RD Sharma Solutions Class 9 Constructions Exercise 17.2

 Q1. Draw an angle and label it as \(\angle BAC\). Construct another angle, equal to \(\angle BAC\)

Steps of construction:

1. Draw an angle ABC and a line segment QR.
2. With center A and any radius, draw an arc which intersects \(\angle BAC\)at E and D.
3. With Q as a centre and same radius draw an arc which intersects QR at S.
4. With S as center and radius equal to DE, draw an arc which intersects the previous arc at T.
5. Draw a line segment joining Q and T.

Therefore \(\angle PQR\)= \(\angle BAC\)

Q2. Draw an obtuse angle. Bisect it. Measure each of the angles so formed.

Steps of construction:

1. Draw an angle \(\angle ABC\) of \(120^{0}\).
2. With B as a centre and any radius, draw an arc which intersects AB at P and BC at Q.
3. With P as center and radius more than half of PQ draw an arc.
4. With Q as a center and same radius draw an arc which cuts the previous arc at R.
5. Join BR.

Therefore \(\angle ABR\)=  \(\angle RBC\)=  \(60^{0}\)

Q3. Using your protractor, draw an angle of \(108^{0}\). With this given angle as given, draw an angle of \(54^{0}\).

Steps of construction:

1. Draw an angle ABC of \(108^{0}\).
2. With B as the center and any radius draw an arc which intersects AB at P and BC at Q.
3. With P as center and radius more than half of PQ draw an arc.
4. With Q as the centre and same radius draw an arc which intersects the previous arc at R.
5. Join BR.

Therefore \(\angle RBC\)=  \(54^{0}\)

Q4. Using the protractor, draw a right angle. Bisect it to get an angle of measure \(45^{0}\).

Steps of construction:

1. Draw an angle ABC of \(90^{0}\).
2. With B as the centre and any radius draw an arc which intersects AB at P and BC at Q.
3. With P as center and radius more than half of PQ draw an arc.
4. With Q as center and same radius draw an arc which intersects the previous arc at R.
5. Join RB.

Therefore \(\angle RBC\)=  \(45^{0}\)

Q5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other.

Steps of construction:

1. Draw two angles DCA and DCB forming linear pair
2. With center C and any radius draw an arc which intersects AC at P and CD at Q and CB at R
3. With center P and Q and any radius draw two arcs which intersect each other at S
4. Join SC
5. With Q and R as center and any radius draw two arcs which intersect each other at T
6. Join TC

Therefore \(\angle SCT\)=  \(90^{0}\).

Q6. Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the same line.

Steps of Construction:

1. Draw a pair of vertically opposite angle \(\angle AOC\) and \(\angle DOB\).
2. Keeping O as the center and any radius draw two arcs which intersect OA at P, OC at Q, OB at S and OD at R.
3. Keeping P and Q as center and radius more than half of PQ draw two arcs which intersect each other at T.
4. Join TO.
5. Keeping R and S as center and radius more than half of RS draw two arcs which intersect each other at U.
6. Join OU.

Therefore TOU is a straight line

Q7. Using rulers and compasses only, draw a right angle.

Steps of construction:

1. Draw a line segment AB.
2. Keeping A as the center and any radius draw an arc which intersects AB at C.
3. Keeping C as center and the same radius draw an arc which intersects the previous arc at D.
4. Keeping D as the center and same radius draw an arc which intersects arc in (2) at E.
5. Keeping E and D as center and radius more than half of ED draw arcs which intersect each other at F.
6. Join FA.

Therefore \(\angle FAB\)=  \(90^{0}\)

Q8.Using rulers and compasses only, draw an angle of measure \(135^{0}\).

Steps of construction:

1. Draw a line segment AB and produce BA to C.
2. Keeping A as the center and any radius draw an arc which intersects AC at D and AB at E.
3. Keeping D and E as center and radius more than half of DE draw arcs which intersect each other at F.
4. Join FA which intersects the arc in (2) at G.
5. Keeping G and D as center and radius more than half of GD draw arcs which intersect each other at H.
6. Join HA.

Therefore \(\angle HAB\)=  \(135^{0}\)

Q9. Using a protractor, draw an angle of measure \(72^{0}\). With this angle as given draw angles of measure \(36^{0}\) and \(54^{0}\).

Steps of construction:

1. Draw an \(\angle ABC\) of \(72^{0}\) with the help of a protractor.
2. Keeping B as center and any radius draw an arc which intersects AB at D and BC at E.
3. Keeping D and E as center and radius more than half of DE draw two arcs which intersect each other at F.
4. Join FB which intersects the arc in (2) at G.
5. Keeping D and G as center and radius more than half of DG draw two arcs which intersect each other at H
6. Join HB

Therefore \(\angle HBC\)=  \(54^{0}\)

\(\angle FBC\)=  \(36^{0}\)

Q10. Construct the following angles at the initial point of a given ray and justify the construction:

1. \(45^{0}\)
2. \(90^{0}\)

Answers

1. \(45^{0}\)

Steps of construction:

1. Draw a line segment AB and produce BA to C.
2. Keeping A as the center and any radius draw an arc which intersects AC at D and AB at E.
3. Keeping D and E as center and radius more than half of DE draw arcs which intersect each other at F.
4. Join FA which intersects the arc in (2) at G.
5. Keeping G and E as center and radius more than half of GE draw arcs which intersect each other at H.
6. Join HA.

