MAT 360 Lecture 8

1. MAT 360 Lecture 8 • Neutral Geometry

2. Remarks • Midterm Grades • Next Tuesday: Sketchpad homework • 11/13: Chapter 4: 1,3,4,5,6 15 (estimated) • 11/20: Sketchpad project • 11/27: Proof project: • Choose one of the theorems we studied and write a detailed proof. • The proof should have at least 10 steps. • Each step must be numbered and justified. • 12/13: Chapter 5: 8 Chapter 6:: 2,3,5,14 (estimated)

3. Euclid’s postulates (modern formulation) • For every point P and every point Q not equal to P there exists a unique line l that passes for P and Q. • For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE. • For every point O and every point A not equal to O there exists a circle with center O and radius OA • All right angles are congruent to each other • For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.

4. In Chapter 3, • we will study results that can be proved using only the axioms of incidence, betweenness, congruence and continuity. • That is, all Hilbert’s axioms with the exception of the axiom of parallelism. • Neutral geometry consists in all the results that can be proved in this way.

5. Exercise: • Let t be a line intersecting the lines l and l’. • Define alternate interior angles.

6. Theorem (Alternate Interior Angle Theorem) • If two distinct lines cut by a transversal have a pair of congruent alternate interior angles then the two lines are parallel.

7. Corollaries • Two distinct lines perpendicular to the same line are parallel. • (consequence of 1.) The perpendicular dropped from a point P not on the line l to l, is unique. • If l is a line and P is a point not on l, then there exists at least one line through P parallel to l.

8. Question Can you prove… • If two lines are parallel and they are intersected by a third line, then the alternate interior angles are congruent. • Compare this statement with the Alternate Interior Angle Theorem: If two distinct lines cut by a transversal have a pair of congruent alternate interior angles then the two lines are parallel.

9. Definition • An angle supplementary to an angle of a triangle is called an exterior angle. • The two angles no adjacent to an exterior angle are called remote interior angles (of that exterior angle).

10. Theorem: Exterior angle theorem • An exterior angle is greater that either remote interior angle.

11. Consequences • (SAA) Given two triangles, ΔABC and ΔDEF. If AC ~ DF, <A ~ <D and <B ~ <E then the two triangles are congruent. • Two right triangles are congruent if the hypotenuse and a leg of one are congruent to the hypotenuse of a leg of the other. • Every segment has a unique midpoint. • Every angle has a unique bisector • Every segment has a unique perpendicular bisector

12. Theorem (Measure of angles) • There is a unique way to assign a degree measure to each angle with the following properties. • (<A)° is a real number and 0°< (<A)° <180° • (<A)°= 90° if and only if <A is a right angle. • (<A)°= (<B)° if and only if <A ~<B • Observe that we denote the measure of an )<A angle by (<A)°

13. Theorem (Measure of angles, cont) • If AC is a ray interior to <DAB then (<DAB)°= (<DAC)°+ (<CAB)°. • For every real number x such that 0<x<180, there exists an angle <A such that (<A)°= x° • If <B is supplementary to <A then (<A)°+(<B)°=180° • (<A)° < (<B)° if and only if (<A) < (<B)

14. Idea of the proof • We “know” which angles measure 90° • We can bisect angles, so we find angles that measure 45° and 90 °+45° • By bisecting again and “adding” we find angles of 22.5°, 45°+22.5°, 90°+22.5°,135°+22.5°, • And so on… • Using continuity one can show that it is possible to “squeeze” all the other angles between two of the ones above

15. Theorem (Measure of segments) • Given a segment OI (called a unit segment) there is a unique way of assigning a length AB to each segment such that the following property holds • OI =1 • AB is a positive real number • AB=CD if and only if AB ~ CD • A*B*C if and only if AC=AB+BC • AB<CD if and only if AB<CD • For every positive real number x, there exist a segment AB such that AB=x.

16. Corollary • The sum of two angles of a triangle is less than 180° • Exercise: Prove it using Exterior Angle Theorem and Measure of Angles Theorem.

17. Saccheri-Legendre Theorem • The sum of the interior angles of a triangle is less than or equal to 180°. • Idea of the proof: Use that given a triangle ΔABC, there exists a triangle ΔDEF, such that <E measures half of <A and the sum of the interior angles of ΔDEF is equal to ΔABC. • Needs Arquimedean property of real numbers.

18. Corollary • The sum of the measures of two angles of a triangle is less than or equal to the degree measure of their remote exterior angle.

19. Definition • Quadrilateral □ABCD is convex if it has a pair of opposite sides, let’s say AB and CD such that • AB is contained in one of the half planes bounded by the line CD • CD is contained in one of the half planes bounded by the line AB. • When conditions 1 and 2 hold we say that segments AB and CD are semiparallel.

20. Corollary of Saccheri-Legendre Theorem • The sum of the interior angles in any convex quadrilateral is less than or equal to 360° • Proof: Step 1: Show that if □ABCD is a convex quadrilateral then both pairs of opposite sides are semiparallel. In other words, AB and CD are semiparallel and BC and AD are semiparallel. • Step 2: Find triangles to apply Saccheri-Legendre.

21. Recall • Euclid’s Postulate V: If two lines are intersected by a transversal in such a way that the sum of the degree measures of the two interior angles on one side of the transversal is less than 180° then the two lines intersect on that side of the transversal. • Hilbert’s Axiom of Parallelism: For every line l and for every point P not in l there exists at most one line m through P such that l is parallel to m.

22. Theorem • If Euclid’s Postulate V holds then Hilbert’s Axiom of Parallelism holds. • If Hilbert’s Axiom of Parallelism then Euclid’s Postulate V holds • In other words • Euclid’s Postulate V if and only if Hilbert’s Axiom of Parallelism

23. All the following statements are equivalent to Hilbert Parallel Postulate • If line intersects one of two parallel lines then it intersects the other. • If two lines are parallel then alt. int. angles are congruent • If t is a transversal to a line l and m, l is parallel to m and t is perpendicular to l then t is perpendicular to m. • If k and l are parallel lines, m is perpendicular to k and n is perpendicular to l the m=n or m is parallel to n.

24. Definition • The angle sum of the triangle ΔABC is • (<A)°+ (<B)°+ (<C)° • The defect of ΔABC, denoted by δABC is • 180°-( (<A)°+ (<B)°+ (<C)°) • Question: Which values can be taken by the defect of a triangle?

25. Theorem: • Let ΔABC be a triangle and let D be a point between A and B. Then • δABC = δACD + δBCD

26. Corollary • Let ΔABC be a triangle and let D be a point between A and B. Then • δABC = 0 if and only if • δACD=0 and δBCD=0.

27. Equivalent definitions of rectangle in Euclidean Geometry • A quadrilateral with four right angles • A quadrilateral with four angles congruent to each other • A parallelogram with at least one right angle.

28. Definition • A rectangle is a quadrilateral whose four angles are right angles.

29. Theorem • If a triangle with an angle sum of 180° exists then a rectangle exists. • If a rectangle exists then every triangle has angle sum of 180°.

30. The steps of the proof • Construct a right triangle with defect 0. • Construct a rectangle • Given a right triangle ΔABC where <ABC is the right angle, construct a rectangle □DEFG such that AB < DE and BC<EF. • Prove that all right triangles have defect 0. • Prove that all triangles have defect 0.