Download Concepts 12-16 Notes Triangle Relationships and Similar Triangles

Concepts 12-16 Notes
Triangle Relationships and Similar Triangles
Concept12–AngleBisectors&PerpendicularBisectors(Section5.6)
AngleBisector:
Theorem5.3-AngleBisectorTheorem
Ifapointisonthebisectorofanangle,thenthepointis
_________________________fromthetwosidesoftheangle.
PerpendicularBisector:
Theorem5.4-PerpendicularBisectorTheorem
C
Ifapointisontheperpendicularbisectorofasegment,thenitis
______________________fromtheendpointsofthesegment.
Concept12–MediansandAltitudes(Section4.6)
MedianofaTriangle:
Altitude:
Example:Namethespecialsegmentineachtriangle.
A
D
B
Concept12–PointsofConcurrency
PointofConcurrency:
Circumcenter:
PropertiesoftheCircumcenter
1)
2)
Incenter:
PropertiesoftheIncenter
1)
2)
Centroid:
PropertiesoftheCentroid
1)
2)
Orthocenter:
Concept14–TriangleInequalities(Section4.7)
Theorem4.10
Ifonesideofatriangleislongerthananotherside,thentheangleoppositethelongersideislarger
than______________________________________________________________.
A
Largest angle = _____
Mid-sized angle = _____
Smallest angle = _____
17
14
C
20
B
Theorem4.11
Ifoneangleofatriangleislargerthananotherangle,thenthesideoppositethelargerangleis
longerthan__________________________________________________________.
D
Longest side = _____
Medium side = _____
Shortest side = _____ F
112
32
o
o
36
Theorem4.12-TriangleInequalityTheorem
o
E
Thesumofthelengthsofanytwosidesofatriangleis___________________________thelength
ofthethirdside.
SimplifiedVersion:
Concept15–RatiosandProportions(Section7.1)
Ratio:
Example:Theperimeterofarectangleis80feet.Theratioofthelengthtothewidthis7:3.Find
thelengthandthewidthoftherectangle.
Proportion:
€
Cross-ProductsProperty a c
If = ,then_________________.
b d
Example:Solvetheproportion.
x +1 2x − 3
=
3
5
Concept15–UseSimilarPolygons(Section7.2)
Similar:
-
Correspondinganglesare______________________
-
Correspondingsidesare________________________
SimilarityStatement:
ScaleFactor:
Theorem7.1–PerimetersofSimilarPolygons
Iftwopolygonsaresimilar,thentheratiooftheirperimetersifequaltotheratiosoftheir
_____________________________________________________.
Sincecorrespondingsidesareproportional,aproportion
canbeusedtofindmissingsidelengths.
Example:Findthemeasureofxandy.
Concept15–ProveTrianglesSimilar(Section7.3-7.4)
Postulate15-Angle-AngleSimilarityPostulate(AA)
If_______________________ofonetrianglearecongruentto__________________________
ofanothertriangle,thenthetrianglesaresimilar.
B
Theorem7.2-Side-Side-SideSimilarityTheorem(SSS)
Ifthecorrespondingsidesoftwotrianglesare
A
________________________,thenthetrianglesaresimilar.
Theorem7.3-Side-Angle-SideSimilarityTheorem(SAS)
Ifoneangleofonetriangleiscongruenttooneangleofanothertriangleand
thesidesincludingthetwoanglesare______________________________,thenthe
trianglesaresimilar.
Y
4
3
5
C
X
12
9
15
Z
Concept16–UseProportionalityTheorems(Section7.5)
Theorem7.4–TriangleProportionalityTheorem
Ifalineparalleltoonesideofatriangleintersectstheothertwosides,
thenitdividesthetwosides_______________________________.
Theorem7.5–ConverseofTriangleProportionalityTheorem
Ifalinedividestwosidesofatriangleproportionally,thenitis__________________
tothethirdside.
MidsegmentofaTriangle:
Theorem7.6-MidsegmentTheorem
Thesegmentconnectingthemidpointsoftwosidesofatriangle
is___________________tothethirdsideandis_________________________asthatside.
ParallelLinesandTransversalsTheorem
Ifthreeparallellinesintersecttwotransversals,thentheydividethe
transversals________________________________.
TriangleAngleBisectorTheorem
Ifaraybisectsanangleofatriangle,thenitdividestheoppositeside
intotwosegmentsthatare_____________________________to
the_______________________________________________.