Question

A. Explain the connections between geometric constructions and two of Euclid’s postulates.

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Answer 1

1. The postulate is "A straight line segment can be drawn joining any two points". While constructing a straight line using a ruler, given any two points we can place the ruler such that it touches the two points and draw a straight line through them.

2. The postulate is "Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center". Now when a line segment is given, we can take a compass and place it on the segment such that the tip touches the ends of the line segment. Rotating the compass in this configuration, we get a circle with the line segment as radius and one of the end points as the center.