MAT 360 Lecture 9

1. MAT 360 Lecture 9 History of the parallel postulate

2. Alternate Interior Angle Theorem If alt. int angle are congruent then lines are parallel. Measure of angles and segments theorem Exterior Angle Theorem Exterior angle is greater than remote interior The sum of the measures of two angles of a triangle is less than 180 Saccheri-Legendre Theorem The sum of the interior angles of a triangle is at most 180 • Hilbert parallelism axiom • Euclid V • Converse to Alt. Int. Angle theo • Sum of int ang of triangle 180 • etc/ Equivalence of parallel postulates: are all equivalent.

3. Some attempts to prove Euclid’s V • Proclus: Measuring distances. • Wallis: Add postulate: “Given any triangle ΔABC, and a segment DE there exists a triangle ΔDEF similar (= with cong. angles)to ΔABC” • Saccheri: Sacheri quadrilaterals □ABCD (A, B are right, C congruent to D). Try prove: If Sach. quadrilateral not rectangle, then contradiction • Clairaut: Add Axiom “Rectangles exist”. • Legendre: “Accute angle” • Lambert: Quadrilaterals with three right angles • Bolyai

4. As Y moves away from P, the segment XY becomes larger and larger (without bound) • Eventually, XY>PQ, then Y and P are on different sides of l. So, l intersects n. Proclus • Let l and m be parallel lines. • Let n be a line that intersects m at P. We want to show that n intersects l. • Let Q be the foot of the perp. to l through P. • If n = PQ , we are done. • Assume n is not PQ . Then there exists Y in n and U in m such ray PY is between the rays PU and PQ. • Let X be the foot of the perp. to m through Y.

5. As Y moves away from P, the segment XY becomes larger and larger (without bound) • Eventually, XY>PQ, then Y and P are on different sides of l. So, l intersects m. • Let Z be the foot of a perpendicular to l through Y. Then • X, Y and Z are collinear • XZ and PQ are congruent. • Then when XY>PQ, XY>XZ. • Thus Z is between X and Y. • So Y and P are on different sides of l. Proclus • Let l and m be parallel lines. • Let n be a line that intersects m at P. We want to show that n intersects l. • Let Q be the foot of the perp. to l through P. • If n = PQ , we are done. • Assume n is not PQ . Then there exists Y in n and U in m such ray PY is between the rays PX and PQ. • Let X be the foot of the perp. to m through Y.

6. Legendre’s Theorem If for any acute angle <BAC, and any point D in the interior of <A there exist a line through D intersecting both rays AB and AC then The sum of the interior angles of a triangle is 180°

8. In problems 2 and 3 you need to find a way to work with the sign, taking in to account that the perfect solution may not exist. • 1.Write a script to construct an inscribed circle in a triangle ΔABC (that is, the incircle of a triangle with vertices A, B and C.) The “Given” information should be only three points, the vertices of a triangle. (You need to do some research to find out the meaning of “inscribed circle” and what the construction is.) • 2.Write a script to illustrate Menelaus’ Theorem (see exercise H-5 in page 287 of the textbook). The “Given” information should be three points A, B and C, and points D, E, F as described on the exercise. • 3.Write a script to illustrate Ceva’s Theorem (see exercise H-6 in page 288 of the textbook). The “Given” information should be three points A, B and C, and points D, E, F as described on the exercise • 4.Choose your favorite geometry theorem and write a script that illustrate. Do this in a separate page of your document and make sure you include the statement of the theorem you are illustrating. 8