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page 325 16. Appendix 1: List of Definitions Definition 1: Interpretation of an axiom system (page 12) Suppose that an axiom system consists of the following four things ο· an undefined object of one type, and a set π΄ containing all of the objects of that type ο· an undefined object of another type, and a set π΅ containing all of the objects of that type ο· an undefined relation β from set π΄ to set π΅ ο· a list of axioms involving the primitive objects and the relation An interpretation of the axiom systems is the following three things ο· a designation of an actual set π΄β² that will play the role of set π΄ ο· a designation of an actual set π΅β² that will play the role of set π΅ ο· a designation of an actual relation ββ² from π΄β² to π΅β² that will play the role of the relation β Definition 2: successful interpretation of an axiom system; model of an axiom system (page 14) To say that an interpretation of an axiom system is successful means that when the undefined terms and undefined relations in the axioms are replaced with the corresponding terms and relations of the interpretation, the resulting statements are all true. A model of an axiom system is an interpretation that is successful. Definition 3: isomorphic models of an axiom system (page 15) Two models of an axiom system are said to be isomorphic if it is possible to describe a correspondence between the objects and relations of one model and the objects and relations of the other model in a way that all corresponding relationships are preserved. Definition 4: consistent axiom system (page 15) An axiom system is said to be consistent if it is possible for all of the axioms to be true. The axiom system is said to be inconsistent if it is not possible for all of the axioms to be true. Definition 5: dependent and independent axioms (page 18) An axiom is said to be dependent if it is possible to prove that the axiom is true as a consequence of the other axioms. An axiom is said to be independent if it is not possible to prove that it is true as a consequence of the other axioms. Definition 6: independent axiom system (page 20) An axiom system is said to be independent if all of its axioms are independent. An axiom system is said to be not independent if one or more of its axioms are not independent. Definition 7: complete axiom system (page 20) An axiom system is said to be complete if any two models of the axiom system are isomorphic. An axiom system is said to be not complete if there exist two models that are not isomorphic. Chapter 16 Appendix 1: List of Definitions page 326 Definition 8: Alternate definition of a complete axiom system (page 22) An axiom system is said to be not complete if it is possible to write an additonal independent statement regarding the primitive terms and relations. (An additional independent statement is a statement S that is not one of the axioms and such that there is a model for the axiom system in which Statement S is true and there is also a model for the axiom system in which Statement S is false.) An axiom system is said to be complete if it is not possible to write such an additional independent statement. Definition 9: passes through (page 29) ο· words: Line L passes through point P. ο· meaning: Point P lies on line L. Definition 10: intersecting lines (page 29) ο· words: Line L intersects line M. ο· meaning: There exists a point (at least one point) that lies on both lines. Definition 11: parallel lines (page 29) ο· words: Line L is parallel to line M. ο· symbol: L||M. ο· meaning: Line L does not intersect line M. That is, there is no point that lies on both lines. Definition 12: collinear points (page 29) ο· words: The set of points {P1, P2, β¦ , Pk} is collinear. ο· meaning: There exists a line L that passes through all the points. Definition 13: concurrent lines (page 30) ο· words: The set of lines {L1, L2, β¦ , Lk} is concurrent. ο· meaning: There exists a point P that lies on all the lines. Definition 14: Abstract Model, Concrete Model, Relative Consistency, Absolute Consistency (page 45) ο· An abstract model of an axiom system is a model that is, itself, another axiom system. ο· A concrete model of an axiom system is a model that uses actual objects and relations. ο· An axiom system is called relatively consistent if an abstract model has been demonstrated. ο· An axiom system is called absolutely consistent if a concrete model has been demonstrated. Definition 15: the concept of duality and the dual of an axiomatic geometry (page 46) Given any axiomatic geometry with primitive objects point and line, primitive relation βthe point lies on the lineβ, and defined relation βthe line passes through the pointβ, one can obtain a new axiomatic geometry by making the following replacements. ο· ο· ο· ο· Replace every occurrence of point in the original with line in the new axiom system. Replace every occurrence of line in the original with point in the new axiom system. Replace every occurrence of lies on in the original with passes through in the new. Replace every occurrence of passes through in the original with lies on in the new. page 327 The resulting new axiomatic geometry is called the dual of the original geometry. The dual geometry will have primitive objects line and point, primitive relation βthe line passes through the pointβ, and defined relation βthe point lies on the line.β Any theorem of the original axiom system can be translated as well, and the result will be a valid theorem of the new dual axiom system. Definition 16: self-dual geometry (page 50) An axiomatic geometry is said to be self-dual if the statements of the dual axioms are true statements in the original geometry. Definition 17: The Axiom System for Neutral Geometry (page 55) Primitive Objects: point, line Primitive Relation: the point lies on the line Axioms of Incidence and Distance <N1> There exist two distinct points. (at least two) <N2> For every pair of distinct points, there exists exactly one line that both points lie on. <N3> For every line, there exists a point that does not lie on the line. (at least one) <N4> (The Distance Axiom) There exists a function π: π« × π« β β, called the Distance Function on the Set of Points. <N5> (The Ruler Axiom) Every line has a coordinate function. Axiom of Separation <N6> (The Plane Separation Axiom) For every line πΏ, there are two associated sets called half-planes, denoted π»1 and π»2 , with the following three properties: (i) The three sets πΏ, π»1 , π»2 form a partition of the set of all points. (ii) Each of the half-planes is convex. (iii) If point π is in π»1 and point π is in π»2 , then segment Μ Μ Μ Μ ππ intersects line πΏ. Axioms of Angle measurement <N7> (Angle Measurement Axiom) There exists a function π: π β (0,180), called the Angle Measurement Function. βββββ be a ray on the edge of the half-plane π». <N8> (Angle Construction Axiom) Let π΄π΅ βββββ with point For every number π between 0 and 180, there is exactly one ray π΄π π in π» such that π(β ππ΄π΅) = π. <N9> (Angle Measure Addition Axiom) If π· is a point in the interior of β π΅π΄πΆ, then π(β π΅π΄πΆ) = π(β π΅π΄π·) + π(β π·π΄πΆ). Axiom of Triangle Congruence <N10> (SAS Axiom) If there is a one-to-one correspondence between the vertices of two triangles, and two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then all the remaining corresponding parts are congruent as well, so the correspondence is a congruence and the triangles are congruent. Definition 18: the unique line passing through two distinct points (page 57) words: line π, π β‘ββββ symbol: ππ usage: π and π are distinct points Chapter 16 Appendix 1: List of Definitions page 328 meaning: the unique line that passes through both π and π. (The existence and uniqueness of such a line is guaranteed by Axiom <N2>.) Definition 19: The set of all abstract points is denoted by the symbol π« and is called the plane. (page 63) Definition 20: abbreviated symbol for the distance between two points (page 63) abbreviated symbol: ππ meaning: the distance between points π and π, that is, π(π, π) Definition 21: Coordinate Function (page 64) Words: π is a coordinate function on line πΏ. Meaning: π is a function with domain πΏ and codomain β (that is, π: πΏ β β) that has the following properties: (1) π is a one-to-one correspondence. That is, π is both one-to-one and onto. (2) π βagrees withβ the distance function π in the following way: For all points π,π on line πΏ, the equation |π(π) β π(π)| = π(π, π) is true. Additional Terminology: In standard function notation, the symbol π(π) denotes the output of the coordinate function π when the point π is used as input. Note that π(π) is a real number. The number π(π) is called the coordinate of point π on line πΏ. Additional Notation: Because a coordinate function is tied to a particular line, it might be a good idea to have a notation for the coordinate function that indicates which line the coordinate function is tied to. We could write ππΏ for a coordinate function on line πΏ. With that notation, the symbol ππΏ (π) would denote the coordinate of point π on line πΏ. But although it might be clearer, we do not use the symbol ππΏ . We just use the symbol π. Definition 22: Distance Function on the set of Real Numbers (page 66) Words: The Distance Function on the Set of Real Numbers Meaning: The function πβ : β × β β β defined by πβ (π₯, π¦) = |π₯ β π¦|. Definition 23: betweenness for real numbers (page 91) words: βπ¦ is between π₯ and π§β, where π₯, π¦, and π§ are real numbers. symbol: π₯ β π¦ β π§, where π₯, π¦, and π§ are real numbers meaning: π₯ < π¦ < π§ or π§ < π¦ < π₯. additional symbol: the symbol π€ β π₯ β π¦ β π§ means π€ < π₯ < π¦ < π§ or π§ < π¦ < π₯ < π€, etc. Definition 24: betweenness of points (page 94) words: βπ is between π and π β, where π, π, π are points. symbol: π β π β π , where π, π, π are points. meaning: Points π, π, π are collinear, lying on some line πΏ, and there is a coordinate function π for line πΏ such that the real number coordinate for π is between the real number coordinates of π and π . That is, π(π) β π(π) β π(π ). remark: By Theorem 14, we know that it does not matter which coordinate function is used on line πΏ. The betweenness property of the coordinates of the three points will be the same regardless of the coordinate function used. additional symbol: The symbol π β π β π β π means π(π) β π(π) β π(π ) β π(π), etc. page 329 Definition 25: segment, ray (page 96) Μ Μ Μ Μ , and βray π΄, π΅β, denoted π΄π΅ βββββ words and symbols: βsegment π΄, π΅β, denoted π΄π΅ usage: π΄ and π΅ are distinct points. meaning: Let π be a coordinate function for line β‘ββββ π΄π΅ with the property that π(π΄) = 0 and π(π΅) is positive. (The existence of such a coordinate function is guaranteed by Theorem 11 (Ruler Placement Theorem).) ο· Segment Μ Μ Μ Μ π΄π΅ is the set Μ Μ Μ Μ π΄π΅ = {π β β‘ββββ π΄π΅ π π’πβ π‘βππ‘ 0 β€ π(π) β€ π(π΅)}. βββββ is the set π΄π΅ βββββ = {π β π΄π΅ β‘ββββ π π’πβ π‘βππ‘ 0 β€ π(π)}. ο· Ray π΄π΅ additional terminology: ο· Points π΄ and π΅ are called the endpoints of segment Μ Μ Μ Μ π΄π΅ . βββββ ο· Point π΄ is called the endpoint of ray π΄π΅ . ο· The length of a segment is defined to be the distance between the endpoints. That is, Μ Μ Μ Μ ) = π(π΄, π΅). As mentioned in Definition 20, many books use the symbol πππππ‘β(π΄π΅ π΄π΅ to denote π(π΄, π΅). Thus we have the following choice of notations: Μ Μ Μ Μ ) = π(π΄, π΅) = π΄π΅ πππππ‘β(π΄π΅ βββββ where π΄ β π΅ β πΆ. (page 97) Definition 26: Opposite rays are rays of the form βββββ π΅π΄ and π΅πΆ Definition 27: angle (page 97) words: βangle π΄, π΅, πΆβ symbol: β π΄π΅πΆ usage: π΄, π΅, πΆ are non-collinear points. βββββ meaning: Angle π΄, π΅, πΆ is defined to be the following set: β π΄π΅πΆ = βββββ π΅π΄ βͺ π΅πΆ βββββ and π΅πΆ βββββ are each additional terminology: Point π΅ is called the vertex of the angle. Rays π΅π΄ called a side of the angle. Definition 28: triangle (page 98) words: βtriangle π΄, π΅, πΆβ symbol: Ξπ΄π΅πΆ usage: π΄, π΅, πΆ are non-collinear points. Μ Μ Μ Μ βͺ Μ Μ Μ Μ meaning: Triangle π΄, π΅, πΆ is defined to be the following set: Ξπ΄π΅πΆ = Μ Μ Μ Μ π΄π΅ βͺ π΅πΆ πΆπ΄ Μ Μ Μ Μ additional terminology: Points π΄, π΅, πΆ are each called a vertex of the triangle. Segments π΄π΅ Μ Μ Μ Μ Μ Μ Μ Μ and π΅πΆ and πΆπ΄ are each called a side of the triangle. Definition 29: segment congruence (page 