Axioms for Euclidean Geometry: Points, Lines, Planes, and Distance, Study notes of Mathematics

Download Axioms for Euclidean Geometry: Points, Lines, Planes, and Distance and more Study notes Mathematics in PDF only on Docsity! Math 3305 Euclidean Axioms Introduction to Axiomatic Systems The SMSG Axioms for Euclidean Geometry Points, Lines, Planes, and Distance Axioms 1, 2, 3 Coordinates Axioms 3, 4, 5, 6, and 7 Polygons Axiom 8 Convexity and Separation Issues Axioms 9 and 10 Angles Axioms 11 – 14 Congruent Triangles Axiom 15 Parallel Lines Axiom 16 Area and Congruent Triangles Axioms 17 – 20 Volumes and Solids Axiom 21 and 22 Answers to exercises Glossary Appendix A Why Does the Geometric Coordinate Formula Work? Appendix B Theorem List 1 Introduction to Axiomatic Systems In studying any geometry, it is important to note the axiomatic framework of the geometry and keep it in mind. Often students are so challenged by the details that they forget that there is a structure to geometry. Each geometry has a framework called its axiomatic system. An outline of a typical axiomatic system is below. Any axiomatic system has four parts: undefined terms axioms (also called postulates) definitions theorems The undefined terms are a short list of nouns and relationships. These terms may be visualized but cannot be defined. Any attempt at a definition ends up circling around the terms and using one to define the other. These are the basic building blocks of the geometry. It is usually a good idea to have a mental image of the undefined terms – a visualization of the objects and how they relate. Axioms (or postulates) are a list of rules that define the basic relationships among the undefined terms and make clear the fundamentals facts about a system. Axioms are always true for the system. No deviation from the facts they state is permitted in working with the system. Definitions and theorems build on the axioms and undefined terms, clarifying relationships and auxiliary facts. We will be using, with slight modification, the set of undefined terms and axioms developed by The School Mathematics Study Group during the 1960’s for this module. This list of axioms is not as brief as one that would be used by graduate students in a mathematics program nor as long as some of those systems in use in middle school textbooks. One definite advantage to the SMSG list is that it is public domain by design. We will be using the Cartesian coordinate plane as our visualization of the undefined terms of Euclidean geometry. Once we have spent time learning the axioms, some definitions and a few theorems we will move to the second module on Euclidean Topics and look at geometric shapes and proofs that require using the axioms, definitions, and theorems in concert We are studying Euclidean Geometry in this module and it is assumed that you know quite a bit already. We will find more axiomatic systems in the Other Geometries module. 2 A10. The Space Separation Postulate: The points of space that do not line in a given plane form two sets such that A. each of the sets is convex, and B. if P is in one set and Q is in the other, then the segment PQ intersects the plane. A11. The Angle Measurement Postulate: To every angle there corresponds a real number between 0 and 180. A12. The Angle Construction Postulate: Let AB be a ray on the edge of the half-plane H. For every r between 0 and 180 there is exactly one ray AP with P in H such that m  PAB = r. A13. The Angle Addition Postulate: If D is a point in the interior of  BAC, then m  BAC = m  BAD + m  DAC. A14. The Supplement Postulate: If two angles form a linear pair, then they are supplementary A15. The SAS Postulate: Given an one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence. A16 The Parallel Postulate: Through a given external point there is at most one line parallel to a given line. A17. To every polygonal region there corresponds a unique positive number called its area. A18. If two triangles are congruent, then the triangular regions have the same area. A19. Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2. A20. The area of a rectangle is the product of the length of its base and the length of its altitude. 5 A21. The volume of a rectangular parallelpiped is equal to the product of the length of its altitude and the area of its base. A22. Cavalieri’s Principal: Given two solids and a plane. If for every plane that intersects the solids and is parallel to the given plane, the two intersections determine regions that have the same area, then the two solids have the same volume. 