Question

$\text{Construct each of the following angles, using ruler and compasses:}\left(i\right){75}^{\circ }\left(ii\right){37.5}^{\circ }\left(iii\right){135}^{\circ }\left(iv\right){22.5}^{\circ }$

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 .