Geometry Vocabulary Cards



GeometryVocabulary Word Wall CardsMathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development.?The cards should be used as an instructional tool for teachers and then as a reference for all students. Table of ContentsReasoning, Lines, andTransformationsBasics of Geometry 1Basics of Geometry 2Geometry NotationLogic NotationSet NotationConditional StatementConverseInverseContrapositive HYPERLINK \l "symbolicrepresentation" Symbolic Representations in Logical ArgumentsConditional Statements and Venn DiagramsDeductive ReasoningInductive ReasoningHYPERLINK \l "directproofs"Direct ProofsProperties of CongruenceLaw of DetachmentLaw of SyllogismCounterexamplePerpendicular LinesParallel Lines Skew LinesTransversalCorresponding AnglesAlternate Interior AnglesAlternate Exterior AnglesConsecutive Interior AnglesParallel LinesMidpoint (definition)Midpoint FormulaFind a Missing EndpointSlope FormulaSlope of Lines in Coordinate PlaneDistance Formula HYPERLINK \l "linesymmetry" Line Symmetry (Examples) HYPERLINK \l "pointsymmetry" Point Symmetry (Examples)Rotation (Origin)ReflectionTranslationDilationPerpendicular BisectorConstructions:A line segment congruent to a given line segmentPerpendicular bisector of a line segmentA perpendicular to a given line from a point not on the lineA perpendicular to a given line at a point on the lineA bisector of an angleAn angle congruent to a given angleA line parallel to a given line through a point not on the given lineAn equilateral triangle inscribed in a circleA square inscribed in a circleA regular hexagon inscribed in a circleTrianglesClassifying Triangles by SidesClassifying Triangles by AnglesTriangle Sum TheoremExterior Angle TheoremPythagorean TheoremAngle and Sides RelationshipsTriangle Inequality TheoremCongruent TrianglesSSS Triangle Congruence PostulateSAS Triangle Congruence PostulateHL Right Triangle Congruence ASA Triangle Congruence PostulateAAS Triangle Congruence TheoremSimilar PolygonsSimilar Polygons and ProportionsAA Triangle Similarity PostulateSAS Triangle Similarity TheoremSSS Triangle Similarity TheoremAltitude of a TriangleMedian of a TriangleConcurrency of Medians of a Triangle30°-60°-90° Triangle Theorem45°-45°-90° Triangle TheoremTrigonometric RatiosInverse Trigonometric RatiosArea of a TrianglePolygons and CirclesPolygon Exterior Angle Sum TheoremPolygon Interior Angle Sum TheoremRegular PolygonProperties of ParallelogramsRectangleRhombusSquareTrapezoidIsosceles TrapezoidCircle HYPERLINK \l "circles" Circles – Inscribed Circle EquationLines and CirclesSecantTangentCentral AngleMeasuring ArcsArc LengthSecants and TangentsInscribed AngleArea of a SectorInscribed Angle Theorem 1Inscribed Angle Theorem 2Inscribed Angle Theorem 3Segments in a CircleSegments of Secants TheoremSegment of Secants and Tangents TheoremThree-Dimensional FiguresConeCylinderPolyhedronSimilar Solids TheoremSphereHemispherePyramidBasics of Geometry 14928870227965 Ppoint P00 Ppoint PPoint – A point has no dimension. It is a location on a plane. It is represented by a dot.15005931449070A B m00A B mLine – A line has one dimension. It is an infinite set of points represented by a line with two arrowheads that extend without end.1799560563614AB or BA or line m00AB or BA or line m4520062921340NABC00NABCPlane – A plane has two dimensions extending without end. It is often represented by a parallelogram.-73025361315plane ABC or plane N00plane ABC or plane NBasics of Geometry 2Line segment – A line segment consists of two endpoints and all the points between them.3462655563245AB or BA00AB or BA1756410730885B00B-50165498475A00A16446546990000Ray – A ray has one endpoint and extends without end in one direction.1557020732155C00C152019070993000336550438150003404235382905BC00BC21894801138555Note: Name the endpoint first.BC and CB are different rays. 00Note: Name the endpoint first.BC and CB are different rays. 1504951314450B00BGeometry NotationSymbols used to represent statements or operations in geometry.6534152667000BCsegment BC6597653556000BCray BC6388101841500BCline BCBClength of BCangle ABCmmeasure of angle ABCtriangle ABC||is parallel tois perpendicular to is congruent tois similar toLogic Notation?or?and→read “implies”, if… then…?read “if and only if”iffread “if and only if”~not∴thereforeSet Notation{ }empty set, null set?empty set, null setx|read “x such that”x:read “x such that”?union, disjunction, or?intersection, conjunction, andConditional Statementa logical argument consisting of a set of premises, hypothesis (p), and conclusion (q)2157730330200hypothesis 00hypothesis 2821305-121031000-4572074930If an angle is a right angle,then its measure is 90.00If an angle is a right angle,then its measure is 90.3160395-696595002609215180975conclusion00conclusionSymbolically:if p, then qp→qConverseformed by interchanging the hypothesis and conclusion of a conditional statement Conditional: If an angle is a right angle, then its measure is 90.-61669237977Converse: If an angle measures 90, then the angle is a right angle.00Converse: If an angle measures 90, then the angle is a right angle.Symbolically:if q, thenpq→pInverseformed by negating the hypothesis and conclusion of a conditional statementConditional: If an angle is a right angle, then its measure is 90.-58420375920Inverse: If an angle is not a right angle, then its measure is not 90.00Inverse: If an angle is not a right angle, then its measure is not 90.Symbolically:if ~p, then ~q~p→~qContrapositiveformed by interchanging and negating the hypothesis and conclusion of a conditional statementConditional: If an angle is a right angle, then its measure is 90.-144780380365Contrapositive: If an angle does not measure 90, then the angle is not a right angle.00Contrapositive: If an angle does not measure 90, then the angle is not a right angle.Symbolically:if ~q, then ~p ~q→~pSymbolic Representations in Logical ArgumentsConditionalif p, then qp→qConverseif q, then pq→pInverseif not p, then not q~p→~qContrapositiveif not q, then not p~q→~pConditional Statements and Venn DiagramsOriginal Conditional StatementConverse - Reversing the Clauses1998773555758mammal00mammal175604150163500If an animal is a dolphin, then it is a mammal. 2056082154527dolphin00dolphinTrue! 1595061343832001938507537801dolphin00dolphinIf an animal is a mammal, then it is a dolphin. 1878972130485mammal00mammalFalse! (Counterexample: An elephant is a mammal but is not a dolphin)Inverse - Negating the ClausesContrapositive - Reversing and Negating the ClausesIf an animal is not a dolphin, then it is not a mammal. 165780845160not mammal00not mammal14007226601000170491874546not dolphin00not dolphinFalse!(Counterexample: A whale is not a dolphin butis still a mammal) If an animal is not a mammal, then it is not a dolphin. 17371833987200203135065833not dolphin00not dolphin196702332075not mammal00not mammalTrue!Deductive Reasoningmethod using logic to draw conclusions based upon definitions, postulates, and theoremsleft239321Example of Deductive Reasoning:Statement A: If a quadrilateral contains only right angles, then it is a rectangle.Statement B: Quadrilateral P contains only right angles. Conclusion: Quadrilateral P is a rectangle.00Example of Deductive Reasoning:Statement A: If a quadrilateral contains only right angles, then it is a rectangle.Statement B: Quadrilateral P contains only right angles. Conclusion: Quadrilateral P is a rectangle.Inductive Reasoningmethod of drawing conclusions from a limited set of observations12759241523Example: Given a pattern, determine the next figure (set of dots) using inductive reasoning. Figure 1 Figure 2 Figure 3The next figure should look like this: Figure 400Example: Given a pattern, determine the next figure (set of dots) using inductive reasoning. Figure 1 Figure 2 Figure 3The next figure should look like this: Figure 4Direct Proofsa justification logically valid and based on initial assumptions, definitions, postulates, and theorems30480143510Example: (two-column proof)Given: 1 2Prove: 2 1StatementsReasons1 2Givenm1 = m2Definition of congruent anglesm2 = m1Symmetric Property of Equality2 1Definition of congruent angles00Example: (two-column proof)Given: 1 2Prove: 2 1StatementsReasons1 2Givenm1 = m2Definition of congruent anglesm2 = m1Symmetric Property of Equality2 1Definition of congruent angles27940552450Example: (paragraph proof)It is given that 1?2. By the Definition of congruent angles, m 1 = m2. By the Symmetric Property of Equality, m2 = m1. By the Definition of congruent angles, 2?1. 