6 Points, Lines, Planes, and Distance First, read the list below and see what comes to mind as you read. Then we’ll look at them a little more closely with a bit more detail. Taken together, the first 8 axioms give us all that we need to get started with Euclidean Geometry. The relationship between points and lines, points and planes, and lines and planes are specified. The basics of distance and location are given and the stage is set for us to use the Cartesian coordinate plane as our model. A1. Given any two distinct points there is exactly one line that contains them. A2. The Distance Postulate: To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points. A3. The Ruler Postulate: The points of a line can be placed in a correspondence with the real numbers such that A. To every point of the line there corresponds exactly one real number. B. To every real number there corresponds exactly one point of the line, and C. The distance between two distinct points is the absolute value of the difference of the corresponding real numbers. A4. The Ruler Placement Postulate: Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive. A5. A. Every plane contains at least three non-collinear points. B. Space contains at least four non-coplanar points. A6. If two points line in a plane, then the line containing these points lies in the same plane. A7. Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane. A8. If two planes intersect, then that intersection is a line. 7 Going back to A2 and looking at a number line: we can select the point associated with the real number 1 and talk about distances. How many numbers are a distance 5 from 1? What are the place names? (−6 and 4) Does have two points at a distance 5 from 1 violate A2? The Ruler Postulate also guarantees us an infinite number of points by stating that there are as many points as there are real numbers. Note that in the earlier axioms we were talking only of two points or some limited number of points. It is necessary for people to know how many points there are and this is the axiom that tells us. One way to distinguish among all the points is to specify points that all have the same characteristic property. For example if we are interested in all the points on a specific line we might say that all the points are collinear. We say that a point A and a point B are collinear if they are on the same line. In algebra and Cartesian coordinate geometry, we use the following notation to denote collinear points: }bmxy)y,x{(  where m and b are specific real numbers. All the points in this set are collinear. We may then speak of the points between A and B. We will say that a point C is between A and B, denoted A – C – B, if and only if AC + CB = AB. This is a test for collinearity. Example: Look at the points A = ( 2, 0), B = ( 5, 0), and C = (1, 1). Are these points collinear? Is A – C – B? (which is to say, is C between A and B or on the same line as them?) If C is, then AC + CB = AB by the definition above. 10 1 AB = 7 AC = 10  3.16 CB = 17  4.12 So AC + CB is 7.28 which isn’t 7. Since the distances don’t add up correctly, C is not between A and B. Check the algebra by graphing the points on a piece of graph paper. Is C between A and B on the graph paper? So C doesn’t share the property “collinearity” with A and B. More definitions: A, B, and all points C such that A – C – B form the segment AB . If we wish to focus solely on the points between A and B we may speak of the open segment AB . We can look at a point D such that A – B  D, thus we have a ray terminating at A and extending past B, denoted AB  . If we mean points D such that D – A – B we should reverse the location of A and B under our ray symbol and say BA  or reverse the ray symbol itself and say AB  . And we note that the union of AB  and BA  is the line AB  . If we are working with a segment and need to extend the segment to a ray or a line, we may do so. Similarly, if we have a line and find the need to focus only on part of it, we may speak of a ray or a segment. Theorem Given a segment AB , there is exactly one point C such that A – C – B and AC = CB. This point is called the midpoint of the segment. In coordinate geometry, a segment is specified by listing the endpoints. You may calculate the midpoint of any segment by using the midpoint formula. If the endpoints are point A = (x1, y1) and point B = (x2, y2), the coordinates of the midpoint C are        2 yy , 2 xx 2121 Midpoint Exercise: Working with the same segment as above with A = ( 1, 2) and B = ( 1, 6), find the coordinates