Therefore \(\angle HAB\)=  \(45^{0}\)

2. \(90^{0}\)

Steps of construction

1. Draw a line segment AB.
2. Keeping A as the center and any radius draw an arc which intersects AB at C.
3. Keeping C as center and the same radius draw an arc which intersects the previous arc at D.
4. Keeping D as the center and same radius draw an arc which intersects arc in (2) at E.
5. Keeping E and D as center and radius more than half of ED draw arcs which intersect each other at F.
6. Join FA.

Therefore \(\angle FAB\)=  \(90^{0}\)

Q11. Construct the angles of the following measurements:

1. \(30^{0}\)
2. \(75^{0}\)
3. \(105^{0}\)
4. \(135^{0}\)
5. \(15^{0}\)
6. \(22\frac{1}{2}^{0}\)

ANSWERS:

1. \(30^{0}\)

Steps of construction:

1. Draw a line segment AB.
2. Keeping A as the centre and any radius draw an arc which intersects AB at C.
3. Keeping C as center and the same radius draw an arc which intersects the previous arc at D.
4. Keeping D and C as center and radius more than half of DC draw arcs which intersect each other at E.
5. Join EA.

Therefore \(\angle EAB\)=  \(30^{0}\)

2. \(75^{0}\)

Steps of construction:

1. Draw a line segment AB
2. Keeping A as center and any radius draw an arc which intersects AB at C
3. Keeping C as center and the same radius draw an arc which intersects the previous arc at D
4. Keeping D as center and same radius draw an arc which intersects arc in (2) at E
5. Keeping E and D as center and radius more than half of ED, draw arcs intersecting each other at F
6. Join FA which intersects arc in (2) at G
7. Keeping G and D as center and radius more than half of GD draw arcs intersecting each other at H
8. Join HA

Therefore \(\angle HAB\)=  \(75^{0}\)

3. \(105^{0}\)

Steps of construction:

1. Draw a line segment AB.
2. Keeping A as the center and any radius draw an arc which intersects AB at C.
3. Keeping C as center and the same radius draw an arc which intersects the previous arc at D.
4. Keeping D as the centre and same radius draw an arc which intersects arc in (2) at E.
5. Keeping E and D as center and radius more than half of ED draw arcs which intersect each other at F.
6. Join FA which intersects arc in (2) at G.
7. Keeping E and G as center and radius more than half of EG draw arcs which intersect each other at H.
8. Join HA.

Therefore \(\angle HAB\)=  \(105^{0}\)

4. \(135^{0}\)

Steps of construction:

1. Draw a line segment AB and produce BA to C.
2. Keeping A as the center and any radius draw an arc which intersects AC at D and AB at E.
3. Keeping D and E as center and radius more than half of DE draw arcs which intersect each other at F.
4. Join FA which intersects the arc in (2) at G.
5. Keeping G and D as center and radius more than half of GD draw arcs which intersect each other at H
6. Join HA.

Therefore \(\angle HAB\)=  \(135^{0}\)

5. \(15^{0}\)

Steps of construction:

1. Draw a line segment AB.
2. Keeping A as the centre and any radius draw an arc which intersects AB at C.
3. Keeping C as center and the same radius draw an arc which intersects the previous arc at D.
4. Keeping D and C as center and radius more than half of DC draw arcs which intersect each other at E.
5. Join EA which intersects arc in (2) at F.
6. Keeping F and C as center and radius more than half of FC draw arcs which intersect each other at G.
7. Join GA.

Therefore \(\angle GAB\)=  \(15^{0}\)

6. \(22\frac{1}{2}^{0}\)

Steps of construction:

1. Draw a line segment AB.
2. Keeping A as the center and any radius draw an arc which intersects AB at C.
3. Keeping C as center and the same radius draw an arc which intersects the previous arc at D.
4. Keeping D as the centre and same radius draw an arc which intersects arc in (2) at E.
5. Keeping E and D as center and radius more than half of ED draw arcs which intersect each at F.
6. Join FA which intersects arc in (2) at G.
7. Keeping G and C as center and radius more than half of GC draw arcs intersecting each other at point H.
8. Join HA which intersects the arc in (2) at a point I.
9. Keeping I and C as center and radius more than half of IC draw arcs intersecting each other at point J.
10. Join JA.

Therefore \(\angle JAB\)=  \(22\frac{1}{2}^{0}\)