00Example: (paragraph proof)It is given that 1?2. By the Definition of congruent angles, m 1 = m2. By the Symmetric Property of Equality, m2 = m1. By the Definition of congruent angles, 2?1. Properties of CongruenceReflexive PropertyAB AB∠A?∠ASymmetric PropertyIf AB?CD , then CD?AB.If ∠A?∠B, then ∠B?∠ATransitive PropertyIf AB?CD and CD?EF, then AB?EF. If ∠A?∠B and ∠B?∠C, then ∠A?∠C.Law of Detachmentdeductive reasoning stating that if the hypothesis of a true conditional statement is true then the conclusion is also true2301875170815A12000A120center46117200Example: If mA > 90°, then A is an obtuse anglemA = 120Therefore, A is an obtuse angle.If pq is a true conditional statement and p is true, then q is true.Law of Syllogismdeductive reasoning that draws a new conclusion from two conditional statements when the conclusion of one is the hypothesis of the other-35938043143100Example:If a rectangle has four congruent sides, then it is a square.If a polygon is a square, then it is a regular polygon.If a rectangle has four congruent sides, then it is a regular polygon.If pq and qr are true conditional statements, then pr is true.Counterexamplespecific case for which a conjecture is false-155575527685Example:Conjecture: “The product of any two numbers is odd.”Counterexample: 2 ? 3 = 600Example:Conjecture: “The product of any two numbers is odd.”Counterexample: 2 ? 3 = 6One counterexample proves a conjecture false. Perpendicular Linestwo lines that intersect to form a right angle1386840347345mn00mnleft1543200Line m is perpendicular to line n.m nPerpendicular lines have slopes that are negative reciprocals.Parallel Linescoplanar lines that do not intersect 00mn00mn-5842027622500m||nLine m is parallel to line n.Parallel lines have the same slope.Skew Lineslines that do not intersect and are not coplanar45174093156756F020000F37073203941899E020000E1805940508587C020000C4996369470223D020000D11322242569817A020000A33573492228708B020000B387667540189150046482003228340388810522066251287780304482546596309302752040255930275403597184106200350940426766330010496559727112128722329956221287249806430013822943313684001344930977900Transversala line that intersects at least two other lines 31496036195txy00txy334327542545tba00tba-5715060515500Line t is a transversal.Corresponding Anglesangles in matching positions when a transversal crosses at least two lines8356601905tab4563217800tab45632178-129941489684Examples:2 and 6 3) 1 and 53 and 7 4) 4 and 800Examples:2 and 6 3) 1 and 53 and 7 4) 4 and 8Alternate Interior Anglesangles inside the lines and on opposite sides of the transversal 118681579375abt234100abt23417283451515745Examples:1 and 42 and 300Examples:1 and 42 and 376200159385000Alternate Exterior Angles2734310845820t00tangles outside the two lines and on opposite sides of the transversal1182370282575ab213400ab2134253682536195001233805631825Examples:1 and 42 and 300Examples:1 and 42 and 35143561849000Consecutive Interior Anglesangles between the two lines and on the same side of the transversal971550762002134tab002134tab852170172720Examples:1 and 23 and 400Examples:1 and 23 and 49525018478500Parallel Lines1010285-38100abt4563217800abt45632178-3098805461000Line a is parallel to line b whenCorresponding angles are congruent1 5, 2 6,3 7, 4 8Alternate interior angles are congruent3 64 5Alternate exterior angles are congruent1 82 7Consecutive interior angles are supplementarym3+ m5 = 180°m4 + m6 = 180°Midpoint(Definition)divides a segment into two congruent segments-7429568580DCM00DCM8211631643000Example: M is the midpoint of CDCM MDCM = MDSegment bisector may be a point, ray, line, line segment, or plane that intersects the segment at its midpoint.403161583074200Midpoint Formula QUOTE x1+ x22, y1+y22 20955097155given points A(x1, y1) and B(x2, y2) midpoint M = (x1+x22,y1+y22) QUOTE x1+ x22, y1+y22 00given points A(x1, y1) and B(x2, y2) midpoint M = (x1+x22,y1+y22) QUOTE x1+ x22, y1+y22 -85062507500Example: Find the midpoint, M, of the segment with endpoints A(4,1) and B(-2,5).M = (4+-22,1+52) = 22, 62=(1,3)Find a Missing 4169839171308200Endpoint QUOTE x1+ x22, y1+y22 20955097155given points A(x1, y1) and B(x2, y2) midpoint M = (x1+x22,y1+y22) QUOTE x1+ x22, y1+y22 00given points A(x1, y1) and B(x2, y2) midpoint M = (x1+x22,y1+y22) QUOTE x1+ x22, y1+y22 -13243048428600Example:Find the endpoint B(x,y) if A(-2,3) and M(3,8).