of C, the midpoint. vnet: Coordinates Coordinates 11 A close reading of Axiom 3 shows that in pure geometry we have one number that specifies a point on a line, not two as in Cartesian Plane. We call this number the geometric coordinate to distinguish it from the Cartesian coordinates of a given point. We will spend some time exploring this notion. We start with a horizontal number line in one dimension. Identify points on the line with the real numbers A = 5, B = 1, C = 0, D = 3, and E = 7. Note that each of these locations is specifically identified with exactly one point that has one real number associated with it. Now in order to calculate the distance from E to B, take the absolute value of the difference of the real numbers in either order: 817  . Note that EB = BE because we are using the absolute value of the difference. This is exactly what the postulate stipulates. This works perfectly well with vertical number lines, too. Try it with points Q = 12 and R = 5. QR = 125  = 512  . There’s no real difference between the number lines except orientation. The horizontal line has slope = 0 and the vertical line has an undefined slope. When we combine the vertical number line and the horizontal number line, intersecting them at the point 0 of each line, we have the structure to move into the Cartesian plane and begin discussing coordinate geometry. NOTE THAT the postulate applies to the lines we are looking at as axes AND to every other line in the Cartesian Plane. In fact, reading it closely shows that we must be able to find a real number for each point on any line. (especially those with slopes that are any real number, m > 0). Further, 12 The points (1, 0) and (0, 3 ) are on the line 3 3y x  . Here’s a sketch of the points and the line done in Math GV: What are the geometric coordinates for the points and the distance between them? 21 2m  which is a very nice number So the geometric coordinate for (1, 0) is 2 and the geometric coordinate for (0, 3 ) is 0 and the distance between the two points is 2 units. Coordinate Exercise: Find the geometric coordinates and the distance between the points (1, 2) and (3, 10), Check your work using the Euclidean distance formula. Application Exercise: An application of this idea is to explore the following situation: Two segments are said to be congruent segments if they have the same length. 15 Are the two segments defined by the following endpoints congruent? Segment A has endpoints ( 5, 2) and (7, 2) Segment B has endpoints (0, 5) and (12, 9) Find the coordinates of each endpoint. Get the distances: What do these segments look like when graphed on the coordinate plane? 16 A4 The Ruler Placement Postulate: Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive. Suppose you have a number line and you have 5 points on the number line: A, B, C, D, and E in this order, left to right. This axiom allows you to choose which point will be associated with the number zero; you may choose any one of these points to be assigned the real number 0 as a coordinate. We have already looked at axioms that say there is a distance between each of these points and that we can assign real number coordinates to each point so that the absolute value of the difference of the numbers is the distance. Ruler Placement Exercise: Suppose we have the following distances between the points: AB = ½ BC = 3 CD = 2.5 DE = 1 What is AE ? (remember that our notation for distance is just the two point names placed side by side) What is BD? Here is a horizontal number line with the points on it. Suppose we pick C and D to be the two points in the axiom above. So C will have the coordinate zero. What are the coordinates of all the other points? 17 A B C D E Looking at A5, part B, suppose you take the shape from A5 part A and put a point 2 inches up from the plane somewhere over the middle of the shape in the plane. What solid do you have? Do you see that we have now moved from 1 dimensional work with numbers on lines through two dimensions and now into 3 dimensions? You should be visualizing a tetrahedron: A tetrahedron is three dimensional figure, commonly called a pyramid, with 4 points, 6 edges and 4 sides. A6. If two points line in a plane, then the line containing these points lies in the same plane. This axiom gives us the relationship between planes and lines. It ties A1 and A5 together. This just makes sure that lines stay put in planes and don’t somehow sag their way into being 3 dimensional. 20 A7. Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane. We say points are “collinear” if they are on the same line. Non-collinear means that one line doesn’t have all the points on it; that there is a second line or a third line has some of the points on it as well. If you take any 3 points, it is possible that they are collinear. How many planes pass through 3 collinear points? We will illustrate this with a piece of paper. Fold the paper in half the long way and put a line with 3 points showing along the fold line. Do you see that the plane of the paper is one plane containing the points? Now hold the paper up and look at it with the fold line pointing in between your eyes, support the line with your fingers and let the sides of the paper drop naturally. Do you see that the halves of the paper can be viewed as the visible parts of two planes, one on the left going straight through the line and one on the right going straight through the line? Now shift the edges of the paper – you have two more planes. How many planes go through the line? Now flatten out the paper we used above and put a point on the paper that is not on the line. Can you find another plane that contains two of the points on the line and this new point? Will all three of these points be in more than one plane if you pick up the paper view it as we did above? 21 Polygons Definition A polygon is a closed figure in a plane. It has 3 or more segments, called sides, that intersect only at their endpoints. Each point of intersection is called a vertex. No two consecutive sides are on the same line. figure B figure A I B C D A F HE G Figure A is a type of polygon called a quadrilateral; figure B is not a polygon because sides EI and IG are on the same line EG . Ditto for the line FIH and the two sides it contains. Two polygons are said to be congruent polygons if corresponding angles are congruent and corresponding sides are congruent. A solid figure formed by joining two congruent polygonal shapes in parallel planes is called a prism. The sides of a prism are parallelograms. Is it necessary that each side of a prism form a 90 angle with the plane of the base? Not according to our definition. Please check the definition of the book you’re teaching from, however, because sometimes authors use a definition different from ours. Take a rectangular box. If you slide the top over a couple of inches horizontally so that it’s no longer over the base while keeping the sides are still attached, you will make a prism. The sides are now parallelograms instead of rectangles. What is the angle between the tabletop that the base is sitting on and the front of the prism? Not 90°, for sure. 22 Convexity for a set is defined: If you take two points in the set and connect them, then all of the points inbetween them lie in same set at the two original points. The interior of a circle is a convex set; so is the interior of a square. Here’s a picture of polygon interior that is NOT convex. Connect points A and B to make a segment to see why. B A These two axioms discuss convexity of the plane and space. It is important that we know the relationship that exists between two halves of any plane and that of space. Axiom 9 says that each half-plane is a convex set and Axiom 10 says that the spaces above and below a plane are convex sets. We do need to deal with one new concept when we move from triangles to polygons of more sides. We need a working definition of convex polygon. Many of the theorems we will encounter refer to convex polygons; these are the polygons in which a segment connecting any two points of the polygon is entirely inside the polygon. Which is to say that the open segment between the two points lies entirely in the interior of the polygon. 25 Angles A11. The Angle Measurement Postulate: To every angle there corresponds a real number between 0 and 180. A12. The Angle Construction Postulate: Let AB be a ray on the edge of the half-plane H. For every r between 0 and 180 there is exactly one ray AP with P in H such that m  PAB = r. A13. The Angle Addition Postulate: If D is a point in the interior of  BAC, then m  BAC = m  BAD + m  DAC. A14. The Supplement Postulate: If two angles form a linear pair, then they are supplementary. A11. The Angle Measurement Postulate: To every angle there corresponds a real number between 0 and 180. This postulate allows us yet another figure in the plane. An angle is a shape created by two rays that share a common endpoint called the vertex of the angle. Each ray is called a side of the angle. The angle divides the plane into three sets of points. Those points on the rays and the vertex form one set. The second set of points is made up of the points that are interior to the angle. The points that are not part of the angle and not part of the interior are called exterior to the angle and form the third set. In order to visualize the interior of an angle, we need to have the angle and a line that intersects the rays or sides of the angle. Definition Any line that intersects two other lines or parts of lines is called a transversal of the lines, rays, or segments intersected. (It is not necessary for the lines so intersected to be parallel lines!) 