### Transcription

Math Handbookof Formulas, Processes and Tricks GeometryPrepared by: Earl L. Whitney, FSA, MAAAVersion 3.2August 28, Earl Whitney, Reno NV.

Geometry HandbookTable of ContentsPageDescription67891011Chapter 1: BasicsPoints, Lines & PlanesSegments, Rays & LinesDistance Between Points (1‐Dimensional, 2‐Dimensional)Distance Formula in “n” DimensionsAnglesTypes of Angles12131415Chapter 2: ProofsConditional Statements (Original, Converse, Inverse, Contrapositive)Basic Properties of Algebra (Equality and Congruence, Addition and Multiplication)Inductive vs. Deductive ReasoningAn Approach to Proofs16171819Chapter 3: Parallel and Perpendicular LinesParallel Lines and TransversalsMultiple Sets of Parallel LinesProving Lines are ParallelParallel and Perpendicular Lines in the Coordinate Plane2021222324Chapter 4: Triangles ‐ BasicTypes of Triangles (Scalene, Isosceles, Equilateral, Right)Congruent Triangles (SAS, SSS, ASA, AAS, CPCTC)Centers of TrianglesLength of Height, Median and Angle BisectorInequalities in Triangles252627Chapter 5: PolygonsPolygons – Basic (Definitions, Names of Common Polygons)Polygons – More Definitions (Definitions, Diagonals of a Polygon)Interior and Exterior Angles of a PolygonCover art by Rebecca Williams.2Page 2 of 82August 28

Geometry HandbookTable of ContentsPageDescription2829303132Chapter 6: QuadrilateralsDefinitions of QuadrilateralsFigures of QuadrilateralsCharacteristics of ParallelogramsParallelogram Proofs (Sufficient Conditions)Kites and Trapezoids333536374041Chapter 7: TransformationsIntroduction to TransformationReflectionRotationRotation by 90⁰ about a Point (x0, y0)TranslationCompositions4243444546474849Chapter 8: SimilarityRatios Involving UnitsSimilar PolygonsScale Factor of Similar PolygonsDilations of PolygonsMore on DilationSimilar Triangles (SSS, SAS, AA)Proportion Tables for Similar TrianglesThree Similar Triangles5051525354555657Chapter 9: Right TrianglesPythagorean TheoremPythagorean TriplesSpecial Triangles (45⁰‐45⁰‐90⁰ Triangle, 30⁰‐60⁰‐90⁰ Triangle)Trigonometric Functions and Special AnglesTrigonometric Function Values in Quadrants II, III, and IVGraphs of Trigonometric FunctionsVectorsOperating with VectorsVersion 3.2Page 3 of 82August 28,

Geometry HandbookTable of ContentsPageDescription5859Chapter 10: CirclesParts of a CircleAngles and Circles606162636465Chapter 11: Perimeter and AreaPerimeter and Area of a TriangleMore on the Area of a TrianglePerimeter and Area of QuadrilateralsPerimeter and Area of General PolygonsCircle Lengths and AreasArea of Composite Figures666768697071727374757677Chapter 12: Surface Area and VolumePolyhedraA Hole in Euler’s TheoremPlatonic SolidsPrismsCylindersSurface Area by DecompositionPyramidsConesSpheresSimilar SolidsSummary of Perimeter and Area Formulas – 2D ShapesSummary of Surface Area and Volume Formulas – 3D Shapes78IndexVersion 3.2Page 4 of 82August 28,

Geometry HandbookTable of ContentsUseful WebsitesWolfram Math World – Perhaps the premier site for mathematics on the Web. This site containsdefinitions, explanations and examples for elementary and advanced math – Developed specifically for math students from Middle School to College, based on theauthor's extensive experience in professional mathematics in a business setting and in mathtutoring. Contains free downloadable handbooks, PC Apps, sample tests, and more. California Standard Geometry Test – A standardized Geometry test released by the state ofCalifornia. A good way to test your m.pdfSchaum’s OutlinesAn important student resource for any high school math student is aSchaum’s Outline. Each book in this series provides explanations of thevarious topics in the course and a substantial number of problems for thestudent to try. Many of the problems are worked out in the book, so thestudent can see examples of how they should be solved.Schaum’s Outlines are available, Barnes & Noble andother booksellers.Version 3.2Page 5 of 82August 28,