98) Two line segments are said to be congruent if they have the same length. The symbol β is Μ Μ Μ Μ ) = πππππ‘β(πΆπ· Μ Μ Μ Μ ). Of course, used to indicate this. For example Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ πΆπ· means πππππ‘β(π΄π΅ this can also be denoted π(π΄, π΅) = π(πΆ, π·) or π΄π΅ = πΆπ·. Definition 30: reflexive property (page 99) words: Relation β is reflexive. usage: β is a relation on some set π΄. meaning: Element of set π΄ is related to itself. abbreviated version: For every π₯ β π΄, the sentence π₯ βπ₯ is true. More concise abbreviaton: βπ₯ β π΄, π₯βπ₯ Chapter 16 Appendix 1: List of Definitions page 330 Definition 31: symmetric property (page 99) words: Relation β is symmetric. usage: β is a relation on some set π΄. meaning: If π₯ is related to π¦, then π¦ is related to π₯. abbreviated version: For every π₯, π¦ β π΄, if π₯βπ¦ is true then π¦βπ₯ is also true. More concise abbreviaton: βπ₯, π¦ β π΄, ππ π₯βπ¦ π‘βππ π¦βπ₯ Definition 32: transitive property (page 99) words: Relation β is transitive. usage: β is a relation on some set π΄. meaning: If π₯ is related to π¦ and π¦ is related to π§, then π₯ is related to π§. abbreviated: For every π₯, π¦, π§ β π΄, if π₯βπ¦ is true and π¦βπ§ is true, then π₯ βπ§ is also true. More concise abbreviaton: βπ₯, π¦ β π΄, ππ π₯βπ¦ πππ π¦βπ§ π‘βππ π₯βπ§ Definition 33: equivalence relation (page 99) words: Relation β is an equivalence relation. usage: β is a relation on some set π΄. meaning: β is reflexive and symmetric and transitive. Definition 34: midpoint of a segment (page 104) Words: π is a midpoint of Segment π΄, π΅. Meaning: π lies on β‘ββββ π΄π΅ and ππ΄ = ππ΅. Definition 35: convex set (page 112) ο· Without names: A set is said to be convex if for any two distinct points that are elements of the set, the segment that has those two points as endpoints is a subset of the set. ο· With names: Set π is said to be convex if for any two distinct points π, π β π, the Μ Μ Μ Μ β π. segment ππ Definition 36: same side, opposite side, edge of a half-plane. (page 113) Two points are said to lie on the same side of a given line if they are both elements of the same half-plane created by that line. The two points are said to lie on opposite sides of the line if one point is an element of one half-plane and the other point is an element of the other. The line itself is called the edge of the half-plane. Definition 37: Angle Interior (page 117) Words: The interior of β π΄π΅πΆ. Symbol: πΌππ‘πππππ(β π΄π΅πΆ) Meaning: The set of all points π· that satisfy both of the following conditions. β‘ββββ . Points π· and π΄ are on the same side of line π΅πΆ Points π· and πΆ are on the same side of line β‘ββββ π΄π΅ . Meaning abbreviated in symbols: πΌππ‘πππππ(β π΄π΅πΆ) = π»π΄π΅ β‘ββββ (πΆ) β© π»π΅πΆ β‘ββββ (π΄) Related term: The exterior of β π΄π΅πΆ is defined to be the set of points that do not lie on the angle or in its interior. page 331 Definition 38: Triangle Interior (page 117) Words: The interior of Ξπ΄π΅πΆ. Symbol: πΌππ‘πππππ(Ξπ΄π΅πΆ) Meaning: The set of all points π· that satisfy all three of the following conditions. β‘ββββ . Points π· and π΄ are on the same side of line π΅πΆ Points π· and π΅ are on the same side of line β‘ββββ πΆπ΄. β‘ββββ Points π· and πΆ are on the same side of line π΄π΅ . Meaning abbreviated in symbols: πΌππ‘πππππ(Ξπ΄π΅πΆ) = π»π΄π΅ β‘ββββ (πΆ) β© π»π΅πΆ β‘ββββ (π΄) β© π»πΆπ΄ β‘ββββ (π΅). Related term: The exterior of Ξπ΄π΅πΆ is defined to be the set of points that do not lie on the triangle or in its interior. Definition 39: quadrilateral (page 123) words: βquadrilateral π΄, π΅, πΆ, π·β symbol: β‘π΄π΅πΆπ· usage: π΄, π΅, πΆ, π· are distinct points, no three of which are collinear, and such that the segments Μ Μ Μ Μ Μ Μ Μ Μ , Μ Μ Μ Μ π΄π΅ , π΅πΆ πΆπ· , Μ Μ Μ Μ π·π΄ intersect only at their endpoints. Μ Μ Μ Μ βͺ π΅πΆ Μ Μ Μ Μ βͺ πΆπ· Μ Μ Μ Μ βͺ π·π΄ Μ Μ Μ Μ meaning: quadrilateral π΄, π΅, πΆ, π· is the set β‘π΄π΅πΆπ· = π΄π΅ additional terminology: Points π΄, π΅, πΆ, π· are each called a vertex of the quadrilateral. Μ Μ Μ Μ and Μ Μ Μ Μ Segments Μ Μ Μ Μ π΄π΅ and π΅πΆ πΆπ· and Μ Μ Μ Μ π·π΄ are each called a side of the quadrilateral. Μ Μ Μ Μ and π΅π· Μ Μ Μ Μ are each called a diagonal of the quadrilateral. Segments π΄πΆ Definition 40: convex quadrilateral (page 126) A convex quadrilateral is one in which all the points of any given side lie on the same side of the line determined by the opposite side. A quadrilateral that does not have this property is called non-convex. Definition 41: The set of all abstract angles is denoted by the symbol π. (page 131) Definition 42: adjacent angles (page 133) Two angles are said to be adjacent if they share a side but have disjoint interiors. That is, the two angles can be written in the form β π΄π΅π· and β π·π΅πΆ, where point πΆ is not in the interior of β π΄π΅π· and point π΄ is not in the interior of β π·π΅πΆ. Definition 43: angle bisector (page 134) An angle bisector is a ray that has its endpoint at the vertex of the angle and passes through a point in the interior of the angle, such that the two adjacent angles created have equal measure. That is, for an angle β π΄π΅πΆ, a bisector is a ray ββββββ π΅π· such that π· β πππ‘πππππ(β π΄π΅πΆ) and such that π(β π΄π΅π·) = π(β π·π΅πΆ). Definition 44: linear pair (page 136) Two angles are said to be a linear pair if they share one side, and the sides that they do not share are opposite rays. That is, if the two angles can be written in the form β π΄π΅π· and β π·π΅πΆ, where π΄ β π΅ β πΆ. Definition 45: vertical pair (page 138) A vertical pair is a pair of angles with the property that the sides of one angle are the opposite rays of the sides of the other angle. Chapter 16 Appendix 1: List of Definitions page 332 Definition 46: acute angle, right angle, obtuse angle (page 141) An acute angle is an angle with measure less than 90. A right angle is an angle with measure 90. An obtuse angle is an angle with measure greater than 90. Definition 47: perpendicular lines (page 141) Two lines are said to be perpendicular if there exist two rays that lie in the lines and whose union is a right angle. The symbol πΏ β₯ π is used to denote that lines πΏ and π are perpendicular. Definition 48: perpendicular lines, segments, rays (page 141) Suppose that Object 1 is a line or a segment or a ray and that Object 2 is a line or a segment or a ray. Object 1 is said to be perpendicular to Object 2 if the line that contains Object 1 is perpendicular to the line that contains Object 2 by the definition of perpendicular lines in the previous definition. The symbol πΏ β₯ π is used to denote that objects πΏ and π are perpendicular. Definition 49: angle congruence (page 144) Two angles are said to be congruent if they have the same measure. The symbol β is used to indicate this. For example β π΄π΅πΆ β β π·πΈπΉ means π(β π΄π΅πΆ) = π(β π·πΈπΉ). Definition 50: symbol for equality of two sets (found on page 149) Words: π πππ’πππ π. Symbol: π = π. Usage: π and π are sets. Meaning: π and π are the same set. That is, every element of set π is also an element of set π, and vice-versa. Definition 51: function, domain, codomain, image, machine diagram, correspondence (found on page 149) Symbol: π: π΄ β π΅ Spoken: βπ is a function that maps π΄ to π΅.β Usage: π΄ and π΅ are sets. Set π΄ is called the domain and set π΅ is called the codomain. Meaning: π is a machine that takes an element of set π΄ as input and produces an element of set π΅ as output. More notation: If an element π β π΄ is used as the input to the function , then the symbol π(π) is used to denote the corresponding output. The output π(π) is called the image of π under the map π. Machine Diagram: π input Domain: the set π΄ π π(π) output Codomain: the set π΅ page 333 Additional notation: If π is both one-to-one and onto (that is, if π is a bijection), then the symbol π: π΄ β π΅ will be used. In this case, π is called a correspondence between the sets π΄ and π΅. Definition 52: Correspondence between vertices of two triangles (found on page 151) Words: βπ is a correspondence between the vertices of triangles Ξπ΄π΅πΆ and Ξπ·πΈπΉ.β Meaning: π is a one-to-one, onto function with domain {π΄, π΅, πΆ} and codomain {π·, πΈ, πΉ}. Definition 53: corresponding parts of two triangles (found on page 152) Words: Corresponding parts of triangles Ξπ΄π΅πΆ and Ξπ·πΈπΉ. Usage: A correspondence between the vertices of triangles Ξπ΄π΅πΆ and Ξπ·πΈπΉ has been given. Meaning: As discussed above, if a correspondence between the vertices of triangles Ξπ΄π΅πΆ and Ξπ·πΈπΉ has been given, then there is an automatic correspondence between the sides of triangle Ξπ΄π΅πΆ and and the sides of triangle Ξπ·πΈπΉ, and also between the angles of triangle Ξπ΄π΅πΆ and the angles of Ξπ·πΈπΉ, For example, if the correspondence between vertices were (π΄, π΅, πΆ) β (π·, πΈ, πΉ), then corresponding parts would be pairs Μ Μ Μ Μ β Μ Μ Μ Μ such as the pair of sides π΄π΅ π·πΈ and the pair of angles β π΄π΅πΆ β β π·πΈπΉ. Definition 54: triangle congruence (found on page 152) To say that two triangles are congruent means that there exists a correspondence between the vertices of the two triangles such that corresponding parts of the two triangles are congruent. If a correspondence between vertices of two triangles has the property that corresponding parts are congruent, then the correspondence is called a congruence. That is, the expression a congruence refers to a particular correspondence of vertices that has the special property that corresponding parts of the triangles are congruent. Definition 55: symbol for a congruence of two triangles (found on page 152) Symbol: Ξπ΄π΅πΆ β Ξπ·πΈπΉ. Meaning: The correspondence (π΄, π΅, πΆ) β (π·, πΈ, πΉ) of vertices is a congruence. Definition 56: scalene, isosceles, equilateral, equiangular triangles (found on page 156) A scalene triangle is one in which no two sides are congruent. An isosceles triangle is one in which at least two sides are congruent. An equilateral triangle is one in which all three sides are congruent. An equiangular triangle is one in which all three angles are congruent. Definition 57: exterior angle, remote interior angle (found on page 162) An exterior angle of a triangle is an angle that forms a linear pair with one of the angles of the triangle. Each of the two other angles of the triangle is called a remote interior angle for that exterior angle. For example, a triangle Ξπ΄π΅πΆ has six exterior angles. One of these is β πΆπ΅π·, where π· is a point such that π΄ β π΅ β π·. For the exterior angle β πΆπ΅π·, the two remote interior angles are β π΄πΆπ΅ and β πΆπ΄π΅. Definition 58: right triangle, and hypotenuse and legs of a right triangle (found on page 172) A right triangle is one in which one of the angles is a right angle. Recall that Theorem 60 states that if a triangle has one right angle, then the other two angles are acute, so there can only be one right angle in a right triangle. In a right triangle, the side opposite the right angle Chapter 16 Appendix 1: List of Definitions page 334 is called the hypotenuse of the triangle. Each of the other two sides is called a leg of the triangle. Definition 59: altitude line, foot of an altitude line, altitude segment (found on page 173) An altitude line of a triangle is a line that passes through a vertex of the triangle and is perpendicular to the opposite side. (Note that the altitude line does not necessarily have to intersect the opposite side to be perpendicular to it. Also note that Theorem 66 in the previous chapter tells us that there is exactly one altitude line for each vertex.) The point of intersection of the altitude line and the line determined by the opposite side is called the foot of the altitude line. An altitude segment has one endpoint at the vertex and the other endpoint at the foot of the altitude line drawn from that vertex. For example, in triangle Ξπ΄π΅πΆ, an β‘ββββ . altitude line from vertex π΄ is a line πΏ that passes through π΄ and is perpendicular to line π΅πΆ β‘ββββ . The The foot of altitude line πΏ is the point π· that is the intersection of line πΏ and line π΅πΆ altitude segment from vertex π΄ is the segment Μ Μ Μ Μ π΄π·. Point π· can also be called the foot of the Μ Μ Μ Μ . altitude segment π΄π· Definition 60: transversal (found on page 177) Words: Line π is transversal to lines πΏ and π. Meaning: Line π intersects πΏ and π in distinct