-2+x2,3+y2=(3,8)-2+x2=3 and 3+y2=8x = 8 and y = 13B (8,13)Slope Formularatio of vertical change tohorizontal changeslope=m=change in y=rise=y2 – y1change in xrunx2 – x1 QUOTE y2 - y1 873996185272AB(x1, y1)(x2, y2)(run)x2 – x1y2 – y1(rise)00AB(x1, y1)(x2, y2)(run)x2 – x1y2 – y1(rise)Slopes of Lines in Coordinate Plane-183515262890Parallel lines have the same slope.Perpendicular lines have slopes whose product is -1.Vertical lines have undefined slope.Horizontal lines have 0 slope.00Parallel lines have the same slope.Perpendicular lines have slopes whose product is -1.Vertical lines have undefined slope.Horizontal lines have 0 slope.2936402248920yxnp00yxnp5549265121123048922021212850-179705789305Example: The slope of line n = -2. The slope of line p =.-2 ? = -1, therefore, n p.00Example: The slope of line n = -2. The slope of line p =.-2 ? = -1, therefore, n p.Distance Formula given points A (x1, y1) and B (x2, y2) 101790513970AB(x1, y1)(x2, y2)x2 – x1y2 – y1 00AB(x1, y1)(x2, y2)x2 – x1y2 – y1 277622030353000-171961494276The distance formula is derived from the application of the Pythagorean Theorem.00The distance formula is derived from the application of the Pythagorean Theorem. Examples of Line Symmetry674370128905003232150242570003670303492500315531517145000MOM207010057023000400494514986000 BXExamples of Point Symmetry62484066040AAˊCCˊP00AAˊCCˊP217551058610500pod S Z13671555143500Rotation1051560556260(Origin) 4126703342900 ′00 ′3095625344008 ′00 ′310470756279 ′00 ′PreimageImageA(-3,0)A(0,3)B(-3,3)B(3,3)C(-1,3)C(3,1)D(-1,0)D(0,1)Pre-image has been transformed by a 90 clockwise rotation about the origin.ReflectionPreimageImageD(1,-2)D(-1,-2)E(3,-2)E(-3,-2)F(3,2)F(-3,2)TranslationPreimageImageA(1,2)A(-2,-3)B(3,2)B(0,-3)C(4,3)C(1,-2)D(3,4)D(0,-1)E(1,4)E(-2,-1)Dilation8804118301800PreimageImageA(0,2)A(0,4)B(2,0)B(4,0)C(0,0)C(0,0)PerpendicularBisector32061151324610s00sa segment, ray, line, or plane that is perpendicular to a segment at its midpoint 2831465379095Z00Z3198495151765003240405160020001021080150495003169920109220004150995279400042551357747000201993510985500193802016192500320865540767000424180010604500218313091440005251450228600Y00Y806450222250X00X2730500210820M00M315595014097000102870019177000-171454445000Example:46120053492500Line s is perpendicular to XY. 4459605698500052692307937500M is the midpoint, therefore XM MY.Z lies on line s and is equidistant from X and Y.Constructions00Traditional constructions involving a compass and straightedge reinforce students’ understanding of geometric concepts. Constructions help students visualize Geometry. There are multiple methods to most geometric constructions. These cards illustrate only one method. Students would benefit from experiences with more than one method, including dynamic geometry software, and should be able to justify each step of geometric constructions.Traditional constructions involving a compass and straightedge reinforce students’ understanding of geometric concepts. Constructions help students visualize Geometry. There are multiple methods to most geometric constructions. These cards illustrate only one method. Students would benefit from experiences with more than one method, including dynamic geometry software, and should be able to justify each step of geometric constructions.Constructsegment CD congruent to segment AB87122059055BA00BA8216901458595CD00CD-146053958590Fig. 200Fig. 2-457201286510Fig. 100Fig. 1Construct8890866140BA00BAa perpendicular bisector of segment AB31369003269615BA00BA9023353624580Fig. 200Fig. 2-660401425575Fig. 100Fig. 130264105253355Fig. 300Fig. 39982201352550AB00ABConstructa perpendicular to a line from point P not on the line-825503391535PFig. 3BA00PFig. 3BA33464503365500BAP00BAP56908705955665Fig. 400Fig. 4292102757170Fig. 100Fig. 157099202736215Fig. 200Fig. 2-21590237490003432175524510BAP00BAP-62230523875BAP00BAPConstructa perpendicular to a line from point P on the line38792153491865 P00 P-558803608705BA P00BA P3644265254000BA P00BA P58623202854325Fig. 200Fig. 21308102679700035731454787900BA00BA59194706050915Fig. 400Fig. 4215906122035Fig. 300Fig. 3209552845435Fig. 100Fig. 1-215901685290BA P00BA PConstructa bisector of A-114303785235A00A35788603803015A00A56946806107430Fig. 