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Geometry - Concepts 9-12 Congruent Triangles and Special Segments Concept9–ParallelLinesandTriangles(Section3.5) ANGLEClassifications SIDEClassifications Acute: Scalene: Obtuse: Isosceles: Right: Equilateral: Equiangular: Theorem3.11–TriangleSumTheorem Thesumofthemeasuresoftheanglesofatriangleis________. Theorem3.12–TriangleExteriorAngleTheorem Themeasureofeachexteriorangleofatriangle equalsthesumofthemeasuresofits two_________________________________________. m∠1 = m∠2 + m∠3 Concept10–CongruentFigures(Section4.1) CongruentPolygons: - Havecorresponding___________thatarecongruent -Havecorresponding____________thatarecongruent CongruenceStatement: -astatementsayingthattwofiguresarecongruent -correspondinganglesmustbelinedup Example:Listallpairsofcongruentanglesandallpairsofcongruentsidesusingthestatementbelow. ΔRST ≅ ΔMNL Theorem4.1-ThirdAngleTheorem Iftwoanglesofonetrianglearecongruenttotwoanglesofanother triangle,thenthethirdangles_______________________________________. Concept10–CongruentTriangles(Sections4.2,4.3,4.6) Side-Side-SidePostulate(SSS) Ifthesidesofonetrianglearecongruenttothe sidesofanothertriangle,then __________________________________. Side-Angle-SidePostulate(SAS) If________________________________________ofatriangle arecongruenttotwosidesandtheincludedangle ofanothertriangle,thenthetrianglesarecongruent. Angle-Side-AnglePostulate(ASA) If________________________________________ofatriangle arecongruenttotwoanglesandtheincludedside ofanothertriangle,thenthetrianglesarecongruent. Angle-Angle-SideTheorem(AAS) Iftwoanglesanda___________________sideofa trianglearecongruenttotwoanglesandthe nonincludedsideofanothertriangles,thenthe trianglesarecongruent. Hypotenuse-LegTheorem(HL) Ifthe_______________________________ofonerighttriangleis congruenttothehypotenuseandlegofanotherrighttriangle, thenthetrianglesarecongruent. BADPROOFS–thesedonotwork Concept10–UsingCorrespondingPartsofCongruentTriangles(4.4) CPCTC:_________________________________________________________________ Example: Concept10–IsoscelesandEquilateralTriangles(Section4.5) PartsofanIsoscelesTriangle Legs: Base: BaseAngles: VertexAngle: Theorem4.3–IsoscelesTriangleTheorem Iftwosidesofatrianglearecongruent,thenthetwoanglesoppositethosesidesare____________________. Theorem4.4–ConverseoftheIsoscelesTriangleTheorem Iftwoanglesofatrianglearecongruent,thenthetwosidesoppositethoseanglesare___________________. CorollarytoTheorem4.3 Ifatriangleisequilateral,thenitis_____________________. CorollarytoTheorem4.4 Ifatriangleisequiangular,thenitis_____________________. Examples:Findthevalueofeachvariable. Concept11–MidsegmentTheorem(Section5.1) MidsegmentofaTriangle: Theorem5.1–TriangleMidsegmentTheorem Ifasegmentjoinsthemidpointsoftwosidesofatriangle,thenthesegmentis__________________tothe thirdsideand_____________________________________. Concept11–Bisectors,Medians,andAltitudes(Section5.2and5.4) PerpendicularBisector: Theorem5.2-PerpendicularBisectorTheorem C Ifapointisontheperpendicularbisectorofasegment,thenitis ______________________fromtheendpointsofthesegment. AngleBisector: Theorem5.4-AngleBisectorTheorem Ifapointisonthebisectorofanangle,thenthepointis _________________________fromthetwosidesoftheangle. MedianofaTriangle: Altitude: A D B Example:Namethespecialsegmentineachtriangle. Concept11–PointsofConcurrency(Section5.3and5.4) Concurrent: PointofConcurrency: Circumcenter: PropertiesoftheCircumcenter 1) 2) Incenter: PropertiesoftheIncenter 1) 2) Centroid: PropertiesoftheCentroid 1) 2) Orthocenter: Concept12-InequalitiesinTriangles(Section5.6) CorollarytotheTriangleExteriorAngleTheorem Themeasureofanexteriorangleofatriangleis_______________________________the measureofeachofitsremoteinteriorangles. Theorem5.10 Iftwosidesofatrianglearenotcongruent,thenthelargerangleliesopposite the_____________________________________________. A Largest angle = _____ Mid-sized angle = _____ Smallest angle = _____ 17 14 C B 20 Theorem5.11 Iftwoanglesofatrianglearenotcongruent,thenthelongersideliesopposite the______________________________________________. D Longest side = _____ Medium side = _____ Shortest side = _____ F Theorem5.12-TriangleInequalityTheorem 112 32 o o 36 o E Thesumofthelengthsofanytwosidesofatriangleis___________________________thelength ofthethirdside. SimplifiedVersion: Example: Iftwosidesofatriangleare3inchesand9incheslong,writearangeofpossiblelengthsforthethird sideofthetriangle.