construct an angle of 45 at the initial point of a given ray & justify the construction

In order to construct an angle of 45°, first of all we have to construct an angle of 90° and then the bisector of 90° and 0° will result in angle of 45°.

1. Draw a ray PQ.

2. With P as centre and any suitable radius, draw an arc, cutting PQ at A.

3. With A as centre and the same radius, draw an arc cutting the arc draw in step 2 at B.

4. With B as centre and the same radius as in step 2 and 3, draw an arc, cutting the arc drawn in step 3 at C.

5. With B as centre and the same radius, draw an arc

6. With C as centre and the same radius, draw an arc cutting the arc drawn in step 5 at D.

7. Draw PD and produce it to O. Now ∠ OPQ is the angle of measure 90°.

8. Draw PE, the bisector of ∠ OPQ.

The ∠ QPE so obtained is the required angle of measure 45°.