400Fig. 4-1816106180455Fig. 300Fig. 34490085655320004791075974090004150360102997000209552465070Fig. 100Fig. 157740552493645Fig. 200Fig. 244881807092950036652201743075A020000A40214555467350040024051889760003556076200001600201647825A020000A51625545148500497205179451000Construct3436620573405A00A1293495506095020000Y congruent to A4481195122364500449961025812750034747201474470Y00Y38842951284605001602105569468000150495041236900014655805246370001367155368427000-292103543300038398454954270001486535513778500480377550914300013468355092700001377315356489000131445213360A00A34556703261360A00A457203308985A00A35032955033645Y00Y361955024120Y00Y41548059017000214122012763500200001885952514600Y020000Y52578026993850058470806450330Fig. 400Fig. 4-673106466205Fig. 300Fig. 3-933452807970Fig. 100Fig. 158312052817495Fig. 200Fig. 2Constructline n parallel to line m through point P not on the line3332480277495mP00mP-93345508635mP00mP-800101263650057804052537460Fig. 200Fig. 2-1441452527935Fig. 100Fig. 1489585021780500421322528575000477520515620Draw a line through point P intersecting line m.00Draw a line through point P intersecting line m.12884152043430001962785863600001270635200596500194500582613500179578055245000110680517100550058089802651125Fig. 400Fig. 4-1054102657475Fig. 300Fig. 3-69215142240mP00mP3464560183515mPn00mPnConstructan equilateral triangle inscribed 38842954508500in a circle 527367573406000114046012553950051562096520002882903709670003838575369252500-1085852051050056572155902325Fig. 400Fig. 4-2070105998210Fig. 300Fig. 3-1441452720975Fig. 100Fig. 157296052689860Fig. 200Fig. 2Constructa square inscribed in a circle 3810635660400004222757112000057569102908300Fig. 200Fig. 2-1498602923540Fig. 100Fig. 1-11684046609000779145100965Draw a diameter.00Draw a diameter.38347651822450047637701771650005226050162115500438975531115000391414031305500393509516630650048037751039495005326380749300004274185850900039217601435100043535601111250019113515240000-2133602638425Fig. 300Fig. 356984902566670Fig. 400Fig. 4Construct378142555245000a regular hexagon inscribed 34353554229000in a circle 17995905497830005210810691515008972551220470001508760520700000205740365442500365760035521900057892956019165Fig. 400Fig. 4-2374906062980Fig. 300Fig. 3-2368552747010Fig. 100Fig. 156775352726055Fig. 200Fig. 2-20129525463500Classifying Triangles by Sides ScaleneIsoscelesEquilateral306070233045001468755977900001466215537845001062990972820001034415515620005727701333500090424041656000135572511791950048641011766550010026651264285001191895803275005429258299450035242530162500No congruent sidesAt least 2 congruent sides3 congruent sidesNo congruent angles2 or 3 congruent angles3 congruent anglesAll equilateral triangles are isosceles.Classifying Trianglesby Angles AcuteRightObtuseEquiangular203203511550030543526733500304800267335008636033274000133604011125200045466011201400087312536512500323850234315003 acute angles1 right angle1 obtuse angle3 congruent angles3 angles, each less than 901 angle equals 901 angle greater than 903 angles, each measures 60Triangle Sum Theorem246761062230B020000B1692910398780001403350325755A020000A4789170815340C020000Cmeasures of the interior angles of a triangle = 180-4889519240500mA + mB + mC = 180Exterior Angle Theorem1029335-96520ABC100ABC1 Exterior angle, m1, is equal to the sum of the measures of the two nonadjacent interior angles.-698514668500m1 = mB + mCPythagorean Theorem94488065405b chypotenusea BAC00b chypotenusea BAC-241301714500If ABC is a right triangle, thena2 + b2 = c2.Conversely, if a2 + b2 = c2, then ABC is a right triangle.Angle and Side Relationships5303520189865A00A34569406223012 8688o54o38oBC0012 8688o54o38oBCA is the largest angle, 15608305588000therefore BC is the longest side.-476255842012 8688o54o38oBCA0012 8688o54o38oBCA579691554610000B is the smallest angle, therefore AC is the shortest side.Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.129730511430ABC00ABC323442336239326 in026 in11717081178448 in08 in24369813055422 in022 in-9356739591400Example:AB + BC > ACAC + BC > AB 8 + 26 > 22 22 + 26 > 8 AB + AC > BC8 + 22 > 26Congruent Triangles-64135215900ABCFD00ABCFD400494590805E020000ETwo possible congruence statements:ABC FED -5651545656500BCA EDF Corresponding Parts of Congruent FiguresA F B EC D-1447802743200ABCFD00ABCFD39535102622550E00ESSS Triangle Congruence Postulate-14859054610000Example:2145665831850029267159017000 If Side AB FE,2896870857250021278858001000 Side AC FD, and 2100580889000028905208890000 Side BC ED , then ABC FED.SAS Triangle Congruence Postulate653415-185420ABCFED00ABCFED-1257305016500Example:2953385901700021456658318500 If Side AB DE, Angle A D, and 2109470889000029083008890000 Side AC DF , then ABC DEF.HL Right Triangle Congruence206375351155RSTXYZ00RSTXYZ-1581151778000Example:4007485863600032397709080500 If Hypotenuse RS XY, and2051050781050027743157810500 Leg ST YZ , then RST XYZ.ASA Triangle Congruence Postulate820420-245745BCFED00BCFED1097915130810A00A-1257305016500Example: If Angle A D,2899410889000021094708890000 Side AC DF , and Angle C F then ABC DEF.AAS Triangle Congruence Theorem34544067945RSTXYZ00RSTXYZ-1060455016500Example:If Angle R X, Angle S Y, and 21151858128000Side ST YZ then RST XYZ.Similar Polygons143510185420ABDCEFGH2461200ABDCEFGH24612ABCD HGFEAnglesSidesA corresponds to HAB corresponds to HGB corresponds to GBC corresponds to GFC corresponds to FCD corresponds to FED corresponds to EDA corresponds to EH-9715526416000Corresponding angles are congruent.Corresponding sides are proportional.Similar Polygons and Proportions1186180-153670ABCHGF1264x00ABCHGF1264xCorresponding vertices are listed in the same order.-501655715000Example: ABC HGF = QUOTE ADHE = The perimeters of the polygons are also proportional.AA Triangle Similarity Postulate117475314325RSTXYZ00RSTXYZ-812801143000Example: If Angle R X andAngle S Y, then RST XYZ.SAS Triangle Similarity Theorem149987093980BB4182745392430EE991870243205141439255703860807735115503352805149853175041389306800854591050836295665367020491490FF3837940783590DD3035300588010CC1364615795020193357511366512121017270187960AA-15557586360000Example:If A D and = then ABC DEF.SSS Triangle Similarity Theorem5245100358775Y00Y2599055134620S00S995045-76200054222654718052.5002.539890702908306.5006.52732405352425500512998451219201300134047490-173990005327650516890Z00Z3459480515620X00X2679700553720T00T111760488315R00R4359275228606006134493097155120012-527053048000Example:If = = then RST XYZ.Altitude of a Trianglea segment from a vertex perpendicular to the line containing the opposite side297180300990GJHaltitudes00GJHaltitudes231775205740003415030337185altitude/heightBCA00altitude/heightBCA122110524511000-6477030416500Every triangle has 3 altitudes.Median of a TriangleA line segment from a vertex to the midpoint of the opposite side163004523495001438910186690aDmedianACB00aDmedianACBleft3159490036787177620000134775155720500D is the midpoint of AB; therefore, CD is a median of ABC.Every triangle has 3 medians.Concurrency of Medians of a Triangle1449705-97790ABCDEF00ABCDEF4680452375196centroid00centroid3086100426085P020000P3308852164923-8293428266300Medians of ABC intersect at P (centroid) andAP = AF,CP = CE,BP = BD.13017530397450030°-60°-90° TriangleTheorem161861520701030°60°x2xxx60°30°0030°60°x2xxx60°30°-6096040576500Given: short leg = xUsing equilateral triangle,hypotenuse = 2 ? x Applying the Pythagorean Theorem,longer leg = x ? 45°-45°-90° TriangleTheorem189136433154x45°x45°xx45°x45°x-10985545339000Given: leg = x, then applying the Pythagorean Theorem; hypotenuse2 = x2 + x2 hypotenuse = xTrigonometric524510821055(side adjacent A)ABCabc(side opposite A)(hypotenuse)00(side adjacent A)ABCabc(side opposite A)(hypotenuse)Ratios511746522225ac00ac189611020955hypotenuseside opposite A00hypotenuseside opposite Asin A = =5183505377190bc00bc1890395389255hypotenuseside adjacent A00hypotenuseside adjacent Acos A = =544004580010ab00ab190563567310side adjacent to Aside opposite AA00side adjacent to Aside opposite AAtan A = =8991602633345ABCabc00ABCabcInverse Trigonometric RatiosDefinitionExampleIf tan A = x, then tan-1 x = mA.tan-1 = mAIf sin A = y, then sin-1 y = mA.sin-1 = mAIf cos A = z, then cos-1 z = mA.cos-1 = mAArea of a Triangle1226820170815hABCa00hABCa2833370732155b020000b sin C = h = a?sin C A = bh (area of a triangle formula)By substitution, A = b(a?sin C)A = ab?sin CPolygon Exterior Angle Sum TheoremThe sum of the measures of the exterior angles of a convex polygon is 360°.1343660140335523410052341-16827514097000Example:m1 + m2 + m3 + m4 + m5 = 360Polygon Interior Angle Sum TheoremThe sum of the measures of the interior angles of a convex n-gon is (n – 2)?180°.S = m1 + m2 + … + mn = (n – 2)?180°1938020117475523410052341-17018044450000Example:If n = 5, then S = (5 – 2)?180°S = 3 ? 