtract them. Therefore, if you began with a 70° angle and constructed a 45° angle inside it sharing a side, the resultant angle would be a 25° angle. This is illustrated in the construction of the difference between two angles

Bisecting an angle ‘halves’ it

By bisecting an angle you obtain two angles of half the measure of the first. This offers you a few more angles to combine as explained above. For instance constructing angle 30° and then bisecting it ill give you two 15° angles.

Complementary and supplementary angles

By constructing the supplementary angle of a given angle, you get another one to combine as above. For example a 60° angle can be used to create a 120° angle by constructing its supplementary angle. This is shown in Constructing a supplementary angle.

Similarly, you can find the complementary angle. For example the complementary angle for 20° is 70°. Finding the complementary angle is shown in Constructing a complementary angle.

The basic constructions are described on these pages:

• Constructing a 30° angle
• Constructing a 45° angle
• Constructing a 60° angle
• Constructing a 90° angle

Geometry construction of Complementary angle with the use of a compass and straightedge

This construction takes a given angle and constructs its complementary angle. Recall that the complementary angle is an angle that makes the given angle become 90°. Therefore, an angle of 30° has a supplementary angle of 90° – 30° = 60°.

In this construction you can extend either leg back. It will yield the same result.

Proof

	Argument	Reason
1	m∠FAC = 90°	Drawn at point A with the use of the construction Perpendicular to a line at a point. See that page for proof.
2	m∠FAB + m∠BAC = ∠FAC	Adjacent angles
3	m∠FAB and m∠BAC are complementary	m∠FAB + m∠BAC = 90° See (2)

– Q.E.D

Complementary angle

This is the step-by-step, printable instruction to guide you in the absence of a computer.

After you have done this	Your work should look like this
Begin with a angle BAC.
1. Extend either leg (here AC) backwards, away from the interior of the angle.
2. With the compass at any convenient width, make an arc each side of A, producing points P and Q.
3. Make the compass width wider, and from P make an arc above A
4. Repeat from Q, producing the point F above A.
5. Draw a line from A, up through F
When you’ve got to this point, you are done. The angle ∠FAB and the given angle ∠BAC are complementary. (I.E. they add to 90°)

Geometry construction of a Supplementary angle with the use of a compass and straightedge

This construction takes a given angle and constructs its supplementary angle. Recall that the supplementary angle is one that makes the given angle become 180°. Therefore, an angle of 45° has a supplementary angle of 180° – 45° = 135°.

In this construction you can extend either leg back. It will produce the same result.

Proof

	The case	Explanation
1	m∠DAB + m∠BAC = 180°	A linear pair, therefore add to 180°
2	∠DAB is the supplementary angle to ∠BAC	From (1)

– Q.E.D

Supplementary angle

This is the step-by-step, printable instruction to assist you

After you have done this	Your work ought to appear like this
Begin with an angle BAC.
1. Extend either leg (here AC) backwards, away from the interior of the angle.
You are done at this stage. The angle ∠DAB and the given angle ∠BAC are supplementary. (I.E. they add to 180°)