BISECTOR OF AN ANGLE BISECTING AN ANGLE

Bisector Of An Angle-Bisecting An Angle

We have provided the step-by-step, printable instructions on how to bisect an angle

After doing this	Your work should look like this
Start with angle PQR that we will bisect.
1. Place the compasses’ point on the angle’s vertex Q.
2. Change the compasses to a medium wide setting. The correct width is not significant.
3. Without altering the compasses’ width, draw an arc across each leg of the angle.
4. The compasses’ width can be altered here if desired. Recommended: leave it the same.
5. Place the compasses on the point where one arc crosses a leg and draw an arc in the interior of the angle.
6. Without altering the compasses setting repeat for the other leg so that the two arcs cross.
7. Making use of a straightedge or ruler, draw a line from the vertex to the point where the arcs cross
Done. This is the bisector of the angle ∠PQR.

Given an angle created by two lines with a common vertex, this page illustrates how to construct another angle from it that has the same angle size with a compass and straightedge or ruler. It works by producing two congruent triangles. A proof is illustrated below.

Printable step-by-step instructions

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

Proof

This construction functions by producing two congruent triangles. The angle to be copied has the same size in both triangles
The image below is the last drawing above with the red items added.

	Argument	Reason
1	Line segments AK, PM are congruent	Both drawn with the same compass width.
2	Line segments AJ, PL are congruent	Both drawn with the same compass width.
3	Line segments JK, LM are congruent	Both drawn with the same compass width.
4	Triangles ∆AJK and ∆PLM are congruent	Three sides congruent (SSS).
6	Angles BAC, RPQ are congruent.	CPCTC.Equivalent parts of congruent triangles are congruent

– Q.E.D

Copying an angle

This is the step-by-step, printable instruction on the process of copying an angle

After doing this	Your work ought to appear like this
Begin with a angle BAC that we will copy.
1. Make a point P that will be the vertex of the new angle.
2. From P, draw a ray PQ. This will turn into one side of the new angle. • This ray can go off in any direction. • It does not have to be parallel to anything else. • It doesn’t have to be the same length as AC or AB.
3. Put 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, put the compasses’ point on P and draw a similar arc there, producing point M as illustrated.
6. Put 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 ray PR from P through L and onwards a little further. The precise length is not significant.
Done. The angle ∠RPQ is congruent(equal in size) to angle ∠BAC.

Construction of a parallel through a point (translated triangle method)

This is the step-by-step, printable version. If you PRINT this page, any ads will not be printed.

Constructing a parallel through a point through angle copy procedure

We have provided below information on how to construct a line parallel to a given line that passes through a given point with compass and straightedge or ruler. It is known as the ‘angle copy procedure’ due to the fact that it works by the use of the fact that a transverse line drawn across two parallel lines produces pairs of equal equivalent angles. It makes use of this in reverse – by producing two equal equivalent angles, it can produce the parallel lines.

Step-by-step guide on how to achieve this

Proof

This construction functions by the use of the fact that a transverse line drawn across two parallel lines produces pairs of equal corresponding angles. It makes use of this in reverse – by producing two equal equivalent angles, it can produce the parallel lines.

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

	Argument	Reason
1.	Line segments AR,BJ are congruent	Both drawn with the same compass width.
2.	Line segments RS,JC are congruent	Both drawn with the same compass width.
3.	Line segments AS,BC are congruent	Both drawn with the same compass width.
4.	Triangles ∆ARS and ∆BJC are congruent	Three sides congruent (sss).
5.	Angles ARS, BJC are congruent.	CPCTC. Equivalent parts of congruent triangles are congruent
6.	The line AJ is a transversal	It is a straight line drawn with a straightedge and cuts across the lines RS and PQ.
7.	Lines RS and PQ are parallel	Angles ARS, BJC are corresponding angles that are equal in size only if the lines RS and PQ are parallel

– Q.E.D

Construct a line parallel to a given line through a given point with compass and straightedge

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

2.(a)Construct a line parallel to AB through Q, and another line parallel to CD and as well through Q

(b).What is the name of the resulting 4-sided shape?