180° = 540°Regular Polygona convex polygon that is both equiangular and equilateral2021840211455Equilateral TriangleEach angle measures 60o.00Equilateral TriangleEach angle measures 60o.48514017462500403860212725SquareEach angle measures 90o.00SquareEach angle measures 90o.4819015158115002047875612140Regular PentagonEach angle measures 108o.00Regular PentagonEach angle measures 108o.56388047561500283845311785Regular HexagonEach angle measures 120o.00Regular HexagonEach angle measures 120o.476758024384000539115631190002117090114300Regular OctagonEach angle measures 135o.00Regular OctagonEach angle measures 135o.Properties of Parallelograms168656014160500Opposite sides are parallel. Opposite sides are congruent.Opposite angles are congruent.Consecutive angles are supplementary.The diagonals bisect each other.147955053340000RectangleA parallelogram with four right angles142494017018000Diagonals are congruent.Diagonals bisect each other.156464021209000RhombusA parallelogram with four congruent sides190055516256000Diagonals are perpendicular.Each diagonal bisects a pair of opposite angles.160020035814000SquareA parallelogram and a rectangle with four congruent sides19856456223000 Diagonals are perpendicular.Every square is a rhombus.213169510160000TrapezoidA quadrilateral with exactly one pair of parallel sides4240180198821409272833311200-63500445135001459887333112001333762270050003497979177122Bases368282440661004445438424487309124570616Legs0Legs3257156410888001459274362147Median0Median385519433845500right-43243500 Two pairs of supplementary angles Median joins the midpoints of the nonparallel sides (legs)Length of median is half the sum of the lengths of the parallel sides (bases)Isosceles TrapezoidA quadrilateral where the two base angles are equal and therefore the sides opposite the base angles are also equal231806023369500Legs are congruentDiagonals are congruentCircleall points in a plane equidistant from a given point called the center152958844704radiusdiameterchordPONMRS00radiusdiameterchordPONMRS-508035560000Point O is the center.2154555547370004876809017000MN passes through the center O and therefore, MN is a diameter.11353808001000255460589535004781558953500OP, OM, and ON are radii and 22879058953500132588089535004781559906000OP OM ON.459105635000016116307302500RS and MN are chords. Circles4445038735000A polygon is an inscribed polygon if all of its vertices lie on a circle.379349032385000A circle is considered“inscribed” if it istangent to each side of the polygon. Circle Equation1513205-13970yx(x,y)xyr00yx(x,y)xyr x2 + y2 = r2circle with radius r and center at the origin -533402794000standard equation of a circle(x – h)2 + (y – k)2 = r2with center (h,k) and radius rLines and Circles179260576835CDAB00CDAB-355603568700016541755905500Secant (AB) – a line that intersects a circle in two points.18929356921500Tangent (CD) – a line (or ray or segment) that intersects a circle in exactly one point, the point of tangency, D.Secant 153225591440y°1x°00y°1x°If two lines intersect in the interior of a circle, then the measure of the angle formed is one-half the sum of the measures of the intercepted arcs.-11303029845000m1 = (x° + y°)Tangent 147828075565QSR00QSRA line is tangent to a circle if and only if the line is perpendicular to a radius drawn to the point of tangency. -56515277495004324353556000QS is tangent to circle R at point Q.Radius RQ QSTangent1228725160020CBA00CBAIf two segments from the same exterior point are tangent to a circle, then they are congruent.