points. Definition 61: alternate interior angles, corresponding angles, interior angles on the same side of the transversal (found on page 177) Usage: Lines πΏ, π, and transversal π are given. Labeled Points: Let π΅ be the intersection of lines π and πΏ, and let πΈ be the intersection of lines π and π. (By definition of transversal, π΅ and πΈ are not the same point.) By Theorem 15, there exist points π΄ and πΆ on line πΏ such that π΄ β π΅ β πΆ, points π· π» π and πΉ on line π such that π· β πΈ β πΉ, and points π· π πΈ πΊ and π» on line π such that πΊ β π΅ β πΈ and π΅ β πΉ πΈ β π». Without loss of generality, we may π΄ π΅ πΆ πΏ assume that points π· and πΉ are labeled such πΊ that it is point π· that is on the same side of line π as point π΄. (See the figure at right.) Meaning: Special names are given to the following eight pairs of angles: ο· The pair {β π΄π΅πΈ, β πΉπΈπ΅} is a pair of alternate interior angles. ο· The pair {β πΆπ΅πΈ, β π·πΈπ΅} is a pair of alternate interior angles. ο· The pair {β π΄π΅πΊ, β π·πΈπΊ} is a pair of corresponding angles. ο· The pair {β π΄π΅π», β π·πΈπ»} is a pair of corresponding angles. ο· The pair {β πΆπ΅πΊ, β πΉπΈπΊ} is a pair of corresponding angles. ο· The pair {β πΆπ΅π», β πΉπΈπ»} is a pair of corresponding angles. ο· The pair {β π΄π΅πΈ, β π·πΈπ΅} is a pair of interior angles on the same side of the transversal. ο· The pair {β πΆπ΅πΈ, β πΉπΈπ΅} is a pair of interior angles on the same side of the transversal. page 335 Definition 62: special angle property for two lines and a transversal (found on page 180) Words: Lines πΏ and π and transversal π have the special angle property. Meaning: The eight statements listed in Theorem 73 are true. That is, each pair of alternate interior angles is congruent. And each pair of corresponding angles is congruent. And each pair of interior angles on the same side of the transversal has measures that add up to 180. Definition 63: circle, center, radius, radial segment, interior, exterior (found on page 187) Symbol: πΆπππππ(π, π) Spoken: the circle centered at π with radius π Usage: π is a point and π is a positive real number. Meaning: The following set of points: πΆπππππ(π, π) = {π π π’πβ π‘βππ‘ ππ = π} Additional Terminology: ο· The point π is called the center of the circle. ο· The number π is called the radius of the circle. ο· The interior is the set πΌππ‘πππππ(πΆπππππ(π, π)) = {π π π’πβ π‘βππ‘ ππ < π}. ο· The exterior is the set πΈπ₯π‘πππππ(πΆπππππ(π, π)) = {π π π’πβ π‘βππ‘ ππ > π}. ο· Two circles are said to be congruent if they have the same radius. ο· Two circles are said to be concentric if they have the same center. Definition 64: Tangent Line and Secant Line for a Circle (found on page 187) A tangent line for a circle is a line that intersects the circle at exactly one point. A secant line for a circle is a line that intersects the circle at exactly two points. Definition 65: chord, diameter segment, diameter, radial segment (found on page 189) ο· A chord of a circle is a line segment whose endpoints both lie on the circle. ο· A diameter segment for a circle is a chord that passes through the center of the circle. ο· The diameter of a circle is the number π = 2π. That is, the diameter is the number that is the length of a diameter segment. ο· A radial segment for a circle is a segment that has one endpoint at the center of the circle and the other endpoint on the circle. (So that the radius is the number that is the length of a radial segment.) Definition 66: median segment and median line of a triangle (found on page 191) A median line for a triangle is a line that passes through a vertex and the midpoint of the opposite side. A median segment for triangle is a segment that has its endpoints at those points. Definition 67: incenter of a triangle (found on page 195) The incenter of a triangle in Neutral Geometry is defined to be the point where the three angle bisectors meet. (Such a point is guaranteed to exist by Theorem 92.) Definition 68: inscribed circle (found on page 196) An inscribed circle for a polygon is a circle that has the property that each of the sides of the polygon is tangent to the circle. Definition 69: tangent circles (found on page 197) Two circles are said to be tangent to each other if they intersect in exactly one point. Chapter 16 Appendix 1: List of Definitions page 336 Definition 70: The Axiom System for Euclidean Geometry (found on page 201) Primitive Objects: point, line Primitive Relation: the point lies on the line Axioms of Incidence and Distance <N1> There exist two distinct points. (at least two) <N2> For every pair of distinct points, there exists exactly one line that both points lie on. <N3> For every line, there exists a point that does not lie on the line. (at least one) <N4> (The Distance Axiom) There exists a function π: π« × π« β β, called the Distance Function on the Set of Points. <N5> (The Ruler Axiom) Every line has a coordinate function. Axiom of Separation <N6> (The Plane Separation Axiom) For every line πΏ, there are two associated sets called half-planes, denoted π»1 and π»2 , with the following three properties: (i) The three sets πΏ, π»1 , π»2 form a partition of the set of all points. (ii) Each of the half-planes is convex. (iii) If point π is in π»1 and point π is in π»2 , then segment Μ Μ Μ Μ ππ intersects line πΏ. Axioms of Angle measurement <N7> (Angle Measurement Axiom) There exists a function π: π β (0,180), called the Angle Measurement Function. βββββ be a ray on the edge of the half-plane π». <N8> (Angle Construction Axiom) Let π΄π΅ βββββ with point For every number π between 0 and 180, there is exactly one ray π΄π π in π» such that π(β ππ΄π΅) = π. <N9> (Angle Measure Addition Axiom) If π· is a point in the interior of β π΅π΄πΆ, then π(β π΅π΄πΆ) = π(β π΅π΄π·) + π(β π·π΄πΆ). Axiom of Triangle Congruence <N10> (SAS Axiom) If there is a one-to-one correspondence between the vertices of two triangles, and two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then all the remaining corresponding parts are congruent as well, so the correspondence is a congruence and the triangles are congruent. Euclidean Parallel Axiom <EPA> (EPA Axiom) For any line πΏ and any point π not on πΏ, there is not more than one line π that passes through π and is parallel to πΏ. Definition 71: circumcenter of a triangle (found on page 207) The circumcenter of a triangle in Euclidean Geometry is defined to be the point where the perpendicular bisectors of the three sides intersect. (Such a point is guaranteed to exist by Theorem 106) Definition 72: a circle circumscribes a triangle (found on page 207) We say that a circle circumscribes a triangle if the circle passes through all three vertices of the triangle. Definition 73: parallelogram (found on page 207) A parallelogram is a quadrilateral with the property that both pairs of opposite sides are parallel. page 337 Definition 74: midsegment of a triangle (found on page 209) A midsegment of a triangle is a line segment that has endpoints at the midpoints of two of the sides of the triangle. Definition 75: medial triangle (found on page 209) Words: Triangle #1 is the medial triangle of triangle #2. Meaning: The vertices of triangle #1 are the midpoints of the sides of triangle #2. Additional Terminology: We will refer to triangle #2 as the outer triangle. Definition 76: Orthocenter of a triangle in Euclidean Geometry (found on page 211) The orthocenter of a triangle in Euclidean Geometry is a point where the three altitude lines intersect. (The existence of such a point is guaranteed by Theorem 113.) Definition 77: Equally-Spaced Parallel Lines in Euclidean Geometry (found on page 212) Words: lines πΏ1 , πΏ2 , β― , πΏπ are equally-spaced parallel lines. Meaning: The lines are parallel and πΏ1 πΏ2 = πΏ2 πΏ3 = β― = πΏπβ1 πΏπ . Definition 78: Centroid of a triangle in Euclidean Geometry (found on page 214) The centroid of a triangle in Euclidean Geometry is the point where the three medians intersect. (Such a point is guaranteed to exist by Theorem 116.) Definition 79: Parallel Projection in Euclidean Geometry (found on page 217) Symbol: πππππΏ,π,π Usage: πΏ, π, π are lines, and π intersects both πΏ and π. Meaning: πππππΏ,π,π is a function whose domain is the set of points on line πΏ and whose codomain is the set of points on line π. In function notation, this would be denoted by the symbol πππππΏ,π,π : πΏ β π. Given an input point π on line πΏ, the output point on line π is denoted πβ². That is, πβ² = πππππΏ,π,π (π). The output point πβ² is determined in the following way: Case 1: If π happens to lie at the intersection of lines πΏ and π, then πβ² is defined to be the point at the intersection of lines π and π. Case 2: If π lies on πΏ but not on π, then there exists exactly one line π that passes through π and is parallel to line π. (Such a line π is guaranteed by Theorem 97). The output point πβ² is defined to be the point at the intersection of lines π and π. Drawing: π πΏ π π πΏ π π πβ² π Case 1: π lies on both πΏ and π. . πβ² π Case 2: π lies on πΏ but not on π. Chapter 16 Appendix 1: List of Definitions page 338 Definition 80: triangle similarity (found on page 222) To say that two triangles are similar means that there exists a correspondence between the vertices of the two triangles and the correspondence has these two properties: ο· Each pair of corresponding angles is congruent. ο· The ratios of the lengths of each pair of corresponding sides is the same. If a correspondence between vertices of two triangles has the two properties, then the correspondence is called a similarity. That is, the expression a similarity refers to a particular correspondence of vertices that has the two properties. Definition 81: symbol for a similarity of two triangles (found on page 223) Symbol: Ξπ΄π΅πΆ~Ξπ·πΈπΉ. Meaning: The correspondence (π΄, π΅, πΆ) β (π·, πΈ, πΉ) of vertices is a similarity. Definition 82: base times height (found on page 230) For each side of a triangle, there is an opposite vertex, and there is an altitude segment drawn from that opposite vertex. The expression "πππ π π‘ππππ βπππβπ‘" or "πππ π β βπππβπ‘" refers to the product of the length of a side of a triangle and the length of the corresponding altitude segment drawn to that side. The expression can be abbreviated π β β. Definition 83: triangular region, interior of a triangular region, boundary of a triangular region (found on page 235) Symbol: β²π΄π΅πΆ Spoken: triangular region π΄, π΅, πΆ Usage: π΄, π΅, πΆ are non-collinear points Meaning: the union of triangle Ξπ΄π΅πΆ and the interior of triangle Ξπ΄π΅πΆ. In symbols, we would write β²π΄π΅πΆ = Ξπ΄π΅πΆ βͺ πΌππ‘πππππ(Ξπ΄π΅πΆ). Additional Terminology: the interior of a triangular region is defined to be the interior of the associated triangle. That is, πΌππ‘πππππ(β²π΄π΅πΆ) = πΌππ‘πππππ(Ξπ΄π΅πΆ). The boundary of a triangular region is defined to be the associated triangle, itself. That is, π΅ππ’πππππ¦(β²π΄π΅πΆ) = Ξπ΄π΅πΆ.. Definition 84: the set of all triangular regions is denoted by β . (found on page 236) Definition 85: the area function for triangular regions (found on page 236) symbol: π΄πππ β΄ spoken: the area function for triangular regions πβ meaning: the function π΄πππ : β β β+ defined by π΄πππ (β²π΄π΅πΆ) = 2 , where π is the β΄ β΄ β΄ length of any side of Ξπ΄π΅πΆ and β is the length of the corresponding altitude segment. (Theorem 132 guarantees that the resulting value does not depend on the choice of base.) Definition 86: polygon (found on page ) words: polygon π1 , π2 , β¦ , ππ symbol: ππππ¦πππ(π1 π2 β¦ ππ ) usage: π1 , π2 , β¦ , ππ are distinct points, with no three in a row being collinear, and such that the segments Μ Μ Μ Μ Μ Μ π1 π2 , Μ Μ Μ Μ Μ Μ π1 π2 , β¦ , Μ Μ Μ Μ Μ Μ ππ π1 intersect only at their endpoints. meaning: ππππ¦πππ(π1 π2 β¦ ππ ) is defined to be the following set: page 339 Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ ππππ¦πππ(π1 π2 β¦ ππ ) = π 1 π2 βͺ π1 π2 βͺ β¦ βͺ ππ π1 additional terminology: Points π1 , π2 , β¦ , ππ are each called a vertex of the polygon. Pairs of vertices of the form {ππ , ππ+1 } and the pair {ππ , π1 } are called adjacent vertices. The π segments Μ Μ Μ Μ Μ Μ π1 π2 , Μ Μ Μ Μ Μ Μ π1 π2 , β¦ , Μ Μ Μ Μ Μ Μ ππ π1 whose endpoints are adjacent vertices are each called a side of the polygon. Segments whose endpoints are non-adjacent vertices are each called a diagonal of