Constructing a parallel through a point by rhombus method

This page shows how to construct a line parallel to a given line through a given point with compass and straightedge or ruler. This construction functions by creating a rhombus. Since we know that the opposite sides of a rhombus are parallel, then we have produced the desired parallel line. This construction is simpler than the traditional angle copy procedure since it is done with merely one single compass setting.

Proof

This construction works by producing a rhombus. Since we know that the opposite sides of a rhombus are parallel, then we have produced the desired parallel lines.

The diagram below is the final drawing above with the green lines added.

	Argument	Reason
1.	Line segments RJ, JE, ES, RS are congruent	They are all drawn with the same compass width.
2.	RJES is a rhombus	A rhombus is a quadrilateral with 4 congruent sides.
3.	Lines RS and JE are parallel	Opposite sides of a rhombus are always parallel.

– Q.E.D

The step-by-step printable version is provided below to make the construction handy even when you are not before the computer.

When you have done this	Your work ought to appear like this
Begin with a line segment PQ and a point R off the line.
1. Put the compasses on point R and set its width to a little more than the distance to the line PQ. The precise distance is not significant.
2. Draw a wide arc from the right of R around in order that it crosses the line PQ at two points. Label the left point J
3. Without altering the compasses’ width, move the compasses to J and draw an arc across the line PQ. Label this point E.
4. Without altering the compasses’ width, move the compasses to E and draw an arc across the large arc to the right of R. Label this point S.
5. Draw a straight line through points R and S.
Done. The line RS is parallel to the line PQ

Construction of a parallel through a point through translated triangle procedure

This page illustrates how to construct a line parallel to a given line through a given point with compass and straightedge or ruler.

This construction works by producing any triangle between the given point and the given line, then copying (translating) that triangle any distance all along the given line. Due to the fact that we are aware that a translation can map the one triangle onto the second congruent triangle, then the lines connecting the equivalent points of each triangle are parallel, and we can produce the desired parallel line by connecting the top vertices of the two triangles.

Printable step-by-step instructions

This printable step-by-step method can be used for making handouts or when you are not before the computer.

Proof

This construction functions by producing a triangle and then translating (sliding) the triangle along the given line. All equivalent vertices of a translated polygon are connected by lines that are congruent and parallel.

This can be seen more vividly in the animation at Translating a Polygon.

	Argument	Reason
1.	Triangle ARB and A’R’B’ are congruent	By construction. A’R’B’ was copied from ARB.
2.	AB and A’B’ are collinear.	All four points lie on PQ
3.	A’R’B’ is a translation of ARB.	The two triangles are congruent (from 1) and not rotated (from 2) and not reflected (by construction).
4.	RR’ is parallel to AA’	Lines linking the corresponding vertices of translated polygons are parallel.
5.	RR’ is parallel to PQ	From (2) – AA’ is parallel to PQ because they are collinear.

– Q.E.D

Construct a line parallel to a given line through a given point with compass and straightedge

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

2.(a).Construct a line parallel to AB through Q, and another line, with the use of a different method, parallel to CD also through Q

b.What is the name of the resulting 4-sided shape?

After doing this	Your work should look like this
Start with a line PQ and a point R off the line. The aim is to draw a line through R that is parallel to PQ.
1. Mark two points A and B anywhere on It will be easiest if they are anywhere close to R, but anywhere will in fact work.the line PQ. It will be easiest if they are anywhere close to R, but anywhere will in fact work.
2. (Optional). Draw the lines RA and RB to produce the triangle RAB. This step is not compulsory due to the fact that the two lines are just there to help you to see the triangle and assist in you understanding of the way the construction
3. Mark a point A’ anywhere along the length of PQ.
4. Set compass width to the distance AB. Then, from A’, make an arc across the line PQ, producing point B’.
5. Set compass width to the distance AR. Then, from A’, make an arc above the line.’.
6. Set compass width to the distance BR. Then, from B’, make an arc above the line, producing R’ where it crosses the preceding arc.
7. Not compulsory). Draw the lines R’A’ and R’B’ to form the triangle R’A’B’. This step is not compulsory due to the fact that the two lines are only there to allow you to see the triangle and assist to understand how the construction works.
8. Draw a line through points R and R’. The line R R’ is parallel to the line PQ