Chapter 1Basic GeometryGeometryPoints, Lines & PlanesItemIllustrationNotationDefinitionPointA location in space.SegmentA straight path that has two endpoints.RayA straight path that has one endpointand extends infinitely in one direction.lLineorm or(points , ,not linear)PlaneA straight path that extends infinitely inboth directions.A flat surface that extends infinitely intwo dimensions.Collinear points are points that lie on the same line.Coplanar points are points that lie on the same plane.In the figure at right: , , , , and are points.l is a linem and n are planes.In addition, note that: , , and are collinear points. , and are coplanar points. , and are coplanar points. Raygoes off in a southeast direction. Raygoes off in a northwest direction. Together, rays Line l intersects both planes m and n.andmake up line l.Note: In geometric figures such as the one above, it isimportant to remember that, even though planes aredrawn with edges, they extend infinitely in the 2dimensions shown.Version 3.2Page 6 of 82An intersection of geometricshapes is the set of points theyshare in common.l and m intersect at point E.l and n intersect at point D.m and n intersect in line.August 28,

Chapter 1Basic GeometryGeometrySegments, Rays & LinesSome Thoughts About Line Segments Line segments are generally named by their endpoints, so thesegment at right could be named eitheror. Segmentcontains the two endpoints (A and B) and all points on linebetween them.that areRays Rays are generally named by their single endpoint,called an initial point, and another point on the ray. Raycontains its initial point A and all points on linein the direction of the arrow. Rays If point O is on lineandthen raysare not the same ray.andand is between points A and B,are called opposite rays. Theyhave only point O in common, and together they make up line.Lines Lines are generally named by either a single script letter(e.g., l) or by two points on the line (e.g.,.). A line extends infinitely in the directions shown by itsarrows. Lines are parallel if they are in the same plane and theynever intersect. Lines f and g, at right, are parallel. Lines are perpendicular if they intersect at a 90⁰ angle. Apair of perpendicular lines is always in the same plane.Lines f and e, at right, are perpendicular. Lines g and e arealso perpendicular. Lines are skew if they are not in the same plane and theynever intersect. Lines k and l, at right, are skew.(Remember this figure is 3‐dimensional.)Version 3.2Page 7 of 82August 28,

Chapter 1Basic GeometryGeometryDistance Between PointsDistance measures how far apart two things are. The distance between two points can bemeasured in any number of dimensions, and is defined as the length of the line connecting thetwo points. Distance is always a positive number.1‐Dimensional DistanceIn one dimension the distance between two points is determined simply by subtracting thecoordinates of the points.Example: In this segment, the distance between ‐2 and 5 is calculated as: 527.2‐Dimensional DistanceIn two dimensions, the distance between two points can be calculated by considering the linebetween them to be the hypotenuse of a right triangle. To determine the length of this line: Calculate the difference in the x‐coordinates of the pointsCalculate the difference in the y‐coordinates of the pointsUse the Pythagorean Theorem.This process is illustrated below, using the variable “d” for distance.Example: Find the distance between (‐1,1) and (2,5). Based on theillustration to the left:x‐coordinate difference: 2y‐coordinate difference: 511Then, the distance is calculated using the formula:3.4.3491625So,If we define two points generally as (x1, y1) and (x2, y2), then a 2‐dimensional distance formulawould be:Version 3.2Page 8 of 82August 28,

Chapter 1Basic GeometryGeometryDistance Formula in “n” DimensionsADVANCEDThe distance between two points can be generalized to “n” dimensions by successive use of thePythagorean Theorem in multiple dimensions. To move from two dimensions to threedimensions, we start with the two‐dimensional formula and apply the Pythagorean Theorem toadd the third dimension.3 DimensionsConsider two 3‐dimensional points (x1, y1, z1) and (x2, y2, z2). Consider first the situationwhere the two z‐coordinates are the same. Then, the distance between the points is 2‐.dimensional, i.e.,We then add a third dimension using the Pythagorean Theorem:And, finally the 3‐dimensional difference formula:n DimensionsUsing the same methodology in “n” dimensions, we get the generalized n‐dimensionaldifference formula (where there are n terms beneath the radical, one for each dimension): Or, in higher level mathematical notation:The distance between 2 points A (a1, a2, , an) and B (b1, b2, , bn) is,Version 3.2 Page 9 of 82August 28,

