2.1 Angles, Lines and Parallels & 2.2 Congruent Triangles and Pasch s Axiom

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1 2 Euclidean Geometry In the previous section we gave a sketch overview of the early parts of Euclid s Elements. While the Elements set the standard for the modern axiomatic approach to mathematics, it has many weaknesses. In particular, Euclid s system is incomplete: many assumptions are made implicitly (existence of objects, continuity of lines, etc.). Over the centuries, mathematicians tried to find a complete axiomatic system for Euclidean Geometry, culminating in the work of Hilbert (1899) and irkhoff (1932). While both systems are in fact incomplete 1 the pursuit cannot be considered wasted. In this section we list Hilbert s axioms and show how he attempted to fix some of the problems inherent in Euclid. Later in the course we shall discuss irkhoff s system for nalytic Geometry (geometry using co-ordinates). 2.1 ngles, Lines and Parallels & 2.2 ongruent Triangles and Pasch s xiom Hilbert s axioms for plane geometry are shown on the next page. The undefined terms consist of two types of objects (points and lines), and three relations (between, on and congruent.) t various places during the list of axioms, definitions need to be made. efinition 2.1. The expression denotes the line through distinct points and. Lines l and m intersect if there exists a point lying on both. Lines are parallel if they do not intersect. Segments and are parallel if and only if the lines and are parallel (similarly for rays). Given a line l and two points,, we say that, lie on the same side of l if the segment does not intersect l. Otherwise and are on opposite sides of l. The segment consists of distinct points and and all the points on the line though and such that lies between and. The ray is the segment together with all points on the line through and such that lies between and. triangle consists of line segments, and where,, are three points not on the same line. Two triangles are congruent if corresponding sides and angles are congruent. n angle with vertex consists of a point together with its sides: two rays and. We denote the angle. It is important to note that, in the usual model of plane geometry, a line extends infinitely in both directions, a segment is bounded, and a ray extends infinitely in one direction. Line Segment Ray ngle 1 consequence of Gödel s First Incompleteness Theorem.

2 Hilbert s xioms for Plane Geometry Undefined terms 1. Points (use capital letters, e.g.,,, ) 2. Lines (use lower case letters, e.g., l) 3. etween (a point lies between points,, written ) 4. On (a point lies on a line l) 5. ongruence = (for segments, angles, triangles, etc.) xioms of Incidence I-1 Through any distinct, there exists a line l. I-2 There is at most one line through distinct,. Notation: line through and I-3 On every line there exist at least two distinct points. There exist at least three points not all on the same line. xioms of Order/etweenness O-1 If, then, and are distinct points on the same line and. O-2 For any distinct points and, there is at least one point on the line through and such that. O-3 If,, are distinct points on the same line, exactly one lies between the other two. efinitions: segment and triangle O-4 (Pasch s xiom) Let be a triangle and l a line not containing any of,,. If l contains a point of the segment, then it also contains a point of either or. xioms of ongruence efinition: ray -1 If, are distinct points and is another point, then for each ray r from there is a unique point on r such that = and =. -2 If = and = EF, then = EF. Moreover, every segment it congruent to itself. -3 If,, = and =, then =. efinitions: angle, side of line -4 Given and a ray, there is a unique ray on a given side of such that =. -5 If = EF and = GHI, then EF = GHI. Moreover, every angle is congruent to itself. -6 (Side-ngle-Side) Given triangles and, if =, =, and =, then the triangles are congruent. xiom of ontinuity Suppose that the points on l are partitioned into two non-empty subsets Σ 1, Σ 2 such that no point of Σ 1 lies between two points of Σ 2, and vice versa. Then there exists a unique point O lying on l such that P 1 O P 2 if and only if O = P 1, O = P 2 and one of P 1 or P 2 lies in Σ 1 and the other in Σ 2. efinition: parallel lines Playfair s xiom Given a line l and a point P not on l, one may construct exactly one line through P parallel to l. 2