the polygon. Definition 87: convex polygon (found on page 239) A convex polygon is one in which all the vertices that are not the endpoints of a given side lie in the same half-plane determined by that side. A polygon that does not have this property is called non-convex. Definition 88: complex, polygonal region, separated, connected polygonal regions (found on page 239) A complex is a finite set of triangular regions whose interiors do not intersect. That is, a set of the form πΆ = {β²1 , β²2 , β¦ , β²π } where each β²π is a triangular region and such that if π β π, then the intersection πΌππ‘πππππ(β²π ) β© πΌππ‘πππππ (β²π ) is the empty set. A polygonal region is a set of points that can be described as the union of the triangular regions in a complex. That is a set of the form π π = β²1 βͺ β²2 βͺ β¦ βͺ β²π = β β²π π=1 We say that a polygonal region can be separated if it can be written as the union of two disjoint polygonal regions. A connected polygonal region is one that cannot be separated into two disjoint polygonal regions. We will often use notation like π πππππ(π1 π2 β¦ ππ ) to denote a connected polygonal region. In that symbol, the letters π1 , π2 , β¦ , ππ are vertices of the region (I wonβt give a precise definition of vertex. You get the idea.) Definition 89: open disk, closed disk (found on page 240) symbol: πππ π(π, π) spoken: the open disk centered at point π with radius π. meaning: the set πΌππ‘πππππ(πΆπππππ(π, π)). That is, the set {π: πππ π‘ππππ(π, π) < π}. another symbol: Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ πππ π(π, π) spoken: the closed disk centered at point π with radius π. meaning: the set πΆπππππ(π, π) βͺ πΌππ‘πππππ(πΆπππππ(π, π)). That is, {π: πππ π‘ππππ(π, π) β€ π}. pictures: π π π π the open disk the closed disk Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ πππ π(π, π) πππ π(π, π) Definition 90: interior of a polygonal region, boundary of a polygonal region (page 241) words: the interior of polygonal region π meaning: the set of all points π in π with the property that there exists some open disk centered at point π that is entirely contained in π Chapter 16 Appendix 1: List of Definitions page 340 meaning in symbols: {π β π π π’πβ π‘βππ‘ βπ > 0 π π’πβ π‘βππ‘ πππ π(π, π) β π } additional terminology: the boundary of polygonal region π meaning: the set of all points π in π with the property that no open disk centered at point π is entirely contained in π . This implies that every open disk centered at point π contains some points that are not elements of the region π . meaning in symbols: {π β π π π’πβ π‘βππ‘ βπ > 0, πππ π(π, π) β π } picture: π π π π is an interior point; π is a boundary point Definition 91: the set of all polygonal regions is denoted by β. (found on page 242) Definition 92: the area function for polygonal regions (found on page 243) spoken: the area function for polygonal regions meaning: the function π΄πππ: β β β+ defined by π π΄πππ(π ) = π΄πππβ΄ (β²1 ) + π΄πππβ΄ (β²2 ) + β― + π΄πππβ΄ (β²3 ) = β π΄πππβ΄ (β²π ) π=1 where πΆ = {β²1 , β²2 , β¦ , β²π } is a complex for region π . (Theorem 133 guarantees that the resulting value does not depend on the choice of complex πΆ.) Definition 93: polygon similarity (found on page 246) To say that two polygons are similar means that there exists a correspondence between the vertices of the two polygons and the correspondence has these two properties: ο· Each pair of corresponding angles is congruent. ο· The ratios of the lengths of each pair of corresponding sides is the same. If a correspondence between vertices of two polygons has the two properties, then the correspondence is called a similarity. That is, the expression a similarity refers to a particular correspondence of vertices that has the two properties. Definition 94: symbol for a similarity of two polygons (found on page 246) Symbol: ππππ¦πππ(π1 π2 β¦ ππ )~ππππ¦πππ(π1 β²π2 β² β¦ ππ β²). Meaning: The correspondence (π1 π2 β¦ ππ ) β (π1 β²π2 β² β¦ ππ β²) of vertices is a similarity. page 341 Definition 95: seven types of angles intersecting circles (found on page 255) Type 1 Angle (Central Angle) πΆ A central angle of a circle is an angle whose rays lie on two secant lines that intersect at the center of the circle. β‘βββ are secant In the picture at right, lines β‘ββββ π΄πΈ and πΆπΉ lines that intersect at the center point π΅ of the circle. Angle β π΄π΅πΆ is a central angle. So are angles β πΆπ΅πΈ, β πΈπ΅πΉ, β πΉπ΅π΄. π΄ π΅ πΈ πΉ Type 2 Angle (Inscribed Angle) An inscribed angle of a circle is an angle whose rays lie on two secant lines that intersect on the circle and such that each ray of the angle intersects the circle at one other point. In other words, an angle of the form β π΄π΅πΆ, where π΄, π΅, πΆ are three points on the circle. πΆ π΅ In the picture at right, angle β π΄π΅πΆ is an inscribed angle. Type 3 Angle Our third type of an angle intersecting a circle is an angle whose rays lie on two secant lines that intersect at a point that is inside the circle but is not the center of the circle. In the picture at right, lines β‘ββββ π΄πΈ and β‘βββ πΆπΉ are secant lines that intersect at point π΅ in the interior of the circle. Angle β π΄π΅πΆ is an angle of type three. So are angles β πΆπ΅πΈ, β πΈπ΅πΉ, β πΉπ΅π΄. π΄ πΆ π΄ πΉ πΆ Type 4 Angle Our fourth type of an angle intersecting a circle is an angle whose rays lie on two secant lines that intersect at a point that is outside the circle and such that each ray of the angle intersects the circle. In the picture at right, angle β π΄π΅πΆ is an angle of type four. π΅ πΈ π΄ π΅ Chapter 16 Appendix 1: List of Definitions page 342 Type 5 Angle Our fifth type of an angle intersecting a circle is an angle whose rays lie on two tangent lines and such that each ray of the angle intersects the circle. Because the rays lie in tangent lines, we know that each ray intersects the circle exactly once. In the picture at right, angle β π΄π΅πΆ is an angle of type five. πΆ π΅ π΄ Type 6 Angle πΆ Our sixth type of an angle intersecting a circle is an angle whose vertex lies on the circle and such that one ray contains a chord of the circle and the other ray lies in a line that is tangent to the circle. In the picture at right, angle β π΄π΅πΆ is an angle of type six. π΅ π΄ πΆ Type 7 Angle Our seventh type of an angle intersecting a circle is an angle whose rays lie on a secant line and tangent line that intersect outside the circle and such that each ray of the angle intersects the circle. In the picture at right, angle β π΄π΅πΆ is an angle of type seven. π΅ π΄ Definition 96: Circular Arc (found on page 257) Μ Symbol: π΄π΅πΆ Spoken: arc π΄, π΅, πΆ Usage: π΄, π΅, πΆ are non-collinear points. Meaning: the set consisting of points π΄ and πΆ and all points of πΆπππππ(π΄, π΅, πΆ) that lie on the β‘ββββ as point π΅. same side of line π΄πΆ Μ = {π΄ βͺ πΆ βͺ (πΆπππππ(π΄, π΅, πΆ) β© π»π΅ )} Meaning in Symbols: π΄π΅πΆ Additional terminology: Μ. ο· Points π΄ and πΆ are called the endpoints of arc π΄π΅πΆ ο· The interior of the arc is the set πΆπππππ(π΄, π΅, πΆ) β© π»π΅ . Μ is ο· If the center π lies on the opposite side of line β‘ββββ π΄πΆ from point π΅, then arc π΄π΅πΆ called a minor arc. β‘ββββ as point π΅,then arc ο· If the center π of πΆπππππ(π΄, π΅, πΆ) lies on the same side of line π΄πΆ Μ π΄π΅πΆ is called a major arc. page 343 Μ is called a semicircle. ο· If the center π lies on line β‘ββββ π΄πΆ , then arc π΄π΅πΆ Picture: π΅ π΅ π΅ π΄ πΆ π΄ π πΆ π π΄ minor arc π πΆ major arc semicircle Definition 97: angle intercepting an arc (found on page 257) We say that an angle intercepts an arc if each ray of the angle contains at least one endpoint of the arc and if the interior of the arc lies in the interior of the angle. Definition 98: the symbol for the set of all circular arcs is πΜ . (found on page 259) Definition 99: the angle measure of an arc (found on page 259) Symbol: π Μ Name: the Arc Angle Measurement Function Meaning: The function π Μ : πΜ β (0,360), defined in the following way: Μ is a minor arc, then π Μ ) = π(β π΄ππΆ), where point π is the ο· If π΄π΅πΆ Μ (π΄π΅πΆ center of the circle. Μ is a major arc, then π Μ ) = 360 β π(β π΄ππΆ), where point π ο· If π΄π΅πΆ Μ (π΄π΅πΆ is the center of the circle. Μ is a semicircle, then π Μ ) = 180. ο· If π΄π΅πΆ Μ (π΄π΅πΆ Picture: π΅ π΅ π΅ π΄ πΆ π΄ πΆ π π π π΄ minor arc Μ ) = π(β π΄ππΆ) π Μ (π΄π΅πΆ πΆ major arc Μ π Μ (π΄π΅πΆ ) = 360 β π(β π΄ππΆ) semicircle Μ ) = 180 π Μ (π΄π΅πΆ Definition 100: cyclic quadrilateral (found on page 269) A quadrilateral is said to be cyclic if the quadrilateral can be circumscribed. That is, if there exists a circle that passes through all four vertices of the quadrilateral. Definition 101: The Euler Line of a non-equilateral triangle (found on page 281) Chapter 16 Appendix 1: List of Definitions page 344 Given a non-equilateral triangle, the Euler Line is defined to be the line containing the orthocenter, centroid, and circumcenter of that triangle. (Existence and uniqueness of this line s guaranteed by Theorem 157.) Definition 102: The three Euler Points of a triangle are defined to be the midpoints of the segments connecting the vertices to the orthocenter. (found on page 281) Definition 103: The nine point circle associated to a triangle is the circle that passes through the midpoints of the three sides, the feet of the three altitudes, and the three Euler points. (The existence of the nine point circle is guaranteed by Theorem 158.) Definition 104: circumference of a circle and area of a circular region (found on page 285) Given a circle of diameter π, for each π = 1,2,3, β¦ define ππππ¦π to be a polygonal region bounded by a regular polygon with 3 β 2π sides, inscribed in the circle. Define ππ and π΄π to be the perimeter and area of the π π‘β polygonal region. The resulting sequences {ππ } and {π΄π } are increasing and bounded above and so they each have a limit. ο· Define the circumference of the circle to be the real number πΆ = lim ππ . ο· πββ Define the area of the circular region to be the real number π΄ = lim π΄π . πββ Definition 105: The symbol ππ, or π, denotes the real number that is the ratio πΆ πππππ’ππππππππ ππππππ‘ππ for any circle. That is, π = π. (That this ratio is the same for all circles is guaranteed by Theorem 159, found on page 287.) (found on page 288) Definition 106: arc length (found on page 293) Μ on a circle of radius π is defined to be the number The length of an arc π΄π΅πΆ Μ )ππ π Μ (π΄π΅πΆ Μ) = πΏΜ(π΄π΅πΆ 180 Definition 107: Rules for computing area (found on page 293) (1) The area of a triangular region is equal to one-half the base times the height. It does not matter which side of the triangle is chosen as the base. (2) The area of a polygonal region is equal to the sum of the areas of the triangles in a complex for the region. (3) The area of a circular region is ππ 2 . Μ Μ is ππ 2 β πΜ(π΄π΅πΆ). (4) More generally, the area of a circular sector bounded by arc π΄π΅πΆ 360 (5) Congruence property: If two regions have congruent boundaries, then the area of the two regions is the same. (6) Additivity property: If a region is the union of two smaller regions whose interiors do not intersect, then the area of the whole region is equal to the sum of the two smaller regions. Definition 108: Image and Preimage of a single element If π: π΄ β π΅ and π β π΄ is used as input to the function π, then the corresponding output π(π) β π΅ is called the image of π. If π: π΄ β π΅ and π β π΅, then the preimage of π, denoted π β1 (π), is the set of all elements of π΄ whose image is π. That is, π β1 (π) = {π β π΄ π π’πβ π‘βππ‘ π(π) = π}. page 345 Definition 109: Image of a Set and Preimage of a Set If π: π΄ β π΅ and π β π΄, then the image of π, denoted π(π), is the set of all elements of π΅ that are images of elements of π. That is, π(π) = {π β π΅ π π’πβ π‘βππ‘ π = π(π) πππ π πππ π β π} If π: π΄ β π΅ and π β π΅, then the preimage of π, denoted π β1 (π), is the set of all elements of π΄ whose images are elements of π. That is, π β1 (π) = {π β π΄ π π’πβ π‘βππ‘ π(π) β π} Definition 110: composition of fuctions, composite function Symbol: π β π Spoken: βπ circle πβ,or βπ after πβ, or βπ composed with πβ, or βπ after πβ Usage: π: π΄ β π΅ and π: π΅ β πΆ Meaning: the function π β π: π΄ β πΆ defined by π β π(π) = π(π(π)). Additional terminology: A function of the form π β π is called a composite function. Definition 111: One-to-One