Parallel Postulate

Home , Lambert quadrilateral, Saccheri quadrilateral

Cristina Hernandez

October 7, 2015

1. Read the excerpts ”History of the Parallel Postulate” and the ”Discovery of Hy- perbolic Gemometry”, from Euclidean and Non-Euclidean Geometries: Development and History by Greenberg.

2. P roof: A Saccheri quadrilateral is a quadrilateral whose base angles are right angles and whose base-adjacent sides are congruent to each other. The summit angles can either be right, acute, or obtuse.

By I1 through I3 we can draw diagonals. With the diagonals we have created two triangles ACD and BDC that have right angles at the base of the quadrilateral that by C6, the two triangles are congruent and have congruent interior angles. Therefore the diagonals AD and BC are congruent. There are two pairs of congruent angles ACD and BDC and then BCD and ADC. Therefore angle ACB and angle BDA are congruent. The triangles ACD and BDC and the triangles CBA and BAB give us the congruent angles CBA and DAB, so the summit angles are congruent. Thus, the summit angles of a Saccheri quadrilateral are congruent.

3. Case 3 cannot happen in Neutral Geometry, it would lead to a contradiction. Accord- ing to the excerpt ”History Of The Parallel Postulate”, Greenberg says that ”if the summit angles were obtuse, the angle sum of the quadrilateral would be more than 360 degrees, contradicting Corollary 2 to the Saccheri-Legendre Theorem.” The Saccheri-Legendre The-

1 orem says that the angle sim of a triangle cannot exceed 180. Therefore, the summit angles of a Saccheri quadrilateral cannot be obtuse.

4. Case 1

In this case we get a regular rectangle. If you create diagonals in this case you get two congruent trianlges and their interior angles add up to 180 degrees. It will hold also hold for Euclid’s 5th postulate.

Case 2

2 In this case the summit angles are acute so if you were to make triangles with the diagonals, they would add up to be less that 180 degrees. We read that this later led to the discovery of Hyperbolid Geometry.

5. The diﬀerence between Saccheri and Lambert Squares.

A Lambert Quadrilateral has three right angles and two equal sides. A Saccheri Quadri- lateral has to right angles at the base and we proved that the two summit angles can be acute or right as well. My ﬁrst encounter with these two was that I didn’t think that quadrilaterals could have non-straight lines. Now that we have learned about diﬀerent geometries, we know that polygons can come in many forms depending on which geometry we are working with.

In homework 2 we constructed a square from a given line segment. This square turned

3 out to be a right saccheri square, a right lambert square, and a right angle rhombus. Here are some examples.

Here we have an acute Lambert Square.

Here we have an acute Saccheri Square.

4 Bibliography

Greenberg. (n.d.). History Of The Parallel Postulate. In Euclidean and Non-Euclidean Geometries (pp. 148-163).