-85090291465004603756858000AB and AC are tangent to the circle at points B and C.23666458636000Therefore, AB AC and AC = AB.Central Anglean angle whose vertex is the center of the circle415290315595ABCminor arc ABmajor arc ADBD00ABCminor arc ABmajor arc ADBD-12827035496500ACB is a central angle of circle C.Minor arc – corresponding central angle is less than 180°Major arc – corresponding central angle is greater than 180°Measuring Arcs1870710236855DBRC70°110°00DBRC70°110°179070032385A020000AMinor arcsMajor arcsSemicirclesm AB = 110°m AB = 110°m BDA = 250°m BDA = 250°m ADC = 180°m ADC = 180°m BC = 70°m BC = 70°m BAC = 290°m BAC = 290°m ABC = 180°m ABC = 180°-13609726646400The measure of the entire circle is 360o.The measure of a minor arc is equal to its central angle.The measure of a major arc is the difference between 360° and the measure of the related minor arc.Arc Length17018001035054 cmABC120°004 cmABC120°14782801714500139065027051000-1727201905000Example:138938088392000Secants and Tangents73660340995Two secants1x°y°00Two secants1x°y°4163695317500Secant-tangent1x°y°00Secant-tangent1x°y°2364105108585Two tangents1x°y°00Two tangents1x°y°-4064022733000m1 = (x°- y°) Inscribed Angleangle whose vertex is a point on the circle and whose sides contain chords of the circle 1508125323850BAC00BAC-342901524000Area of a Sector154178084455000region bounded by two radii and their intercepted arc1809759525cm00cm-5461045021500-66675419735Example:020000Example:Inscribed Angle Theorem 11748155228600ADBC00ADBC25298402190750021285205715000If two inscribed angles of a circle intercept the same arc, then the angles are congruent.3048021526500BDC BACInscribed Angle Theorem 28089900OACB00OACB381003892550047428156858000mBAC = 90° if and only if BC is a diameter of the circle. Inscribed Angle Theorem 3153352519304088929585MJTHA0088929585MJTHA-3004185609600920092-850630859400M, A, T, and H lie on circle J if and only ifmA + mH = 180° and mT + mM = 180°.(opposite angles are supplementary)-3287395219075850085 Segments in a Circle191770082550abcd00abcdIf two chords intersect in a circle,then a?b = c?d.-5143533909000354647514605126x900126x9363537528638500357441534163000Example:12(6) = 9x72 = 9x 8 = xSegments of Secants Theorem1388110234315BACDE00BACDEAB ? AC = AD ? AE-70485190500055943535623596x160096x16Example:6(6 + x) = 9(9 + 16) 36 + 6x = 225 x = 31.5Segments of Secants and Tangents Theorem152908025400000145923048958500129095559055A00A2198370168910B00B195326071120001498600245745004513580299720C00C1782445772795E00EAE2 = AB ? AC-1352551092200037230054845052520x002520xExample: 252 = 20(20 + x)625 = 400 + 20x x = 11.25Conesolid that has one circular base, an apex, and a lateral surface3426460386715002564130109220apex00apex1813560430530004011295656590slant height (l)00slant height (l)285115273050lateral surface(curved surface of cone)00lateral surface(curved surface of cone)1790065209550001950085830580radius(r)00radius(r)3201670121920height (h)00height (h)2837180396240base00base49570199962V = 13r2hL.A. (lateral surface area) = rlS.A. (surface area) = r2 + rl00V = 13r2hL.A. (lateral surface area) = rlS.A. (surface area) = r2 + rlCylindersolid figure with two congruent circular bases that lie in parallel planes1807210309245height (h)radius (r)basebase00height (h)radius (r)basebase476250666115V = r2hL.A. (lateral surface area) = 2rhS.A. (surface area) = 2r2 + 2rh00V = r2hL.A. (lateral surface area) = 2rhS.A. (surface area) = 2r2 + 2rhPolyhedronsolid that is bounded by polygons, called faces4239895225425003225803930650024555452857500 Similar Solids TheoremIf two similar solids have a scale factor of a:b, then their corresponding surface areas have a ratio of a2: b2, and their corresponding volumes have a ratio of a3: b3. cylinder A cylinder BExamplescale factora : b3:2ratio ofsurface areasa2: b29:4ratio of volumesa3: b327:8-179705269875A00A1217295216535B00B Spherea three-dimensional surface of which all points are equidistant from a fixed point164084050800radius00radius573405407670V = 43r3S.A. (surface area) = 4r200V = 43r3S.A. (surface area) = 4r2Hemispherea solid that is half of a sphere with one flat, circular side28898501606466r020000r37793772374157r020000r30984299258300031146752677531116034289429900 4812633801344V = 23r3S.A. (surface area) = 3r200V = 23r3S.A. (surface area) = 3r2Pyramidpolyhedron with a polygonal base and triangular faces meeting in a common vertex-69368109680vertexbaseslant height (l)height (h)area of base (B)perimeter of base (p)00vertexbaseslant height (l)height (h)area of base (B)perimeter of base (p)6085491365796V (volume) = 13BhL.A. (lateral surface area) = S.A. (surface area) = + B00V (volume) = 13BhL.A. (lateral surface area) = S.A. (surface area) = + B ................
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