Function Words: The function π: π΄ β π΅ is one-to-one. Alternate Words: The function π: π΄ β π΅ is injective. Meaning in Words: Different inputs always produce different outputs. Meaning in Symbols: βπ₯1 , π₯2 , ππ π₯1 β π₯2 π‘βππ π(π₯1 ) β π(π₯2 ). Contrapositive: If two outputs are the same, then the inputs must have been the same. Contrapositive in Symbols: βπ₯1 , π₯2 , ππ π(π₯1 ) = π(π₯2 ), π‘βππ π₯1 = π₯2 . Definition 112: Onto Function Words: The function π: π΄ β π΅ is onto. Alternate Words: The function π: π΄ β π΅ is surjective. Meaning in Words: For every element of the codomain, there exists an element of the domain that will produce that element of the codomain as output. Meaning in Symbols: βπ¦ β π΅, βπ₯ β π΄ π π’πβ π‘βππ‘ π(π₯) = π¦. Definition 113: Bijection, One-to-One Correspondence Words: βThe function π is a bijectionβ, or βthe function π is bijectiveβ. Alternate Words: The function π is a one-to-one correspondence. Meaning: The function π is both one-to-one and onto. Definition 114: Inverse Functions, Inverse Relations Words: Functions π and π are inverses of one another. Usage: π: π΄ β π΅ and π: π΅ β π΄ Meaning: π and π satisfy the following two properties, called inverse relations: βπ β π΄, π β π(π) = π βπ β π΅, π β π(π) = π Additional Symbols and Terminology: Another way of saying that functions π and π are inverses of one another is to say that π is the inverse of π. Instead of using different letters for a function and its inverse, it is common to use the symbol π β1 to denote the inverse of a function π. With this notation, we would say that π: π΄ β π΅ and π β1 : π΅ β π΄, and the inverse relations become: βπ β π΄, π β1 β π(π) = π βπ β π΅, π β π β1 (π) = π Chapter 16 Appendix 1: List of Definitions page 346 Definition 115: The plane is defined to be the set π« of all points. Definition 116: A map of the plane is defined to be a function π: π« β π«. Definition 117: A transformation of the plane is defined to be a bijective map of the plane. The set of all transformations of the plane is denoted by the symbol π. Definition 118: Isometry of the Plane Words: π is an isometry of the plane. Meaning: π is a distance preserving map of the plane. That is, for all points π and π, the distance from π to π is the same as the distance from π(π) to π(π). Meaning in symbols: βπ, π β π«, π(π, π) = π(π(π), π(π)). Definition 119: The Identity Map of the Plane is the map ππ: π« β π« defined by ππ(π) = π for every point π. Definition 120: a Fixed Point of a Map of the Plane Words: π is a fixed point of the map π. Meaning: π(π) = π Definition 121: The Dilation of the Plane Symbol: π·πΆ,π Spoken: The dilation centered at πΆ with scaling factor π Usage: πΆ is a point, called the center of the dilation, and π is a positive real number. Meaning: The map π·πΆ,π : π« β π« defined as follows The point πΆ is a fixed point of π·πΆ,π . That is, π·πΆ,π (πΆ) = πΆ. When a point π β πΆ is used as input to the map π·πΆ,π , the output is the unique point π β² = π·πΆ,π (π) that has these two properties: βββββ ο· Point πβ² lies on ray πΆπ ο· The distance π(πΆ, πβ²) = ππ(πΆ, π) (The existence and uniqueness of such a point π β² is guaranteed by the Congruent Segment Construction Theorem, (Theorem 24).) Definition 122: The Reflection of the Plane Symbol: ππΏ Spoken: The reflection in line πΏ Usage: πΏ is a line, called the line of reflection Meaning: The map ππΏ : π« β π« defined as follows Every point on the line πΏ is a fixed point of ππΏ . That is, if π β πΏ then ππΏ (π) = π. When a point π not on line πΏ is used as input to the map ππΏ , the output is the unique Μ Μ Μ Μ Μ . point π β² = ππΏ (π) such that line πΏ is the perpendicular bisector of segment ππβ² β² (The existence and uniqueness of such a point π is can be proven using the axioms and theorems of Neutral Geometry. You are asked to provide details in an exercise.) Definition 123: A binary operation on a set π is a function β: π × π β π. Definition 124: associativity, associative binary operation page 347 Words: ββ is associativeβ or ββ has the associativity propertyβ Usage: β is a binary operation β on some set π. Meaning: βπ, π, π β π, π β (π β π) = (π β π) β π Definition 125: identity element, binary operation with an identity element Words: ββ has an identity element.β Usage: β is a binary operation β on some set π. Meaning:.There is an element βπ β π with the following property: βπ β π, π β π = π β π = π Meaning in symbols: βπ β π: βπ β π, π β π = π β π = π Additional Terminology: The element βπ β π is called the identity for operation β. Definition 126: binary operation with inverses Words: ββ has inverses.β Usage: β is a binary operation β on some set π. Meaning: For each element π β π, there exists is an π β1 β π such that π β πβ1 = π and πβ1 β π = π. Meaning in symbols: βπ β π, βπβ1 β π: π β πβ1 = π β1 β π = π Additional Terminology: The element πβ1 β π is called the inverse of π. Definition 127: commutativity, commutative binary operation Words: ββ is commutativeβ or ββ has the commutative propertyβ Usage: β is a binary operation β on some set π. Meaning: βπ, π, π β π, π β π = π β π Definition 128: Group A Group is a pair (πΊ,β) consisting of a set πΊ and a binary operation β on πΊ that has the following three properties. (1) Associativity (Definition 124) (2) Existence of an Identity Element (Definition 125) (3) Existence of an Inverse for each Element (Definition 126) Definition 129: Commutative Group, Abelian Group A commutative group (or abelian group) is a group (πΊ,β) that has the commutativity property (Definition 127). . Chapter 16 Appendix 1: List of Definitions . page 348 page 349 17. Appendix 2: List of Theorems Theorem 1: In Neutral Geometry, if πΏ and π are distinct lines that intersect, then they intersect in only one point. (page 56) Theorem 2: In Neutral Geometry, there exist three non-collinear points. (page 56) Theorem 3: In Neutral Geometry, there exist three lines that are not concurrent. (page 56) Theorem 4: In Neutral Geometry, for every point π, there exists a line that does not pass through π. (page 56) Theorem 5: In Neutral Geometry, for every point π, there exist at least two lines that pass through π. (page 56) Theorem 6: In Neutral Geometry, given any points π and π that are not known to be distinct, there exists at least one line that passes through π and π. (page 57) Theorem 7: about how many points are on lines in Neutral Geometry (page 64) In Neutral Geometry, given any line πΏ, the set of points that lie on πΏ is an infinite set. More precisely, the set of points that lie on πΏ can be put in one-to-one correspondence with the set of real numbers β. (In the terminology of sets, we would say that the set of points on line πΏ has the same cardinality as the set of real numbers β.) Theorem 8: The Distance Function on the Set of Points, the function π, is Positive Definite. (page 68) For all points π and π, π(π, π) β₯ 0, and π(π, π) = 0 if and only if π = π. That is, if and only if π and π are actually the same point. Theorem 9: The Distance Function on the Set of Points, the function π, is Symmetric. (page 68) For all points π and π, π(π, π) = π(π, π). Theorem 10: (Ruler Sliding and Ruler Flipping) Lemma about obtaining a new coordinate function from a given one (page 81) Suppose that π: πΏ β β is a coordinate function for a line πΏ. (A) (Ruler Sliding) If π is a real number constant and π is the function π: πΏ β β defined by π(π) = π(π) + π, then π is also a coordinate function for line πΏ. (B) (Ruler Flipping) If π is the function π: πΏ β β defined by π(π) = βπ(π), then π is also a coordinate function for line πΏ. Theorem 11: Ruler Placement Theorem (page 83) If π΄ and π΅ are distinct points on some line πΏ, then there exists a coordinate function β for line πΏ such that β(π΄) = 0 and β(π΅) is positive. Theorem 12: facts about betweenness for real numbers (page 92) (A) If π₯ β π¦ β π§ then π§ β π¦ β π₯. Chapter 17 Appendix 2: List of Theorems page 350 (B) If π₯, π¦, π§ are three distinct real numbers, then exactly one is between the other two. (C) Any four distinct real numbers can be named in an order π€, π₯, π¦, π§ so that π€ β π₯ β π¦ β π§. (D) If π and π are distinct real numbers, then (D.1) There exists a real number π such that π β π β π. (D.2) There exists a real number π such that π β π β π Theorem 13: Betweenness of real numbers is related to the distances between them. (page 92) Claim: For distinct real numbers π₯, π¦, π§, the following are equivalent (A) π₯ β π¦ β π§ (B) |π₯ β π§| = |π₯ β π¦| + |π¦ β π§|. That is, πβ (π₯, π§) = πβ (π₯, π¦) + πβ (π¦, π§). Theorem 14: Lemma about betweenness of coordinates of three points on a line (page 93) If π, π, π are three distinct points on a line πΏ, and π is a coordinate function on line πΏ, and the betweenness expression π(π) β π(π) β π(π ) is true, then for any coordinate function π on line πΏ, the expression π(π) β π(π) β π(π ) will be true. Theorem 15: Properties of Betweenness for Points (page 94) (A) If π β π β π then π β π β π. (B) For any three distinct collinear points, exactly one is between the other two. (C) Any four distinct collinear points can be named in an order π, π, π , π such that π β π β π β π. (D) If π and π are distinct points, then (D.1) There exists a point π such that π β π β π . (D.2) There exists a point π such that π β π β π. Theorem 16: Betweenness of points on a line is related to the distances between them. (page 95) Claim: For distinct collinear points π, π, π , the following are equivalent (A) π β π β π (B) π(π, π ) = π(π, π) + π(π, π ). Theorem 17: Lemma about distances between three distinct, collinear points. (page 96) If π, π, π are distinct collinear points such that π β π β π is not true, then the inequality π(π, π ) < π(π, π) + π(π, π ) is true. Theorem 18: (Corollary) Segment Μ Μ Μ Μ π΄π΅ is a subset of ray βββββ π΄π΅ . (page 97) Theorem 19: about the use of different second points in the symbol for a ray. (page 97) If βββββ π΄π΅ and πΆ is any point of βββββ π΄π΅ that is not π΄, then βββββ π΄π΅ = βββββ π΄πΆ . Theorem 20: Segment congruence is an equivalence relation. (page 100) Theorem 21: About a point whose coordinate is the average of the coordinates of the endpoints. (page 105) Given Μ Μ Μ Μ π΄π΅ , and Point πΆ on line β‘ββββ π΄π΅ , and any coordinate function π for line β‘ββββ π΄π΅ , the following are equivalent: page 351 (i) The coordinate of point πΆ is the average of the coordinates of points π΄ and π΅. π(π΄)+π(π΅) That is, π(πΆ) = . 2 Μ Μ Μ Μ . That is, πΆπ΄ = πΆπ΅. (ii) Point πΆ is a midpoint of segment π΄π΅ Theorem 22: Corollary of Theorem 21. (page 106) Μ Μ Μ Μ , and Point πΆ on line π΄π΅ β‘ββββ , and any coordinate functions π and π for line π΄π΅ β‘ββββ , the Given π΄π΅ following are equivalent: π(π΄)+π(π΅) (i) π(πΆ) = . 