This is perhaps the most important exercise in this book. It is a payoff for all

Question

This is perhaps the most important exercise in this book. It is a payoff for all the work you have done. Come back to this exercise as you do subsequent exercises and read further in the book. Your assignment in this exercise is to make a long list of geometric statements that are equivalent to the Euclidean parallel postulate in the sense that they hold in real Euclidean planes and do not hold in real hyperbolic planes. The statements proved in neutral geometry are valid in both Euclidean and hyperbolic planes, so ignore them. To get you started, here are 10 statements that qualify. They do not say anything about parallel lines, so you might have been surprised before studying this subject that they are equivalent to the Euclidean parallel postulate. Every triangle has a circumscribed circle. Wallis' postulate on the existence of similar triangles. A rectangle exists. Clavius' axiom that the equidistant locus on one side of a line is the set of points on a line is the set of points on a line. Some triangle has an angle sum equal to 180degree. An angle inscribed in a semicircle is a right angle. The Pythagorean equation holds for right triangles. A line cannot lie entirely in the interior of an angle. Any point in the interior of an angle lies on a segment with endpoints on the sides of the angle. Areas of triangles are unbounded.

Explanation / Answer

There is at most one line that can be drawn parallel to another given one through an external point. The sum of the angles in every triangle is 180°. There exists a triangle whose angles add up to 180°. The sum of the angles is the same for every triangle. There exists a pair of similar, but not congruent, triangles. Every triangle can be circumscribed. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. There exists a quadrilateral in which all angles are right angles, that is, a rectangle. There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the same line are also parallel to each other. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. There is no upper limit to the area of a triangle. The summit angles of the Saccheri quadrilateral are 90°. If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.