Lesson 8, Topic 1
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# Bisection of lines & Angles

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### 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.

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 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 105o

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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