2 (ii) π(πΆ) = π(π΄)+π(π΅) 2 . Theorem 23: Every segment has exactly one midpoint. (page 106) Theorem 24: Congruent Segment Construction Theorem. (page 107) Given a segment Μ Μ Μ Μ π΄π΅ and a ray βββββ πΆπ·, there exists exactly one point πΈ on ray βββββ πΆπ· such that Μ Μ Μ Μ Μ Μ Μ Μ πΆπΈ β π΄π΅ . Theorem 25: Congruent Segment Addition Theorem. (page 107) Μ Μ Μ Μ Μ Μ and π΅πΆ Μ Μ Μ Μ Μ Μ then π΄πΆ Μ Μ Μ Μ Μ . Μ Μ Μ Μ β π΄β²π΅β² Μ Μ Μ Μ β π΅β²πΆβ² Μ Μ Μ Μ β π΄β²πΆβ² If π΄ β π΅ β πΆ and π΄β² β π΅ β² β πΆβ² and π΄π΅ Theorem 26: Congruent Segment Subtraction Theorem. (page 107) Μ Μ Μ Μ Μ Μ . Μ Μ Μ Μ β Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ β Μ Μ Μ Μ Μ Μ Μ Μ Μ β π΅β²πΆβ² If π΄ β π΅ β πΆ and π΄β² β π΅ β² β πΆβ² and π΄π΅ π΄β²π΅β² and π΄πΆ π΄β²πΆβ² then π΅πΆ Theorem 27: Given any line, each of its half-planes contains at least three non-collinear points. (page 114) Theorem 28: (Paschβs Theorem) about a line intersecting a side of a triangle between vertices (page 116) If a line intersects the side of a triangle at a point between vertices, then the line also intersects the triangle at another point that lies on at least one of the other two sides. Theorem 29: about a line intersecting two sides of a triangle between vertices (page 116) If a line intersects two sides of a triangle at points that are not vertices, then the line cannot intersect the third side. Theorem 30: about a ray with an endpoint on a line (page 118) If a ray has its endpoint on a line but does not lie in the line, then all points of the ray except the endpoint are on the same side of the line. Theorem 31: (Corollary of Theorem 30) about a ray with its endpoint on an angle vertex (p. 119) If a ray has its endpoint on an angle vertex and passes through a point in the angle interior, then every point of the ray except the endpoint lies in the angle interior. Theorem 32: (Corollary of Theorem 30.) about a segment that has an endpoint on a line (p. 119) If a segment that has an endpoint on a line but does not lie in the line, Chapter 17 Appendix 2: List of Theorems page 352 then all points of the segment except that endpoint are on the same side of the line. Theorem 33: (Corollary of Theorem 32.) Points on a side of a triangle are in the interior of the opposite angle. (page 119) If a point lies on the side of a triangle and is not one of the endpoints of that side, then the point is in the interior of the opposite angle. Theorem 34: The Z Lemma (page 120) β‘ββββ , then ray π΄πΆ βββββ does not intersect ray π΅π· ββββββ . If points πΆ and π· lie on opposite sides of line π΄π΅ Theorem 35: The Crossbar Theorem (page 120) Μ Μ Μ Μ at a point between π΄ and πΆ. ββββββ intersects π΄πΆ If point π· is in the interior of β π΄π΅πΆ, then π΅π· Theorem 36: about a ray with its endpoint in the interior of a triangle (page 122) If the endpoint of a ray lies in the interior of a triangle, then the ray intersects the triangle exactly once. Theorem 37: about a line passing through a point in the interior of a triangle (page 122) If a line passes through a point in the interior of a triangle, then the line intersects the triangle exactly twice. Theorem 38: Three equivalent statements about quadrilaterals (page 124) For any quadrilateral, the following statements are equivalent: (i) All the points of any given side lie on the same side of the line determined by the opposite side. (ii) The diagonal segments intersect. (iii) Each vertex is in the interior of the opposite angle. Theorem 39: about points in the interior of angles (page 133) Given: points πΆ and π· on the same side of line β‘ββββ π΄π΅ . Claim: The following are equivalent: (I) π· is in the interior of β π΄π΅πΆ. (II) π(β π΄π΅π·) < π(β π΄π΅πΆ). Theorem 40: Every angle has a unique bisector. (page 135) Theorem 41: Linear Pair Theorem. (page 136) If two angles form a linear pair, then the sum of their measures is 180. Theorem 42: Converse of the Linear Pair Theorem (page 138) If adjacent angles have measures whose sum is 180, then the angles form a linear pair. That is, if angles β π΄π΅π· and β π·π΅πΆ are adjacent and π(β π΄π΅π·) + π(β π·π΅πΆ) = 180, then π΄ β π΅ β πΆ. Theorem 43: Vertical Pair Theorem (page 138) If two angles form a vertical pair then they have the same measure. page 353 Theorem 44: about angles with measure 90 (page 140) For any angle, the following two statements are equivalent. (i) There exists another angle that forms a linear pair with the given angle and that has the same measure. (ii) The given angle has measure 90. Theorem 45: If two intersecting lines form a right angle, then they actually form four. (page 142) Theorem 46: existence and uniqueness of a line that is perpendicular to a given line through a given point that lies on the given line (page 142) For any given line, and any given point that lies on the given line, there is exactly one line that passes through the given point and is perpendicular to the given line. Theorem 47: Angle congruence is an equivalence relation. (page 144) Theorem 48: Congruent Angle Construction Theorem (page 144) Let βββββ π΄π΅ be a ray on the edge of a half-plane π». For any angle β πΆπ·πΈ, there is exactly one βββββ ray π΄π with point π in π» such that β ππ΄π΅ β β πΆπ·πΈ. Theorem 49: Congruent Angle Addition Theorem (page 144) If point π· lies in the interior of β π΄π΅πΆ and point π·β² lies in the interior of β π΄β²π΅β²πΆβ², and β π΄π΅π· β β π΄β²π΅β²π·β² and β π·π΅πΆ β β π·β²π΅β²πΆβ², then β π΄π΅πΆ β β π΄β²π΅β²πΆβ². Theorem 50: Congruent Angle Subtraction Theorem (page 144) If point π· lies in the interior of β π΄π΅πΆ and point π·β² lies in the interior of β π΄β²π΅β²πΆβ², and β π΄π΅π· β β π΄β²π΅β²π·β² and β π΄π΅πΆ β β π΄β²π΅β²πΆβ², then β π·π΅πΆ β β π·β²π΅β²πΆβ². Theorem 51: triangle congruence is an equivalence relation (page 152) Theorem 52: the CS ο¨ CA theorem for triangles (the Isosceles Triangle Theorem) (page 156) In Neutral geometry, if two sides of a triangle are congruent, then the angles opposite those sides are also congruent. That is, in a triangle, if CS then CA. Theorem 53: (Corollary) In Neutral Geometry, if a triangle is equilateral then it is equiangular. (page 157) Theorem 54: the ASA Congruence Theorem for Neutral Geometry (page 157) In Neutral Geometry, if there is a one-to-one correspondence between the vertices of two triangles, and two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, then all the remaining corresponding parts are congruent as well, so the correspondence is a congruence and the triangles are congruent. Theorem 55: the CA ο¨ CS theorem for triangles in Neutral Geometry (page 159) In Neutral geometry, if two angles of a triangle are congruent, then the sides opposite those angles are also congruent. That is, in a triangle, if CA then CS. Chapter 17 Appendix 2: List of Theorems page 354 Theorem 56: (Corollary) In Neutral Geometry, if a triangle is equiangular then it is equilateral. (page 159) Theorem 57: (Corollary) The CACS theorem for triangles in Neutral Geometry. (page 160) In any triangle in Neutral Geometry, congruent angles are always opposite congruent sides. That is, CA ο³ CS. Theorem 58: the SSS congruence theorem for Neutral Geometry (page 160) In Neutral Geometry, if there is a one-to-one correspondence between the vertices of two triangles, and the three sides of one triangle are congruent to the corresponding parts of the other triangle, then all the remaining corresponding parts are congruent as well, so the correspondence is a congruence and the triangles are congruent. Theorem 59: Neutral Exterior Angle Theorem (page 162) In Neutral Geometry, the measure of any exterior angle is greater than the measure of either of its remote interior angles. Theorem 60: (Corollary) If a triangle has a right angle, then the other two angles are acute. (page 164) Theorem 61: the BS ο¨ BA theorem for triangles in Neutral Geometry (page 164) In Neutral Geometry, if one side of a triangle is longer than another side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side. That is, in a triangle, if BS then BA. Theorem 62: the BA ο¨ BS theorem for triangles in Neutral Geometry (page 164) In Neutral Geometry, if the measure of one angle is greater than the measure of another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. That is, in a triangle, if BA then BS. Theorem 63: (Corollary) The BABS theorem for triangles in Neutral Geometry. (page 165) In any triangle in Neutral Geometry, bigger angles are always opposite bigger sides. That is, BA ο³ BS. Theorem 64: The Triangle Inequality for Neutral Geometry (page 166) In Neutral Geometry, the length of any side of any triangle is less than the sum of the lengths of the other two sides. That is, for all non-collinear points π΄, π΅, πΆ, the inequality π΄πΆ < π΄π΅ + π΅πΆ is true. Theorem 65: The Distance Function Triangle Inequality for Neutral Geometry (page 167) The function π satisfies the Distance Function Triangle Inequality. That is, for all points π, π, π , the inequality π(π, π ) β€ π(π, π) + π(π, π ) is true. Theorem 66: existence and uniqueness of a line that is perpendicular to a given line through a given point that does not lie on the given line (page 170) For any given line and any given point that does not lie on the given line, there is exactly one line that passes through the given point and is perpendicular to the given line. page 355 Theorem 67: The shortest segment connecting a point to a line is the perpendicular. (page 172) Theorem 68: In any right triangle in Neutral Geometry, the hypotenuse is the longest side. (page 173) Theorem 69: (Lemma) In any triangle in Neutral Geometry, the altitude to a longest side intersects the longest side at a point between the endpoints. (page 173) Given: Neutral Geometry triangle Ξπ΄π΅πΆ, with point π· the foot of the altitude line β‘ββββ . drawn from vertex πΆ to line π΄π΅ Μ Μ Μ Μ is the longest side (that is, if π΄π΅ > π΅πΆ and π΄π΅ > π΅πΆ), then π΄ β π· β π΅. Claim: If π΄π΅ Theorem 70: the Angle-Angle-Side (AAS) Congruence Theorem for Neutral Geometry (p.175) In Neutral Geometry, if there is a correspondence between parts of two right triangles such that two angles and a non-included side of one triangle are congruent to the corresponding parts of the other triangle, then all the remaining corresponding parts are congruent as well, so the correspondence is a congruence and the triangles are congruent. Theorem 71: the Hypotenuse Leg Congruence Theorem for Neutral Geometry (page 176) In Neutral Geometry, if there is a one-to-one correspondence between the vertices of any two right triangles, and the hypotenuse and a side of one triangle are congruent to the corresponding parts of the other triangle, then all the remaining corresponding parts are congruent as well, so the correspondence is a congruence and the triangles are congruent. Theorem 72: the Hinge Theorem for Neutral Geometry (page 177) In Neutral Geometry, if triangles Ξπ΄π΅πΆ and Ξπ·πΈπΉ have Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ π·πΈ and Μ Μ Μ Μ π΄πΆ β Μ Μ Μ Μ π·πΉ and π(β π΄) > π(β π·), then π΅πΆ > πΈπΉ. Theorem 73: Equivalent statements about angles formed by two lines and a transversal in Neutral Geometry (page 178) Given: Neutral Geometry, lines πΏ and π and a transversal π, with points π΄, β― , π» labeled as in Definition 61, above. Claim: The following statements are equivalent: (1) The first pair of alternate interior angles is congruent. That is, β π΄π΅πΈ β β πΉπΈπ΅. (2) The second pair of alternate interior angles is congruent. That is, β πΆπ΅πΈ β β π·πΈπ΅. (3) The first pair of corresponding angles is congruent. That is, β π΄π΅πΊ β β π·πΈπΊ. (4) The second pair of corresponding angles is congruent. That is, β π΄π΅π» β β π·πΈπ». (5) The third pair of corresponding angles is congruent. That is, β πΆπ΅πΊ β β πΉπΈπΊ. (6) The fourth pair of corresponding angles is congruent. That is, β πΆπ΅π» β β πΉπΈπ». (7) The first pair of interior angles on the same side of the transversal has measures that add up to 180. That is, π(β π΄π΅πΈ) + π(β π·πΈπ΅) = 180. (8) The second pair of interior angles on the same side of the transversal has measures that add up to 180. That is, π(β πΆπ΅πΈ) + π(β πΉπΈπ΅) = 180. Theorem 74: The Alternate Interior Angle Theorem for Neutral Geometry (page 180) Given: Neutral Geometry, lines πΏ and π and a transversal π Claim: If a pair of alternate interior angles is congruent, then lines πΏ and π are parallel. Chapter 17 Appendix 2: List of Theorems page 356 Contrapositive: If πΏ and π are not parallel, then a pair of alternate interior angles are not congruent. Theorem 75: Corollary of The Alternate Interior Angle Theorem for Neutral Geometry (p. 181) Given: Neutral Geometry, lines πΏ and π and a transversal π Claim: If any of the statements of Theorem 73 are true (that is, if lines πΏ, π, π have the special angle property), then πΏ and π are parallel . Contrapositive: If πΏ and π are not parallel, then all of the statements of Theorem 73 are false (that is, lines πΏ, π, π do not have the special angle property). Theorem 76: Existence of a parallel through a point π not on a line πΏ in Neutral Geometry. (page 181) In Neutral Geometry, for any line πΏ and any point π not on πΏ, there exists at least one line π that passes through π and is parallel to πΏ. Theorem 77: In Neutral Geometry, the Number of Possible Intersection Points for a Line and a Circle is 0, 1, 2. (page 187) Theorem 78: In Neutral Geometry, tangent lines are perpendicular to the radial segment. (p.188) Given: A segment Μ Μ Μ Μ π΄π΅ and a line πΏ passing through point π΅. Claim: The following statements are equivalent. (i) Line πΏ is perpendicular to segment Μ Μ Μ Μ π΄π΅ . (ii) Line πΏ is tangent to πΆπππππ(π΄, π΄π΅) at point π΅. That is, πΏ only intersects πΆπππππ(π΄, π΄π΅) at point π΅. Theorem 79: (Corollary of Theorem 78) For any line tangent to a circle in Neutral Geometry, all points on the line except for the point of tangency lie in the circleβs exterior. (page 189) Theorem 80: about points on a