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

**i.** Open your compasses to a radius greater than half PQ. Place the point of your compasses on P, draw arcs, above and below line PQ.

**ii.** With centre Q, using the same radius, draw arcs above and below line PQ to cut the first two arcs at A and B.

**iii.** Draw a line to join A and B. Label the point where this line cuts PQ at O. Since the line AB bisects PQ at right angles, AB is called the **perpendicular** **bisector** of PQ, and point O is the mid-point of PQ.

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

**i. **Draw an angle, âˆ 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 âˆ XYZ.

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

To copy angle, âˆ 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 âˆ XYZ is similar to âˆ LMN

### 4. Constructing Perpendiculars:

**(a)** To construct an angle of 90Â° 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 âˆ DRQ to verify it is 90Â°.

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

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

### 5. To construct an angle of 45Â°

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

### 6. To construct an angle of 60Â°

**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 60Â°

### 7. To construct an angle of 30Â°

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

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

### 8. To construct an angle of 120Â°

**i.** Recall 120Â° = 2 x 60Â°, hence first construct angle 60Â° 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 60Â° angle, draw another arc to cut at J.

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

**Alternate Method:**

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

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

### 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 âˆ QOL=90Â°

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

### 10. To construct an angle of 75Â°

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

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

### 11. To construct angle 135Â°

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

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

### 12. To construct angle 150Â°

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

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

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.

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