3 asic Theorems Theorem 2.2. If two lines are not parallel, then they meet in exactly one point. Proof. Suppose that, are distinct points of intersection. y axiom I-2, there is exactly one line through and. ontradiction. Theorem 2.3 (ngle-side-ngle). Suppose that triangles and EF satisfy = EF, = E, = EF Then the triangles are congruent. Proof. Given the assumptions, axiom -1 tells us that we can construct a point G on the ray EF such that EG =. If G = F, we are done: = EF allows axiom -6 (side-angle-side) to tell us that the triangles are congruent. Suppose instead, for a contradiction, that G = F. xiom -6 says that = EG. Then EG = = EF, by assumption. Since F and G lie on the same side of E, axiom -4 says that they lie on the same ray through. ut then F and G both lie on two distinct lines: contradicts axiom I-2. EF and G. This G F E Length and angle measure bsolute length and angle measure are not present in the geometry of Euclid, nor are they mentioned in Hilbert s axioms. In a strict interpretation of either system, length and angle can only be compared: line segments or angles may be congruent, greater or lesser in magnitude to one another, or even come in ratios of magnitudes. Example Suppose that a point lies between points and in such a way that =. We would say that the lengths of and are equal, that is double the length of and that is the midpoint of. This is tedious. We are much more comfortable assuming the existence of a common scale and comparing everything to that. To facilitate this, we add a few postulates about distance: 2 1. To every pair of distinct points, there corresponds a unique positive number called the distance between the points. 2. = =. 2 These are a modified version of the School Mathematics Study Group axioms for distance developed in the 1960 s for teaching high school geometry. 3

4 3. For any line l, there is a bijective function p : l R such that the distance between points and on l is p() p(). 4. Given distinct points and on a line, p can be chosen such that p() = 0 and p() > 0. efinition 2.4. triangle is equilateral if the lengths of its three sides are equal. Let O and R be distinct points. The circle with center O and radius OR is the set of all points such that O = OR. point P is inside the circle if P = O or OP < OR. point Q is outside if OQ > OR. point is the midpoint of a segment if lies between and and =. R Equilateral Triangle M Midpoint P O ircle bisector of an angle is a ray such that intersects at a midpoint M. ngles with a common vertex and whose sides form two lines are called vertical. If two angles share a common side and the other sides lie in opposite directions on the same line, then the angles are supplementary. right angle is an angle which is congruent to any supplementary angle. Intersecting lines are perpendicular if one (both) of the angles made at the intersection is a right angle. Q isector Vertical ngles Supplementary angles Right angles ll these definitions could be stated more awkwardly in terms of congruent segments. ngle Measure This can be defined similarly to length: 1. To every angle there corresponds a degree measure m : a real number between 0 and m = m EF = EF. 3. Suppose that lies on one side of a line such that =. The degree measure provides a bijective correspondence between such points lying on the same side of the line and the interval (0, 180). 4

5 4. Suppose P is a point lying between the rays and in the sense that there exist points, E on these rays with P E. Then m + m = m. 5. Supplementary angles are assumed to sum to 180. None of this is strictly necessary, though it makes everything much easier for us and simplifies several common definitions. Working without the definitions is entirely possible. For a challenge, see if you can fix some of the following discussion by removing all references to length and angle measure... side: The xiom of ontinuity To do much with Hilbert s geometry, a major weakness of Euclid has to be overcome: the issue of continuity. Many proofs rely on the intersections of lines and circles, without which Euclid s ruler and compass constructions useless. If you ve studied edekind cuts in analysis, you might find the phrasing of the axiom familiar. Theorem (Elementary ontinuity Principle) If is a circle such that P is inside and Q is outside the circle, then the segment PQ intersects in exactly one point. 2. (ircular ontinuity Principle) If and are circles such that contains at least one point P inside, and one point Q outside,, then the circles intersect in exactly two points. oth statements can be proved from the xiom of ontinuity. The rough idea of the first is to consider all the points Σ 1 on the segment PQ which lie inside or on, and to let Σ 2 be the points on the segment outside of. One shows that Σ 1 and Σ 2 satisfy the assumptions of the xiom: the unique point Õ whose existence is guaranteed by the xiom is then shown to lie on the circle itself. full discussion lies well beyond the difficulty of this class! In the standard model of Euclidean geometry, where lines and circles are what we pictorially think of as lines and circles, the discussion is irrelevant; both continuity principles are pictorially trivial. The challenge is that the principles must to hold in any model. P Σ Q 2 Õ Σ 1 rmed with our definitions, our concepts of length and circle, we can now prove the first proposition in the Elements using a fixed version of Euclid s proof. Theorem 2.6 (Euclid I.1). We may construct an equilateral triangle on a given segment. Proof. onstruct the circle α centered at passing through. Let be the point on such that is the midpoint of. onstruct the circle β centered at passing through. β contains (inside α) and (outside α). y the ircular ontinuity Principle, the circles intersect in precisely two points P, Q. Since P lies on both circles, it follows that = P = P, whence is equilateral. α P β Q 5