Secant line lying in the interior or exterior in Neutral Geometry (page 189) Given: πΆπππππ(π΄, π) and a secant line πΏ passing through points π΅ and πΆ on the circle Claim: (i) If π΅ β π· β πΆ, then π· is in the interior of the circle. (ii) If π· β π΅ β πΆ or π΅ β πΆ β π·, then π· is in the exterior of the circle. Theorem 81: (Corollary of Theorem 80) about points on a chord or radial segment that lie in the interior in Neutral Geometry (page 190) In Neutral Geometry, all points of a segment except the endpoints lie in the interior of the circle. Furthermore, one endpoint of a radial segment lies on the circle; all the other points of a radial segment lie in the interior of the circle. Theorem 82: In Neutral Geometry, if a line passes through a point in the interior of a circle, then it also passes through a point in the exterior. (page 190) Theorem 83: In Neutral Geometry, if a line passes through a point in the interior of a circle and also through a point in the exterior, then it intersects the circle at a point between those two points. (page 190) page 357 Theorem 84: (Corollary) In Neutral Geometry, if a line passes through a point in the interior of a circle, then the line must be a secant line. That is, the line must intersect the circle exactly twice. (page 190) Theorem 85: about special rays in isosceles triangles in Neutral Geometry (page 191) Given: Neutral Geometry, triangle Ξπ΄π΅πΆ with Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ π΄πΆ and ray βββββ π΄π· such that π΅ β π· β πΆ. Claim: The following three statements are equivalent. (i) Ray βββββ π΄π· is the bisector of angle β π΅π΄πΆ. Μ Μ Μ Μ . (ii) Ray βββββ π΄π· is perpendicular to side π΅πΆ Μ Μ Μ Μ . That is, point π· is the midpoint of side π΅πΆ Μ Μ Μ Μ . (iii) Ray βββββ π΄π· bisects side π΅πΆ Theorem 86: about points equidistant from the endpoints of a line segment in Neutral Geometry (page 192) In Neutral Geometry, the following two statements are equivalent (i) A point is equidistant from the endpoints of a line segment. (ii) The point lies on the perpendicular bisector of the segment. Theorem 87: In Neutral Geometry, any perpendicular from the center of a circle to a chord bisects the chord (page 192) Theorem 88: In Neutral Geometry, the segment joining the center to the midpoint of a chord is perpendicular to the chord. (page 192) Theorem 89: (Corollary of Theorem 86) In Neutral Geometry, the perpendicular bisector of a chord passes through the center of the circle. (page 192) Theorem 90: about chords equidistant from the centers of circles in Neutral Geometry (p. 193) Μ Μ Μ Μ in πΆπππππ(π΄, π) and chord ππ Μ Μ Μ Μ in πΆπππππ(π, π) with the Given: Neutral Geometry, chord π΅πΆ same radius π. Claim: The following two statements are equivalent: Μ Μ Μ Μ to center π΄ is the same as the distance from (i) The distance from chord π΅πΆ Μ Μ Μ Μ to center π. chord ππ Μ Μ Μ Μ β ππ Μ Μ Μ Μ . (ii) The chords have the same length. That is, π΅πΆ Theorem 91: about points on the bisector of an angle in Neutral Geometry (page 194) Given: Neutral Geometry, angle β π΅π΄πΆ, and point π· in the interior of the angle Claim: The following statements are equivalent (i) π· lies on the bisector of angle β π΅π΄πΆ. (ii) π· is equidistant from the sides of angle β π΅π΄πΆ. Theorem 92: in Neutral Geometry, the three angle bisectors of any triangle are concurrent at a point that is equidistant from the three sides of the triangle. (page 194) Theorem 93: about tangent lines drawn from an exterior point to a circle in Neutral Geometry (page 195) Chapter 17 Appendix 2: List of Theorems page 358 Given: Neutral Geometry, πΆπππππ(π, π), point π΄ in the exterior of the circle, and lines β‘ββββ π΄π΅ β‘ββββ and π΄πΆ tangent to the circle at points π΅ and πΆ. Claim: Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ π΄πΆ and β ππ΄π΅ β β ππ΄πΆ. Theorem 94: In Neutral Geometry, if three points lie on a circle, then they do not lie on any other circle. (page 195) Theorem 95: in Neutral Geometry, every triangle has exactly one inscribed circle. (page 196) Theorem 96: In Neutral Geometry, if one circle passes through a point that is in the interior of another circle and also passes through a point that is in the exterior of the other circle, then the two circles intersect at exactly two points. (page 197) Theorem 97: (Corollary) In Euclidean Geometry, the answer to the recurring question is exactly one line. (page 202) In Euclidean Geometry, for any line πΏ and any point π not on πΏ, there exists exactly one line π that passes through π and is parallel to πΏ. Theorem 98: (corollary) In Euclidean Geometry, if a line intersects one of two parallel lines, then it also intersects the other. (page 202) In Euclidean Geometry, if πΏ and π are parallel lines, and line π intersects π, then π also intersects πΏ. Theorem 99: (corollary) In Euclidean Geometry, if two distinct lines are both parallel to a third line, then the two lines are parallel to each other. (page 203) In Euclidean Geometry, if distinct lines π and π are both are parallel to line πΏ, then π and π are parallel to each other. Theorem 100: Converse of the Alternate Interior Angle Theorem for Euclidean Geometry (page 203) Given: Euclidean Geometry, lines πΏ and π and a transversal π Claim: If πΏ and π are parallel, then a pair of alternate interior angles is congruent Theorem 101: (corollary) Converse of Theorem 75. (page 204) Given: Euclidean Geometry, lines πΏ and π and a transversal π Claim: If πΏ and π are parallel, then all of the statements of Theorem 73 are true (that is, lines πΏ, π, π have the special angle property). Theorem 102: (corollary) In Euclidean Geometry, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. That is, if lines πΏ and π are parallel, and line π is perpendicular to π, then π is also perpendicular to πΏ. (page 204) Theorem 103: In Euclidean Geometry, the angle sum for any triangle is 180. (page 204) Theorem 104: (corollary) Euclidean Exterior Angle Theorem. (page 205) page 359 In Euclidean Geometry, the measure of any exterior angle is equal to the sum of the measure of its remote interior angles Theorem 105: (corollary) In Euclidean Geometry, the angle sum of any convex quadrilateral is 360. (page 205) Theorem 106: In Euclidean Geometry, the perpendicular bisectors of the three sides of any triangle are concurrent at a point that is equidistant from the vertices of the triangle. (This point will be called the circumcenter.) (page 206) Theorem 107: (corollary) In Euclidean Geometry, every triangle can be circumscribed. (p. 207) Theorem 108: equivalent statements about convex quadrilaterals in Euclidean Geometry (p.208) In Euclidean Geometry, given any convex quadrilateral, the following statements are equivalent (TFAE) (i) Both pairs of opposite sides are parallel. That is, the quadrilateral is a parallelogram. (ii) Both pairs of opposite sides are congruent. (iii) One pair of opposite sides is both congruent and parallel. (iv) Each pair of opposite angles is congruent. (v) Either diagonal creates two congruent triangles. (vi) The diagonals bisect each other. Theorem 109: (corollary) In Euclidean Geometry, parallel lines are everywhere equidistant. (page 208) In Euclidean Geometry, if lines πΎ and πΏ are parallel, and line π is a transversal that is perpendicular to lines πΎ and πΏ at points π΄ and π΅, and line π is a transversal that is perpendicular to lines πΎ and πΏ at points πΆ and π·, then π΄π΅ = πΆπ·. Theorem 110: The Euclidean Geometry Triangle Midsegment Theorem (page 209) In Euclidean Geometry, if the endpoints of a line segment are the midpoints of two sides of a triangle, then the line segment is parallel to the third side and is half as long as the third side. That is, a midsegment of a triangle is parallel to the third side and half as long. Theorem 111: Properties of Medial Triangles in Euclidean Geometry (page 210) (1) The sides of the medial triangle are parallel to sides of outer triangle and are half as long. (2) The altitude lines of the medial triangle are the perpendicular bisectors of the sides of the outer triangle. (3) The altitude lines of the medial triangle are concurrent. Theorem 112: In Euclidean Geometry any given triangle is a medial triangle for some other. (page 210) Theorem 113: (Corollary) In Euclidean Geometry, the altitude lines of any triangle are concurrent. (page 211) Theorem 114: about π distinct parallel lines intersecting a transversal in Euclidean Geometry (page 212) Given: in Euclidean Geometry, parallel lines πΏ1 , πΏ2 , β― , πΏπ intersecting a transversal π at points π1 , π2 , β― , ππ such that π1 β π2 β β― β ππ . Chapter 17 Appendix 2: List of Theorems page 360 Claim: The following are equivalent (i) Lines πΏ1 , πΏ2 , β― , πΏπ are equally spaced parallel lines. (ii) The lines cut congruent segments in transversal π. That is, Μ Μ Μ Μ Μ Μ π1 π2 β Μ Μ Μ Μ Μ Μ π2 π3 β β― β Μ Μ Μ Μ Μ Μ Μ Μ Μ ππβ1 ππ . Theorem 115: (Corollary) about π distinct parallel lines cutting congruent segments in transversals in Euclidean Geometry (page 212) If a collection of π parallel lines cuts congruent segments in one transversal, then the n parallel lines must be equally spaced and so they will also cut congruent segments in any transversal. Theorem 116: about concurrence of medians of triangles in Euclidean Geometry (page 212) In Euclidean Geometry, the medians of any triangle are concurrent at a point that can be called the centroid. Furthermore, the distance from the centroid to any vertex is 2/3 the length of the median drawn from that vertex. Theorem 117: Parallel Projection in Euclidean Geometry is one-to-one and onto. (page ) Theorem 118: Parallel Projection in Euclidean Geometry preserves betweenness. (page 220) If πΏ, π, π are lines, and π intersects both πΏ and π, and π΄, π΅, πΆ are points on πΏ with π΄ β π΅ β πΆ, then π΄β² β π΅ β² β πΆβ². Theorem 119: Parallel Projection in Euclidean Geometry preserves congruence of segments. (page 220) If πΏ, π, π are lines, and π intersects both πΏ and π, and π΄, π΅, πΆ, π· are points on πΏ with Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ πΆπ·, then π΄β²π΅β² β πΆβ²π·β². Theorem 120: Parallel Projection in Euclidean Geometry preserves ratios of lengths of segments. (page 221) If πΏ, π, π are lines, and π intersects both πΏ and π, and π΄, π΅, πΆ, π· are points on πΏ with πΆ β π·, π΄β²π΅β² π΄π΅ then πΆβ²π·β² = πΆπ·. Theorem 121: (corollary) about lines that are parallel to the base of a triangle in Euclidean Geometry. (page 221) Μ Μ Μ Μ of triangle Ξπ΄π΅πΆ and intersects rays In Euclidean Geometry, if line π is parallel to side π΅πΆ π΄π· π΄πΈ βββββ π΄π΅ and βββββ π΄πΆ at points π· and πΈ, respectively, then π΄π΅ = π΄πΆ . Theorem 122: The Angle Bisector Theorem. (page 222) In Euclidean Geometry, the bisector of an angle in a triangle splits the opposite side into two segments whose lengths have the same ratio as the two other sides. That is, in Ξπ΄π΅πΆ, if π· is π·π΄ π΅π΄ the point on side Μ Μ Μ Μ π΄πΆ such that ray ββββββ π΅π· bisects angle β π΄π΅πΆ, then π·πΆ = π΅πΆ . Theorem 123: triangle similarity is an equivalence relation (page 223) Theorem 124: The Angle-Angle-Angle (AAA) Similarity Theorem for Euclidean Geometry (page 225) If there is a one-to-one correspondence between the vertices of two triangles, and each pair of corresponding angles is a congruent pair, then the ratios of the lengths of each pair of page 361 corresponding sides is the same, so the correspondence is a similarity and the triangles are similar. Theorem 125: (Corollary) The Angle-Angle (AA) Similarity Theorem for Euclidean Geometry (page 226) If there is a one-to-one correspondence between the vertices of two triangles, and two pairs of corresponding angles are congruent pairs, then the third pair of corresponding angles is also a congruent pair, and the ratios of the lengths of each pair of corresponding sides is the same, so the correspondence is a similarity and the triangles are similar. Theorem 126: (corollary) In Euclidean Geometry, the altitude to the hypotenuse of a right triangle creates two smaller triangles that are each similar to the larger triangle. (page 226) Theorem 127: The Side-Side-Side (SSS) Similarity Theorem for Euclidean Geometry (p. 226) If there is a one-to-one correspondence between the vertices of two triangles, and the ratios of lengths of all three pairs of corresponding sides is the same, then