Construct each of the following angles, using ruler and compasses:
(i) 75°
(ii) 37.5°
(iii) 135°
(iv) 105°
(v) 22.5°
(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 $\angle Y O R=90^{\circ}$.
5. Draw the bisector of $\angle Y O R=90^{\circ}$ cutting the semi circle at point $S$.
6. With $S$ and $T$ as centres draw two arcs intersecting at point $A$.
$\angle \mathrm{AOY}=75^{\circ}$
(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 $\angle \mathrm{YOR}=90^{\circ}$.
5. Draw the bisector of $\angle Y O R=90^{\circ}$ 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
8.
(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
5. Draw $A B$, bisector of $\angle X A C$.
Thus, $\angle Y A B=135^{\circ}$
(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 $\angle \mathrm{YOS}=90^{\circ}$.
5. Draw $\mathrm{RO}$, bisector of $\angle \mathrm{XOS}$.
6. Draw AO, bisector of $\angle \mathrm{ROS}$.
$\angle \mathrm{AOY}=105^{\circ}$ is the required angle.
(v) 22.5°
Steps of construction:
1. Draw a ray AB.
2. Draw an angle $\angle B A E=45^{\circ}$.
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, $\angle B A C$ is the required angle of measure $22.5^{\circ}$.
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