Question

Construct each of the following angles, using ruler and compasses:
(i) 75°
(ii) 37.5°
(iii) 135°
(iv) 105°
(v) 22.5°

Answer 1

(i) 75°
Steps of construction
1. Draw a line XY.
2. Take a point O on XY.
3. With O as centre, draw a semi circle, cutting XY at P and Q.
4. Construct $\angle$YOR = 90$°$.
5. Draw the bisector of $\angle$YOR = 90$°$ cutting the semi circle at point S.
6. With S and T as centres draw two arcs intersecting at point A.
$\angle$AOY = 75°.

(ii) 37.5°

Steps of construction
1. Draw a line XY.
2. Take a point O on XY.
3. With O as centre, draw a semi circle, cutting XY at P and Q.
4. Construct $\angle$YOR = 90$°$.
5. Draw the bisector of $\angle$YOR = 90$°$ cutting the semi circle at point S.
6. With S and T as centres draw two arcs intersecting at point A.
7. Draw the angle bisector of $\angle$AOY.
8. $\angle$BOY is the required angle of 37.5°.

(iii) 135°


Steps of construction:
1. Draw a line XY.
2. Take a point A on XY.
3. With A as centre, draw a semi circle, cutting XY at P and Q.
4. Construct $\angle$YAC = 90$°$.
5. Draw AB, bisector of $\angle$XAC.
Thus, $\angle$YAB = 135$°$

(iv) 105°

Steps of construction

1. Draw a line XY.
2. Take a point O on XY.
3. With O as centre, draw a semi circle, cutting XY at P and Q.
4. Construct $\angle$YOS = 90$°$.
5. Draw RO, bisector of $\angle$XOS.
6. Draw AO, bisector of $\angle$ROS.
$\angle$AOY = 105° is the required angle.

(v) 22.5°


Steps of construction:
1. Draw a ray AB.
2. Draw an angle $\angle$BAE = 45$°$.
3. With A as the centre and a small radius, draw an arc cutting AB at P and AE at Q.
4. With P as the centre and a radius more than half of PQ, draw an arc.
5. With Q as the centre and the same radius as above, draw another arc cutting the previously drawn arc at D.
6. Join AD.
Thus, $\angle$BAC is the required angle of measure 22.5^o.