The Greeks called those figures that could be drawn with only a straight edge and compass "constructible." Not all figures can be. Those that couldn't be were regarded as inferior.

Instructions: Work in pairs. Together, find a way to construct each of the following with only compass and straightedge. One of the two of you should be doing all the drawing on the first problem; switch after each problem.

1. Any line segment can be copied to another position. To prove this, draw any segment, then draw a ray somewhere else and copy the segment to the ray with one end at the vertex.
2. Any line segment can be bisected. Remember, this means cutting into two equal segments, or finding the midpoint. Draw a line segment and bisect it.
3. Any angle can be bisected. Draw an angle and bisect it.
4. Any angle can be copied to any ray. Draw an angle, then a ray somewhere else. The ray must be one edge of the new angle. Construct the other ray so you have a copy of the angle.
5. Given the dimensions of any triangle, it is always possible to construct it. Carry out the following construction to see that this is true. First, draw three segments of random length, then construct a triangle with those lengths as sides. What do you have to be careful about?
6. From any given point on a line it is possible to construct a perpendicular to the line at that point. Draw a line and choose any point on it. Then draw the perpendicular from that point.
7. From a given line and a given point not on the line it is possible to construct a perpendicular from that point to the line. Draw any line and any point not on it, then draw the perpendicular.
8. Given two line segments, positioned anywhere, one can construct a segment with a length the sum of the lengths of the two segments or the difference of the lengths. Show both.
9. Given any two angles, positioned anywhere, one can always construct a third angle anywhere whose size is the sum or difference of the angles of the given two. Show both.
10. Construct a line parallel to a given line passing through any given point not on the line.
11. Construct a "ruler." That is, on a line, mark off some equal segments. Call each segment one "unit" or one "Smoot" or whatever. Mark off the ruler in 1/2 units, 1/4 units, and 1/8 units.
12. Construct an equilateral triangle.
13. Construct a square.
14. Draw a circle and construct an equilateral triangle inside it so all vertices are on the circle.
15. Draw a circle and construct a square inside it so all vertices are on the circle.
16. Draw a scalene triangle and construct a circle that passes through all three points.
17. Divide a line segment into n congruent line segments. Demonstrate this by dividing it into 5 congruent segments.
18. Construct the center point of a given circle.
19. Construct a circle through three given points.