all three pairs of corresponding angles are congruent pairs, so the correspondence is a similarity and the triangles are similar. Theorem 128: The Side-Angle-Side (SAS) Similarity Theorem for Euclidean Geometry (p. 227) If there is a one-to-one correspondence between the vertices of two triangles, and the ratios of lengths of two pairs of corresponding sides is the same and the corresponding included angles are congruent, then the other two pairs of corresponding angles are also congruent pairs and the ratios of the lengths of all three pairs of corresponding sides is the same, so the correspondence is a similarity and the triangles are similar. Theorem 129: About the ratios of lengths of certain line segments associated to similar triangles in Euclidean Geometry. (page 228) In Euclidean Geometry, if Ξ~Ξβ², then πππππ‘β ππ π πππ πππππ‘β ππ πππ‘ππ‘π’ππ πππππ‘β ππ πππππ πππ πππ‘ππ πππππ‘β ππ ππππππ = = = πππππ‘β ππ π πππβ² πππππ‘β ππ πππ‘ππ‘π’ππβ² πππππ‘β ππ πππππ πππ πππ‘ππβ² πππππ‘β ππ ππππππβ² Theorem 130: The Pythagorean Theorem of Euclidean Geometry (page 229) In Euclidean Geometry, the sum of the squares of the length of the two sides of any right triangle equals the square of the length of the hypotenuse. That is, in Euclidean Geometry, given triangle Ξπ΄π΅πΆ with π = π΅πΆ and π = πΆπ΄ and π = π΄π΅, if angle β πΆ is a right angle, then π2 + π 2 = π 2 . Theorem 131: The Converse of the Pythagorean Theorem of Euclidean Geometry (page 230) In Euclidean Geometry, if the sum of the squares of the length of two sides of a triangle equals the square of the length of the third side, then the angle opposite the third side is a right angle. That is, in Euclidean Geometry, given triangle Ξπ΄π΅πΆ with π = π΅πΆ and π = πΆπ΄ and π = π΄π΅, if π2 + π 2 = π 2 , then angle β πΆ is a right angle. Theorem 132: In Euclidean Geometry, the product of πππ π β βπππβπ‘ in a triangle does not depend on which side of the triangle is chosen as the base. (page 230) Chapter 17 Appendix 2: List of Theorems page 362 Theorem 133: (accepted without proof) Given any polygonal region, any two complexes for that region have the same area sum. (page 242) If π is a polygonal region and πΆ1 and πΆ2 are two complexes for π , then the sum of the areas of the triangular regions of complex πΆ1 equals the sum of the areas of the triangular regions of complex πΆ2 . Theorem 134: Properties of the Area Function for Polygonal Regions (page 243) Congruence: If π 1 and π 2 are triangular regions bounded by congruent triangles, then π΄πππ(π 1 ) = π΄πππ(π 2 ). Additivity: If π 1 and π 2 are polygonal regions whose interiors do not intersect, then π΄πππ(π 1 βͺ π 2 ) = π΄πππ(π 1 ) + π΄πππ(π 2 ). Theorem 135: about the ratio of the areas of similar triangles (page 245) The ratio of the areas of a pair of similar triangles is equal to the square of the ratio of the lengths of any pair of corresponding sides. Theorem 136: polygon similarity is an equivalence relation (page 246) Theorem 137: about the ratio of the areas of similar n-gons (page 248) The ratio of the areas of a pair of similar n-gons (not necessarily convex) is equal to the square of the ratio of the lengths of any pair of corresponding sides. Theorem 138: If two distinct arcs share both endpoints, then the sum of their arc angle measures Μ and π΄π·πΆ Μ are distinct, then π Μ) + π Μ ) = 360. (page 260) is 360. That is, if π΄π΅πΆ Μ (π΄π΅πΆ Μ (π΄π·πΆ Theorem 139: Two chords of a circle are congruent if and only if their corresponding arcs have the same measure.(page 260) Theorem 140: The Arc Measure Addition Theorem (page 260) Μ and πΆπ·πΈ Μ and π΄πΆπΈ Μ are arcs, then π Μ) = π Μ) + π Μ ). If π΄π΅πΆ Μ (π΄πΆπΈ Μ (π΄π΅πΆ Μ (πΆπ·πΈ Theorem 141: the angle measure of an inscribed angle (Type 2) is equal to half the arc angle measure of the intercepted arc. (page 262) Theorem 142: (Corollary) Any inscribed angle that intercepts a semicircle is a right angle. (page 264) Theorem 143: the angle measure of an angle of Type 3. (page 264) The angle measure of an angle of Type 3 is equal to the average of the arc angle measures of two arcs. One arc is the arc intercepted by the angle, itself. The other arc is the arc intercepted by the angle formed by the opposite rays of the original angle. Theorem 144: the angle measure of an angle of Type 4. (page 265) The angle measure of an angle of Type 4 is equal to one half the difference of the arc angle measures of the two arcs intercepted by the angle. (The difference computed by subracting the smaller arc angle measure from the larger one.) page 363 Theorem 145: the angle measure of an angle of Type 5. (page 265) The angle measure of an angle of Type 5 can be related to the arc angle measures of the arcs that it intersects in three useful ways: (i) The angle measure of the angle is equal to 180 minus the arc angle measure of the smaller intercepted arc (ii) The angle measure of the angle is equal to the arc angle measure of the larger intercepted arc minus 180 (iii) The angle measure of the angle is equal to half the difference of the arc angle measures of the two arcs intercepted by the angle. (The difference computed by subracting the smaller arc angle measure from the larger one.) Theorem 146: the angle measure of an angle of Type 6. (page 266) The angle measure of an angle of Type 6 is equal to one half the arc angle measure of the arc intercepted by the angle. Theorem 147: the angle measure of an angle of Type 7. (page 267) The angle measure of an angle of Type 7 is equal to one half the difference of the arc angle measures of the two arcs intercepted by the angle. (The difference computed by subracting the smaller arc angle measure from the larger one. Theorem 148: In Euclidean Geometry, in any cyclic quadrilateral, the sum of the measures of each pair of opposite angles is 180. That is, if ππ’ππ(π΄π΅πΆπ·) is cyclic, then π(β π΄) + π(β πΆ) = 180 and π(β π΅) + π(β π·) = 180. (page 269) Theorem 149: about angles that intercept a given arc (page 269) Μ and a point π on the opposite side of line π΄πΆ β‘ββββ from In Euclidean geometry, given an arc π΄π΅πΆ point π΅, the following are equivalent: (1) Point π lies on πΆπππππ(π΄, π΅, πΆ). Μ) Μ (π΄π΅πΆ π (2) π(β π΄ππΆ) = 2 . Theorem 150: In Euclidean Geometry, in any convex quadrilateral, if the sum of the measures of either pair of opposite angles is 180, then the quadrilateral is cyclic. (page 270) Theorem 151: The Intersecting Secants Theorem (p. 271) In Euclidean geometry, if two secant lines intersect, then the product of the distances from the intersection point to the two points where one secant line intersects the circle equals the product of the distances from the intersection point to the two points where the other secant line intersects the circle. That is, if secant line πΏ passes through a point π and intersects the circle at points π΄ and π΅ and secant line π passes through a point π and intersects the circle at points π· and πΈ, then ππ΄ β ππ΅ = ππ΄ β ππ΅. Theorem 152: about intersecting secant and tangent lines. (page 273) In Euclidean geometry, if a secant line and tangent line intersect, then the square of the distance from the point of intersection to the point of tangency equals the product of the distances from the intersection point to the two points where the secant line intersects the circle. That is, if a secant line passes through a point π and intersects the circle at points π΄ and π΅ and a tangent line passes through π and intersects the circle at π·, then (ππ·)2 = ππ΄ β ππ΅ Chapter 17 Appendix 2: List of Theorems page 364 Theorem 153: Miquelβs Theorem (page 279) If points π΄, π΅, πΆ, π·, πΈ, πΉ are given such that points π΄, π΅, πΆ are non-collinear and π΄ β π· β π΅ and π΅ β πΈ β πΆ and πΆ β πΉ β π΄, then πΆπππππ(π΄, π·, πΉ) and πΆπππππ(π΅, πΈ, π·) and πΆπππππ(πΆ, πΉ, πΈ) exist and there exists a point πΊ that lies on all three circles. Theorem 154: Menelausβs Theorem (page 279) Given: points π΄, π΅, πΆ, π·, πΈ, πΉ such that π΄, π΅, πΆ are non-collinear and π΄ β π΅ β π· and π΅ β πΈ β πΆ and πΆ β πΉ β π΄ Claim: The following are equivalent (1) πΉ β πΈ β π· π΄π· π΅πΈ πΆπΉ (2) π·π΅ β πΆπΈ β π΄πΉ = β1 Theorem 155: Cevaβs Theorem (found on page 280) Given: points π΄, π΅, πΆ, π·, πΈ, πΉ such that π΄, π΅, πΆ are non-collinear and π΄ β π· β π΅ and π΅ β πΈ β πΆ and πΆ β πΉ β π΄ Claim: The following are equivalent (1) Lines β‘ββββ π΄πΈ , β‘ββββ π΅πΉ , β‘ββββ πΆπ· are concurrent π΄π· π΅πΈ πΆπΉ (2) π·π΅ β πΆπΈ β π΄πΉ = 1 Theorem 156: (Corollary to Cevaβs Theorem) (found on page 280) If π΄, π΅, πΆ are non-collinear points and π΄ β π· β π΅ and π΅ β πΈ β πΆ and πΆ β πΉ β π΄ are the points of β‘ββββ are concurrent. β‘ββββ , πΆπ· tangency of the inscribed circle for Ξπ΄π΅πΆ, then lines β‘ββββ π΄πΈ , π΅πΉ Theorem 157: Collinearity of the orthcenter, centroid, and circumcenter (found on page 281) If a non-equilateral triangleβs orthcenter, centroid, and circumcenter are labeled π΄, π΅, πΆ, Then π΄, π΅, πΆ are collinear, with π΄ β π΅ β πΆ and π΄π΅ = 2π΅πΆ. Theorem 158: Existence of a circle passing through nine special points associated to a triangle (found on page 282) For every triangle, there exists a single circle that passes through the midpoints of the three sides, the feet of the three altitudes, and the three Euler points. Theorem 159: The ratio πππππ’ππππππππ ππππππ‘ππ is the same for all circles. (page 287) Theorem 160: The circumference of a circle is π = ππ. The area of a circle is π΄ = ππ 2 . (p. 290) Theorem 161: Function composition is associative. (page 298) For all functions π: π΄ β π΅ and π: π΅ β πΆ and β: πΆ β π·, the functions β β (π β π) and (β β π) β π are equal. Theorem 162: Bijective functions have inverse functions that are also bijective. (page 302) If π: π΄ β π΅ is a bijective function, then π has an inverse function π β1 : π΅ β π΄. The inverse function is also bijective. page 365 Theorem 163: If a function has an inverse function, then both the function and its inverse are bijective. (page 303) If functions π: π΄ β π΅ and π: π΅ β π΄ are inverses of one another (that is, if they satisfy the inverse relations), then both π and π are bijective. Theorem 164: about the inverse of a composition of functions (page 304) If functions π: π΄ β π΅ and π: π΅ β πΆ are both bijective, then their composition π β π will be bijective. The inverse of the composition will be (π β π)β1 = π β1 β πβ1 . Theorem 165: the pair (π,β) consisting of the set of Transformations of the Plane and the operation of composition of functions, is a group. (page 310) Theorem 166: The composition of two isometries of the plane is also an isometry of the plane. (page 312) Theorem 167: Every isometry of the plane is one-to-one. (page 312) Theorem 168: For three distinct points, betweenness is related to distance between the points. (page 313) For distinct points π΄, π΅, πΆ, the following two statements are equivalent. (i) π΄ β π΅ β πΆ (ii) π(π΄, π΅) + π(π΅, πΆ) = π(π΄, πΆ) Theorem 169: Isometries of the plane preserve collinearity. (page 313) If π΄, π΅, πΆ are distinct, collinear points and π is an isometry of the plane, then π(π΄), π(π΅), π(πΆ) are distinct, collinear points. Theorem 170: Every isometry of the plane is onto. (page 314) Theorem 171: (corollary) Every isometry of the plane is also a transformation of the plane. (page 315) Theorem 172: Every isometry of the plane has an inverse that is also an isometry. ( page 316) Theorem 173: the pair (πΌ,β) consisting of the set of Isometries of the Plane and the operation of composition of functions, is a group. (page 316) Theorem 174: Isometries of the plane preserve lines. (page 317) If πΏ is a line and π: π« β π« is an isometry, then the image π(πΏ) is also a line. Theorem 175: Isometries of the plane preserve circles. (page 318) If π: π« β π« is an isometry, then the image of a circle is a circle with the same radius. More specifically, the image π(πΆπππππ(π, π)) is the circle πΆπππππ(π(π), π). Theorem 176: If an isometry has three non-collinear fixed points, then the isometry is the identity map. (page 319) Chapter 17 Appendix 2: List of Theorems page 366 Theorem 177: If two isometries have the same images at three non-collinear fixed points, then the isometries are in fact the same isometry. (page 319) If π: π« β π« and π: π« β π« are isometries and π΄, π΅, πΆ are non-collinear points such that π(π΄) = π(π΄) and π(π΅) = π(π΅) and π(πΆ) = π(πΆ), then π = π.