Chapter 1Basic GeometryGeometryAnglesParts of an AngleAn angle consists of two rays with a commonendpoint (or, initial point). Each ray is a side of the angle. The common endpoint is called the vertex ofthe angle.Naming AnglesAngles can be named in one of two ways: Point‐vertex‐point method. In this method, the angle is named from a point on oneray, the vertex, and a point on the other ray. This is the most unambiguous method ofnaming an angle, and is useful in diagrams with multiple angles sharing the same vertex.In the above figure, the angle shown could be named or . Vertex method. In cases where it is not ambiguous, an angle can be named based solelyon its vertex. In the above figure, the angle could be named .Measure of an AngleThere are two conventions for measuring the size of an angle: In degrees. The symbol for degrees is ⁰. There are 360⁰ in a full circle. The angle abovemeasures approximately 45⁰ (one‐eighth of a circle). In radians. There are 2 radians in a complete circle. The angle above measuresapproximatelyradians.Some Terms Relating to AnglesAngle interior is the area between the rays.Angle exterior is the area not between the rays.Adjacent angles are angles that share a ray for a side. in the figure at right are adjacent angles.andCongruent angles area angles with the same measure.Angle bisector is a ray that divides the angle into two congruentangles. RayVersion 3.2bisects in the figure at right.Page 10 of 82August 28,

Chapter 1Basic GeometryGeometryTypes of AnglesACBDSupplementary AnglesComplementary AnglesAngles A and B are supplementary.Angles C and D are complementary.Angles A and B form a linear pair. 90⁰180⁰Angles which are opposite each other whentwo lines cross are vertical angles.Angles E and G are vertical angles.Angles F and H are vertical angles.FEG H In addition, each angle is supplementary tothe two angles adjacent to it. For example:Vertical AnglesAngle E is supplementary to Angles F and H.An acute angle is one that is less than 90⁰. Inthe illustration above, angles E and G areacute angles.A right angle is one that is exactly 90⁰.AcuteObtuseRightStraightAn obtuse angle is one that is greater than90⁰. In the illustration above, angles F and Hare obtuse angles.A straight angle is one that is exactly 180⁰.Version 3.2Page 11 of 82August 28,

Chapter 2ProofsGeometryConditional StatementsA conditional statement contains both a hypothesis and a conclusion in the following form:If hypothesis, then conclusion.For any conditional statement, it is possible to create three relatedconditional statements, as shown below. In the table, p is the hypothesisof the original statement and q is the conclusion of the original statement.Type of Conditional StatementOriginal Statement: FALSEIf not p, then not q. ( )Example: If a number is not divisible by 6, then it is not divisible by 3.The inverse statement is always true when the converse is true andfalse when the converse is false.Contrapositive Statement:TRUEIf q, then p. ( )Example: If a number is divisible by 3, then it is divisible by 6.The converse statement may be either true or false, and this does notdepend on whether the original statement is true or false.Inverse Statement:ExampleStatement is:If p, then q. ( )Example: If a number is divisible by 6, then it is divisible by 3.The original statement may be either true or false.Converse Statement:Statements linkedbelow by red arrowsmust be either bothtrue or both false.FALSEIf not q, then not p. ( )Example: If a number is not divisible by 3, then it is not divisible by 6.The Contrapositive statement is always true when the originalstatement is true and false when the original statement is false.TRUENote also that: When two statements must be either both true or both false, they are called equivalentstatements.o The original statement and the contrapositive are equivalent statements.o The converse and the inverse are equivalent statements. If both the original statement and the converse are true, the phrase “if and only if”(abbreviated “iff”) may be used. For example, “A number is divisible by 3 iff the sum ofits digits is divisible by 3.”Version 3.2Page 12 of 82August 28,

Chapter 2ProofsGeometryBasic Properties of AlgebraProperties of Equality and Congruence.Definition for EqualityDefinition for CongruenceFor any real numbers a, b, and c:For any geometric elements a, b and c.(e.g., segment, angle, triangle)Property Reflexive PropertySymmetric Property, ,,Transitive PropertySubstitution Property If, then either can besubstituted for the other in anyequation (or inequality). , If , then either can besubstituted for the other in anycongruence expression.More Properties of Equality. For any real numbers a, b, and c:PropertyDefinition for EqualityAddition Property,Subtraction Property,Multiplication Property, 0,Division PropertyProperties of Addition and Multiplication. For