6 isectors and Isosceles Triangles We discuss the following result on isosceles triangles: meaning equal legs, so isosceles triangles have two sides congruent. Theorem 2.7 (Euclid I.5). The base angles of an isosceles triangle are congruent. Euclid s proof relies on a complicated construction following from the Side-ngle-Side theorem (Proposition I.4). Unfortunately his proof of SS relies on superposition: i.e. placing one triangle on top of the other. This concept is not supported by his axioms, whence Euclid s proof is broken. One could attempt to fix this in Euclidean style by using the proof of Proposition 10. Theorem 2.8 (Euclid I.9,10). Given an angle (segment), the bisector of that angle (segment) may be constructed. Here is a sketch of a fixed proof of the isosceles triangle theorem. isect the angle via a ray where is inside the triangle. Let the intersection of the bisector with be M. Side-ngle-Side implies that M = M. Therefore = M = M =. There are three problems with this: M Euclid s system should proceed in such a fashion that earlier results do not depend upon later ones! Using Proposition 9 to prove Proposition 5 is no good! What does inside the triangle mean? How do we know that the bisector of the angle actually intersects the segment? This second and third problems are common in Euclid. The bisector starts off inside the triangle and ends up outside, so it must cross, correct? Euclid makes no such assumption in his axioms, and does not define inside. The sketched argument is therefore broken. Or is it... Problems such as these led to the creation of Pasch s xiom 3 which was incorporated into Euclidean Geometry as Hilbert s xiom O-4. It states that if a line passes through one side of a triangle and doesn t pass through any of the corners, then it must also pass through one (but not both) of the other sides. With careful use of Pasch s xiom, the inside of a triangle can be properly defined and one can show that a line passing through a corner lying partly inside a triangle must also cross an edge. s a fix for Proposition I.5, this is a heavy lift! Here are some of the details. 3 fter Moritz Pasch in

7 Proof of Theorem 2.7. Suppose WLOG we have = so that is isosceles. onsider a new triangle = where =, = and =. We have: = =. This is a consequence of axiom -4. Thus =. = = = and =. y xiom -6 (Side-ngle-Side) we have =. onsequently the angles and = are congruent. = = The proof is very sneaky: just relabel the original triangle and use SS! Proof of Theorem 2.8. To bisect an angle, first construct the point on the ray such that = (axiom -1, or equivalently construct the circle centered at passing through... ) Now draw circles centered at and with radii. These intersect at two points, call one of them E. raw the ray E. y construction, =, E = E. We have two isosceles triangles ( E and ), and thus congruent base angles by the previous result. It follows (sum angles) that E = E. SS then says that E = E, whence angles E and E are congruent. ecause of Pasch s axiom, the sketch proof above is now valid. E Triangle ongruence Theorems We have already seen SS (xiom -6) and S (Theorem 2.3). The remaining angle theorems are SSS and S. It is worth noting that the remaining combinations ( and SS) are not theorems. Theorem 2.9. onsider triangles and EF. Side-Side-Side If = E, = EF and = F then the triangles are congruent. ngle-ngle-side If = E, = EF and = EF then the triangles are congruent. Proofs of these results in Hilbert s system require a little more work, though it is worth noting that they do not depend on Playfair s xiom (Euclid s Parallel Postulate). Euclid similarly proves all four triangle congruence theorems before invoking the parallel postulate for the first time. This implies that congruence theorems are valid in any model satisfing all the axioms except the parallel postulate. The equivalence of Playfair s xiom and Euclid s Parallel Postulate were discussed in hapter 1. 7