Construct an angle of measure 22.5∘ using compass and ruler.

To construct an angle of 22.5° using a compass and ruler, follow these steps:

Draw a base line: Use the ruler to draw a straight horizontal line. Label the endpoint as O.

Construct a 45° angle:

Place the compass needle at point O.
Open the compass to a convenient radius and draw an arc that intersects the base line.
Without changing the radius, place the compass at the intersection point on the arc and draw another arc above the line.
Repeat this process from the new intersection point, ensuring that the arcs intersect at two new points.
Now, use the ruler to draw a straight line from O through the point where the two arcs intersect above the base line. This line will create a 45° angle with the base line.
Bisect the 45° angle:

Now, place the compass needle at point O again, but this time open the compass to a slightly smaller radius.
Draw an arc from O that intersects both arms of the 45° angle.
Keeping the same radius, place the compass at the two points where the arc intersects the arms of the angle and draw two arcs that intersect each other.
Use the ruler to draw a straight line from O through the point where the two arcs intersect. This line will bisect the 45° angle, splitting it into two equal angles of 22.5° each.
Label the angle: The newly created angle between the base line and the bisected line is 22.5°.

This method uses basic geometric principles of angle bisection to construct the required 22.5° angle accurately.