26 The points that are between the points of intersection of the rays and the transversal are the points in the interior of an angle. Angle ABC, denoted  ABC is shown with transversal AC as a dashed line. Which parts of AC are interior points of  ABC and which are exterior? Note, that since an angle measure cannot equal 0 nor 180 we have no ambiguity about how to find an angle’s interior. Note that an angle itself is not a convex set, The points between A and C are NOT part of the angle while A and C themselves are part the angle. An angle’s interior, on the other hand, is a convex set. 27 B A C And they get that by connecting two opposite corners and noting that the quadrilateral gets broken down into two triangles and that the sum of the interior angles of the two triangles is 180° each…hence 360° Illustration: What happened to the polygon preceding? There’s a subtle but real difference between these two quadrilaterals. One is convex and the other isn’t. The problem is with  CDA. Note that the interior of this angle is NOT the same set of points as interior of the quadrilateral. We are required to measure the angle with an angle measure between 0 and 180 so we end up measuring it “outside” the quadrilateral. In other areas of mathematics it is possible to have an angle that measures more than 180 but not in a geometry that uses the axioms that we’re using. So we will restrict ourselves to working with polygons that are convex. With convex polygons, a line segment from one point on the side of the polygon to another point on a side of the polygon is made up entirely of points from the interior of the polygon except where the segment touches the polygon. When we talk about the measures of the interior angles of a polygon, we mean those angles whose interiors coincide with the interiors of the angles themselves. Polygons constructed from congruent segments and having congruent interior angles are called regular polygons. 30 Regular polygon exercise: What is the common name for a regular polygon with 3 sides? With 4 sides? Discovery Lesson on Interior Angles of a Convex Polygon We will attempt to come up with a formula for the sum of the interior angles of any convex polygon. A. What is the sum of the interior angles of a triangle? How do you know this? How many sides? How many triangles? Sum: B. Use A19 to come up with the sum of the interior angles of an arbitrary convex quadrilateral? (hint decompose it!) How many sides? How many triangles? Sum: Will this be true of a square (the common name for a regular quadrilateral)? C. Use Sketchpad and A19 both in separate tries to come up with the sum of the interior angles of a convex pentagon. How many sides? 31 How many triangles? Sum? Will this be true of a regular pentagon? D. Fill in the following chart until you see the pattern: object how many sides? how many triangles? sum triangle 3 1 180 quadrilateral 4 pentagon 5 hexagon heptagon octagon nonagon decagon ngon formula: Definitions An angle with a measure less than 90 is called an acute angle. An angle with a measure of 90 is called a right angle. An angle with a measure greater than 90 is called an obtuse angle. 32 Definitions, continued Two angles, A and B, with the same measure are called congruent angles. We will use the notation  A   B. Adjacent angles are a pair of angles with a common vertex, a common side, and no interior points in common. Adjacent angles whose non-shared rays form a straight line are called a linear pair. ABD and DBC are a linear pair 35 Angle ABC and Angle CBC are adjacent B D CA A C D B Two angles that are not adjacent and yet are formed by the intersection of two straight lines are called vertical angles. ABE and DBC are vertical angles. ABD and CBE are vertical angles. 