Chapter 11: Constructions Exercise – 11.2
Question: 1
Draw an angle and label it as ∠BAC. Construct another angle, equal to ∠BAC.
Solution:
Steps of construction:
1. Draw an angle, ABO and a Line segment QR
2. With center A and any radius, draw an arc which intersects ∠BAC at E and O
3. With center Q and same radius draw arc which intersect QR at S.
4. With center S and radius equal to DE, draw an arc which intersect previous arc at T
5. Draw a line segment joining Q and T
∴ ∠PQR = ∠BAC
Question: 2
Draw an obtuse angle, Bisect it. Measure each of the angles so obtained.
Solution:
Steps of construction:
1. Draw angle ABC of 120°
2. With center B and any radius, draw an arc which intersects AB at P and BC at Q
3. With center P and Q and radius more than 1/2 PQ, draw two arcs, with intersect each other at R.
4. Join BR
∴ ∠ABB = ∠RBC = 60°
Question: 3
Using your protractor, thaw an angle of measure 108°. With this angle as given, draw an angle of 54°.
Solution:
Steps of construction:
1. Draw an angle ABC of 108°
2. With center B and any radius, draw an arc which intersects AB at P and BC at Q
3. With center P and Q and radius more than 2 PQ, draw two arcs, which intersect each
other at R.
4. Join BR
∴ ∠RBC = 54°
Question: 4
Using protractor, draw a right angle. Bisect it to get an angle of measure 45°.
Solution:
Steps of construction:
1. Draw an angle ABC of 90°
2. With center B and any radius, draw an arc which intersects AB at P and BC at Q
3. With center P and Q and radius more than 1/2 PQ, draw two arcs, which intersect each 2 other at R.
4. Join RB
∴ ∠RBC = 45°
Question: 5
Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other.
Solution:
Steps of construction:
1. Draw two angle DCA and DCB forming Linear pair
2. With center C and any radius, draw an arc which intersects AC at P, CD at Q and CB at R.
3. With center P and Q and any radius draw two arcs which interest each other at S
4. Join SC
5. With center Q and R any radius draw two arcs, which intersect each other at T.
6. Join TC
∠SCT = 90° [By using protractor]
Question: 6
Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the same line.
Solution:
Steps of construction:
1. Draw a pair of vertically opposite angle AOC and DOB
2. With center 0 and any radius drawn two arcs which intersect OA at P, Q - OB at S and OD at R.
3. With center P and Q and radius more than 1/2 PQ, draw two arcs which intersect each other at U.
4. Join to
5. With center R and S radius more than 1/2 RS, draw two arcs which intersect each other at U.
6. Join OU.
∴ TOU is a straight line
Question: 7
Using ruler and compasses only, draw a right angle.
Solution:
Steps of construction:
1. Draw a line segment AB
2. With center A and any radius draw arc which intersect AB at C.
3. With center C and same radius draw an arc which intersects AB at C.
4. With center D and same radius draw arc which intersect arc in (2) at E.
5. With centers E and C and any radius, draw two arcs which intersect each other at F.
6. Join FA
∠FAB = 90°
Question: 8
Using ruler and compasses only, draw an angle of measure 135°.
Solution:
Steps of construction:
1. Draw a line segment AB and produce BA to point C.
2. With center A and any radius draw arc which intersect AC at D and AB at E.
3. With center D and E and radius more than 1/2 DE, draw two arcs which intersect each other at F.
4. Join FA which intersect the arc in (2) at G.
5. With centers G and D and radius more than 1/2 GD, draw two arcs which intersect each other at H.
6. Join HA
∴ ∠HAB = 135°
Question: 9
Using a protractor, draw an angle of measure 72°. With this angle as given, draw angles of measure 36° and 54°.
Solution:
Steps of construction:
1. Draw an angle ABC of 72° with the help of protractor.
2. With center B and any radius, draw an arc which intersect AB at D and BC at E.
3. With center D and E and radius more than 1/2 DE, draw two arcs which intersect each other at F.
4. Join FB which intersect the arc in (2) at G.
5. With centers D and G and radius more than 1/2 DE, draw two arcs which intersect each other at F.
6. With centers D and G and radius more than n 1/2 DG draw two arcs which intersect each other at H
7. Join HB
∴ ∠HBC = 54°
∠FBC = 36°
Question: 10
Construct the following angles at the initial point of a given ray and justify the construction:
(i) 45°
(ii) 90°
Solution:
(i) Steps of construction:
1. Draw a line segment AB and produce BA to point C.
2. With center A and any radius drawn an arc which intersect AC at D and AB at E.
3. With center D and E and radius more than 1/2 DE, draw arcs cutting each other at F.
4. Join FA which intersect arc in (2) at G.
5. With centers G and E and radius more than 1/2 GE, draw arcs cutting each other at H.
6. Join HA
∴ ∠HAB = 45°
(ii) Steps of construction:
1. Draw a line segment AB.
2. With center A and any radius draw in arc which intersect AB at C.
3. With center C and same radius thaw an arc which intersects previous arc at D.
4. With centers D same radius draw an arc which intersects are in (2) at E.
5. With centers E and D same radius more than 1/2 ED draw an arc cutting each other at F.
6. Join FA
∴ ∠FAB = 90°