Bisection of Lines and Angles

1. To bisect a given line: To divide a line into two equal parts

i. Draw a line AB (e.g. AB = 10 cm)

ii. Open your compass to a radius greater than half AB. Place the point of your compass on A and draw arcs above and below line AB.

iii. With centre B, using the same radius, draw arcs above and below line AB to cut the first two arcs at C and D.

iii. Draw a line to join C and D. Label the point where this line cuts AB as O. Since the line CD bisects AB at a right angle, CD is called the perpendicular bisector of AB, and point O is the mid-point of PQ.

2. To bisect an angle: To divide an angle into two equal parts e.g. ∠XYZ

i. Draw an angle, ∠XYZ

ii. With centre Y, at any convenient radius, use your compasses to draw an arc to cut YX and YZ at A and B respectively.

iii. With centre A using radius more than half AB, draw an arc, with centre B using the same radius, draw another arc to cut the first at C.

iv. Join YC. Thus YC bisects ∠XYZ.

3. To copy an angle: To draw an angle similar to it

To copy angle, ∠LMN

i. Draw the given angle with centre M and any convenient radius, draw an arc to cut ML at P and MN at Q.

ii. Draw the line YZ with centre Y and the same radius (i.e. MP) draw an arc to cut YZ at B.

iii. With centre B and radius equal to the distance of PQ, draw another arc to cut the previous one at A.

iv. Join YA and produce to a suitable point X. Thus ∠XYZ is similar to ∠LMN

4. Constructing Perpendiculars:

(a) To construct an angle of 90° at a point on a given line. Given a line PQ and a point R on the line, we can draw a perpendicular to PQ at R as follows:

i. With centre R and any convenient radius, draw two arcs to cut line PQ at L and M.

ii. With L and M as centres and a radius more than LR or RM, use your compasses to draw arcs above line PQ to intersect at D.

iii. Join RD to obtain the required perpendicular measure ∠DRQ to verify it is 90°.

(b) To draw a perpendicular to a line from a given point outside it.

1. With centre R and at any convenient radius, draw arcs to cut line PQ at point L and M.
2. With points L and M as centres and a radius more than half of LM, draw arcs to intersect each other at point D.
3. Join RD to cut PQ at O. Thus RD is perpendicular to PQ.

5. To construct an angle of 45°

To construct angle 45°, first construct angle 90°. In this case, construct ∠DRQ. Then bisect it to obtain 45°.

6. To construct an angle of 60°

Steps:

i. Draw line DE

II. Indicate point F anywhere on DE

iii. Choose a convenient radius and draw an arc from centre F to cut DE at G

iv. With centre G and the same radius you chose in the previous step, draw an arc to cut the previous arc at H.

v. Draw a line from F through H, i.e line FI. Angle IFG is 60°

7. To construct an angle of 30°

i. Construct an angle of 60° like we did above.

ii. Bisect the 60° angle. Angle JFE is 30°

8. To construct an angle of 120°

i. Recall 120° = 2 x 60°, hence first construct angle 60° as described before. (This time let the arc cut at G and then D on the other side, forming a semi circle)

ii. With centre H, and the same radius used to construct the 60° angle, draw another arc to cut at J.

iii. Draw a line from F through J, i.e line JF. JFE is a 120° angle.

Alternate Method:

Recall sum of angles on a straight line is 180°.

As given above construction ∠HFG is 60° on the left hand side (LHS), thus the ∠IFD is 120°.

9. To construct an angle of 105^o

i. With centre O and any convenient radius draw a semicircle to meet PQ at A and B

ii. Using A and B as centres and the same radius, draw two arcs to cut the semicircle at C and E.

iii. Using C and E as centres and the same radius, draw two arcs to cut each other at M.

iv. Join MO. Thus ∠QOL=90°

v. But ∠LOF = 30°. Bisect ∠LOF to obtain 15°. Hence ∠QOH = 15° + 90° = 105°

10. To construct an angle of 75°

Recall 75° + 105° = 180° (sum of angles on a straight line).

Therefore, construct angle 105° as above on the right hand side (RHS), i.e. ∠HOQ = 105°, hence the angle on the left hand side (LHS) is 75° i.e. ∠HOP = 75°

11. To construct angle 135°

Recall 45° + 135° = 180° (Sum of angles on a straight line).

Thus, by constructing angle 45° on the left hand side (LHS) i.e. ∠GOP=45°, therefore the angle on the right hand side (RHS) is 135° i.e. ∠GOQ= 135°

12. To construct angle 150°

Note that 150° + 30° = 180° (sum of angles on a straight line)

Hence, we can construct angle 30° on the LHS as given previously i.e. ∠GOP = 30°, thus the angle on the right hand side (RHS) is 150° i.e. ∠GOQ = 150°

Hint: It is important to know how to construct other angles by bisecting/combining/subtracting these angles depending on the side the angle is required.