36 B A C D E Theorem and proof: Vertical angles are congruent. Let AB be any line, P be a point on it such that A – P  B. Let Q be a point not on AB . There is a line joining P and Q (Postulate 1). We can select point D on QL such that Q  L  D and speak of line QD (Theorem after Postulate 1). By definition, QPA and  BPD are vertical angles as are  QPB and  APD. Now, by Postulate 14 E1 m  QPA + m  QPB = 180 E2 m  QPB + m  BPD = 180 E3 m  BPD + m  APD = 180 E4 m  BPD + m  QPB = 180 We may subtract E2 from E1 giving m  QPA = m  BPD so  QPA   BPD Similarly we may subtract E4 from E3 giving m  APD = m  QPB and  APD   QPB. 37 P A B Q D A14. The Supplement Postulate: If two angles form a linear pair, then they are supplementary. Note that you need an axiom that says this even though it’s “obvious” from the sketch. In math, sketches are not proofs and, believe it or not, nobody can prove this. It has to be an axiom, a basic “given”. ABD and DBC are a linear pair mABC + mDBC = 180° Looking at a special situation, suppose we have two lines making vertical angles and one of the angles measures 90. Then by an earlier theorem above, the vertical angle is a right angle and by the linear pair postulate the angle paired with each vertical angle is a right angle. We call these lines perpendicular lines. If we regard a segment or a ray on each line with an endpoint of each segment at the point of intersection of the original lines, we have a single right angle formed of the segments or the rays. We then say that the segments or rays are perpendicular. Theorem and proof: 40 A C D B The angle bisectors of a linear pair are perpendicular. To understand this theorem, we will take an arbitrary linear pair (there is always a temptation to use the special pair that has a perpendicular shared ray, but then we will have illustrated a single special case rather than choosing one of the more numerous general cases) ACB and  BCD are the linear pair. By P14, we know that mACB + m BCD = 180. The angle bisector of  ACB makes two angles of ½ m  ACB which we will call x. This means that m ACB = 2x. Similarly the angle bisector of  BCD makes two congruent angles of that are ½ m  BCD which we will call y so m BCD = 2y. Using substitution we have 2x + 2y = 180 which is equivalent to x + y = 90. This means that  ECF is a right angle which means the rays are perpendicular. 41 y yx x C B A D E F Congruent Triangles A15. The SAS Postulate: Given an one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence. A one-to-one correspondence associates each vertex and side of one triangle with a particular vertex and side of a second triangle. The order of the vertices in the list gives the correspondence. If we have ABC and we set up a correspondence with a triangle that has vertices X, Y, and Z, one correspondence is to say ABC corresponds to XYZ. A totally different correspondence would be ABC corresponds to YXZ. A one-to-one correspondence has six parts: 3 vertex pairs and 3 side pairs. If under some correspondence two triangles have all six pairs with equal measures, then the triangles are said to be congruent. It is true, then that corresponding parts of congruent triangles are congruent. This fact generalizes to the familiar dictum: Congruent parts of congruent figures are congruent. This is always true and we will use the acronym: CPCF when we mean to say this in a proof. Axiom A15 allows us to declare two triangles congruent by checking only 2 sides and the included angle of each…thereby cutting our work in half. I will use the symbol:  to denote congruence. Theorem: ASA congruence If two angles and the included side of one triangle are congruent to the corresponding two and included side of another triangle, then the triangles are congruent. Theorem: SSS congruence If, under some correspondence, three sides of a triangle are congruent to the three sides of another triangle, then the triangles are congruent. 42 Parallel Lines A16. The Parallel Postulate: Through a given external point there is at most one line parallel to a given line. Here’s the Euclidean setup. At most one line through C is parallel to line AB. A B C This is not true in many other geometries – we’ll see a couple at the end of the semester. It is necessary to have a line to compare against to determine if two lines are parallel. The candidates for being parallel and a transversal for the comparison is the usual set up. There are many ways to prove that two lines are parallel when you have a transversal for comparing measures. 