Constructing an angle and triangles with compasses and ruler

Construction problems are solved with compasses and an unmarked ruler only. The ruler draws a line through one or two points; the compasses draw a circle with a given centre and a radius equal to a given segment. Each problem has four steps: analysis, construction, proof, investigation.

An angle equal to a given angle. Given angle ABC and ray DK. Draw a circle of radius r with centre B (it cuts the arms of the angle) and a circle of the same radius with centre D. Call M where it cuts ray DK. Then draw a circle with centre M and radius equal to the distance between the two points where the first circle cut the arms. It cuts the second circle at E. Angle MDE is the one: the triangles are equal by three sides.

A triangle from two sides and the angle between them, or from a side and two adjacent angles. First lay the angle (or the two sides), then mark the lengths on the sides with the compasses. Each such triangle is unique up to congruence (the first and second congruence tests).

A right triangle from the hypotenuse and a leg. Draw two perpendicular lines m and n, meeting at C. On m lay the leg CA = b. Draw a circle with centre A and radius equal to the hypotenuse c. It cuts n at B1 and B2. Both triangles ACB1 and ACB2 work and are congruent, so there is one solution.