Download Geometry - Concepts 9-12

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.