45 Definitions The nonadjacent angles on opposite sides of the transversal and on the exterior of the two lines are called alternate exterior angles. The nonadjacent angles on opposite sides of the transversal but on the interior of the two lines are called alternate interior angles. The nonadjacent angles on the same side of the transversal and in corresponding locations with respect to the non  transversal lines are called corresponding angles. The nonadjacent angles on the same side of the transversal that are in the interior of the two lines are called interior angles on the same side. Parallel Lines Vocabulary Exercise: Here is a pair of horizontal parallel lines and a transversal. Pick out pairs of alternate exterior angles corresponding angles alternate interior angles interior angles on the same side 46 hg fe dc ba vnet: if and only if Here’s a fact about math and communication: information compression is considered a plus and all of the complicated symbols used by mathematicians are to compress facts to a small size and to communicate in information bursts. One fairly typical compressing statement is “if and only if”. Given two symmetrically stated facts or properties in a implication, which is to say that you have a pair of If …then… statements with the statements reversed, mathematicians will write them as one sentence. Here’s an example: Suppose we know: If A, then B and also if B, then A. Mathematicians will quickly write: A if and only if B. (even shorter, among friends: A iff B.) It’s all about saying a whole lot briefly. Here is a theorem that is written as an “iff” single statement. Let’s pick it apart and find out what we can. Two lines are parallel if and only if a pair of alternate exterior angles around a transversal are congruent. First rewrite it as two theorems (ie, undo the if and only if). Then analyze it. If two lines are parallel, then a pair of alternate exterior angles around a transversal are congruent. At the same time: If a pair of alternate exterior angles around a transversal are congruent, then the two lines are parallel. Note that the first statement is a fact about parallel lines: If you know the lines are parallel then you know something. 47 m  BAC + m  ABC + m  BCA = 180, as desired. 50 Sum of the Angles Exercise: In CAI and in BIN,  C   BNI. Prove that  A   NBI. What do you know about the two triangles? Do you recognize a fact from the Trigonometry section? 51 N A C I B Exterior Angle Theorem. An exterior angle to triangle is created when one side is extended to a ray. The adjacent triangle side at the extension point and the ray form an exterior angle. How many exterior angles are there at each vertex? How many exterior angles are there in all for a triangle? A D B C  BAC and  ABC are called remote interior angles and  BCA is called the adjacent interior angle to the exterior angle  BCD. Note that an exterior angle and it’s adjacent interior angle are a linear pair so the sum of their measures is 180. From the theorem immediately preceding we know that the sum of the interior angles of the triangle is also 180. Thus: m  ABC + m  BCA + m  BAC = m  BCA + m  BCD Subtracting the m  BCA from both sides we find that the sum of the remote interior angles is the same as the measure of the exterior angle. m  ABC + m  BAC = m  BCD This is known as the Exterior Angle Theorem: The sum of the measures of the two remote interior angles is the same as the measure of the exterior angle across from them. 52 The simplest polygon has three sides and is a triangle. The formula for the area of a triangle is A = heightbase 2 1  = sinab 2 1 . The height, recall is an altitude line. examples The dotted lines being used as altitudes. Neither is the only choice for the example. What would one other choice of base and height be for each triangle? Note that EF is perpendicular to the extension of the side HG . It is not a requirement that an altitude be located on the interior of a triangle. . Note, though, that the axiom does NOT discuss formulas. It just stipulates that there is some number called area and that it’s positive and that there’s exactly one number for the property area…a polygon doesn’t have two areas only one. We’ll be doing a lot with areas in the Topics module coming up next. 55 H F E C D A B G A18. If two triangles are congruent, then the triangular regions have the same area. If two triangles are congruent, then they’ll have exactly the same numbers for the trigonometric version of the area formula: ½ ab sin  This is because the side lengths and the angle measures for corresponding parts are congruent. In your homework you have to answer the question: Could this be an “iff” axiom? Is it true that: If two triangles have the same area, then they are congruent? A19. Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2. This axiom gets a little closer to telling you how to find the area of a two dimensional shape. If you can break the unknown shape into a few pieces for which you know the areas you can add up the known pieces to get the unknown area. A very handy theorem. Pentagon area exercise: What is the area of this regular convex pentagon? How many points and lines are shared by decomposing this regular pentagon into congruent triangles? Does this number fit the scenario established in Axiom 19? What is the area of this figure? Is Axiom 19 restricted to convex polygons? No, it’s not. 56 A'''''A = 2.00 cm A''''' A''''A''' A'' A' A Arbitrary shape area exercise: Use A19 to find the area enclosed by the following non-convex figure: Get the exact area first and then calculate an approximation to one significant digit. 57 The curved portion is a semicircle BC = 5.16 cm BE = 4.48 cm DE = 2.46 cm CD = 4.04 cm mDEB = 18.15 mBCD = 25.05 C B E D Answers to Selected Exercises A1 Exercise: Two points determine a line. Midpoint Exercise: 1 1 2 6 , (0,4) 2 2         Coordinate Exercise: coordinates : 17, 3 17, distance 2 17 Application Exercise: Segment A: 10 10 5 and 7 3 3  Length of Segment A: 10 12 3 Segment B: 10 0 and 12 3 Length of Segment B: same 60 Ruler Placement Exercise: AE = ½ + 3 + 2.5 +1 = 7 units BD = 3 + 2.5 = 5.5 units With the coordinate of B = 0, we have A = −1/2, B = 0, C = 3, D = 5.5, E = 6.5 Note that AE is still 7 and BD is still 5.5 Ruler Placement Centering Exercise: Using the geometric coordinate formula, the 0 for each line with a non-zero slope will be where it intersects the y axis ( the y-intercept) – the x coordinate is zero there. No the Ruler Placement Postulate says you may move it up or down the line as you please. Regular Polygon Exercise: equilateral triangle; square Interior of an angle exercise: A, B, C are angle points E is in the exterior and D is in the interior Angles Exercise 1:  A is the complement of  C and  C is the complement of  B. m  A = (x + 30) and m  B = (9  6x). Sketch this scenario and find x and m  C. mA + mC = 90=mC + m B now subtract the measure of angle C from both sides. x + 30 = 9  6x x = 3 mA = 27 = mB 61 B = 9 - 6x C A = x + 30 C Congruence Exercise: Illustration 1: ∆ADC  ∆ABC by ASA (AC is the side; it is shared) So AD  AB (not 2 cm longer) and DC  BC (not a cm longer). Illustration 2: ∆DBC  ∆EBA (note the vertices lists, corresponding) because vertical angles are congruent and the sides are bisected (ie cut in perfect halves), so DC  AE and is not longer. Illustration 3: Since ABB’A’ is a parallelogram, opposite sides are congruent. Thus B and A’ are corresponding parts and must measure the same. Congruence Proof Exercise: Since C is the midpoint segments AE and BD, we know that AC CE and BC CD  . Since vertical angles are congruent, ACB  DCE. This means that ∆ACB  ∆EDC by SAS. Note that ∆ACB is NOT congruent to ∆DCE because A and D are not corresponding angles. The order of the vertices in the list matters. Parallel Lines Vocabulary Exercise: alternate exterior angles: a & h; b& g corresponding angles: a & e; b & f; c & g; d & h alternate interior angles: c & f; d & e interior angles on the same side: c & e; d & f IFF exercise: 62 Using ∆SAT, note that 1 is an exterior angle. Thus: m1 = m3 + mS Note, too, that 1 and 2 are supplements so m1 + m2 = 180° Now in ∆SAT: m3 + m2 +mS = 180° = m1 + m2 Subtract m2 from both sides to find that m3 + mS = m1 Subtract m3 from both sides and you have it. Pentagon area exercise: What is the area of this regular convex pentagon? 65 A A'''''A = 2.00 cm A''''' A''''A''' A'' A' A How many points and lines are shared by decomposing this regular pentagon into congruent triangles? By connecting A to each vertex of the pentagon we create 5 regions. Each region shares one segment and two points with an adjacent region. This satisfies the requirements for Axiom 19. 5(1/2)4sin 72° cm squared is the exact area. 9.51 cm squared is the approximate area. Arbitrary shape area exercise: Use A19 to find the area enclosed by the following non-convex figure: 66 Region 1 is the semicircle. It shares one segment and two points with ∆BDC, region 2, which in turn shares one segment and two points with region 3, ∆BDE. Get the exact area first and then calculate an approximation to one significant digit. Exact: Approximate: Region 1: 2(2.58) 48.9 cm squared + Region 2: 5.16(4.04)sin 25.05° + Region 3: 2.46(4.48)sin18.15° Appendix A Why Does the Geometric Coordinate Formula Work? 67 The curved portion is a semicircle BC = 5.16 cm BE = 4.48 cm DE = 2.46 cm CD = 4.04 cm mDEB = 18.15 mBCD = 25.05 C B E D