TRIANGLES 8

8/7/2019 TRIANGLES 8

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Govt. Ser. Sec. Model School P.A.U, Ludhiana

Students Names(9th)

Indermohan Singh,Sandeep Kumar,

Pooja Bhatt,

Ranjana

Teacher Trainer:-

Harinder kaur

Student Trainer:-

Rachna

Presented by:- MRS. KUSUM LATA


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Definition of triangleDefinition of triangle

Triangle Triangle (geometry),(geometry), geometric geometric figurefigureconsistingconsisting ofof threethree points, points, calledcalled vertices,vertices,connectedconnected byby threethree sidessides.. InIn EuclideanEuclidean planeplanegeometry,geometry, thethe sidessides areare straightstraight lineline segmentssegments..InIn sphericalspherical geometry,geometry, thethe sidessides areare arcsarcs ofofgreatgreat circlescircles.. TheThe termterm triangletriangle isis sometimessometimes

usedused toto describedescribe aa geometric geometric figure figure havinghavingthreethree verticesvertices andand sidessides thatthat areare arbitraryarbitrarycurvescurves..


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Types of triangleTypes of triangle A triangle is a plane figurebounded by three straight lines. A scalene triangle has threesides of unequal lengths, an

isosceles triangle has two equalsides, and an equilateraltriangle has three equal sides.

In the isosceles triangle theangles opposite the equal sidesare equal, and in an equilateraltriangle all three angles are

equal.


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Theorems & Properties ofTriangleTheorems & Properties ofTriangle

OnOn thethe Basis Basis ofof SidesSides && AnglesAnglesThereThere areare SixSix Types Types ofof TriangleTriangle..

Let Let usus deducededuce thisthis importantimportantpropertiesproperties ofof triangletriangle.. 1. Sum of the three angles of a

triangle is 180 degree.

2. If a side of a triangle is produced,the exterior angle so formed is equalto the sum of the two interior opposite

angles.

2nd Theorem


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Theorems of congruenceTheorems of congruence TheoremTheorem 11.. Two Two triangletriangle areare congruentcongruent ifif anyany

twotwo sideside andand thethe includedincluded angleangle ofof oneone triangletriangleareare equalequal toto anyany twotwo sidessides andand thethe includedincluded anglesangles

ofof thethe otherother triangletriangle.. ThisThis relationrelation isis referredreferred totoasas (SAS)(SAS) sideside angleangle sideside axiomaxiom..


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Theorem 2Theorem 2ndndof congruenceof congruence

TwoTwo trianglestriangles areare congruentcongruent ifif anyany twotwo anglesanglesandand includedincluded sideside ofof oneone triangletriangle areare equalequal toto thethe

twotwo anglesangles andand thethe includedincluded sideside ofof thethe otherothertriangletriangle.. This This isis calledcalled ASA ASA (( AngleAngle--SideSide--Angle)Angle) criterioncriterion forfor congruencecongruence ofof trianglestriangles..


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Theorem 3Theorem 3rdrdof congruenceof congruence

TwoTwo trianglestriangles areare congruentcongruent ifif thethe threethree sidessides ofofoneone triangletriangle areare equalequal toto thethe threethree sidessides ofof thethe

otherother triangletriangle.. ItIt isis calledcalled SSSSSS CongruenceCongruence ofoftriangletriangle.. AsAs shownshown belowbelow::--


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Theorem 4Theorem 4ththof congruenceof congruence

TwoTwo trianglestriangles areare congruentcongruent ifif thethe threethree anglesanglesofof oneone triangletriangle areare equalequal toto thethe threethree anglesangles ofof

thethe otherother triangletriangle.. ItIt is is calledcalled AAAAAACongruenceCongruence ofof triangletriangle.. AsAs shownshown belowbelow::--


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Theorem 5Theorem 5ththof congruenceof congruence

Two Two Right Right trianglestriangles areare congruentcongruent ifif thethehypotenusehypotenuse andand aa sideside ofof oneone triangletriangle areare

respectivelyrespectively equalequal toto thethe hypotenusehypotenuse andand aa sideside ofofotherother triangletriangle.. ItIt isis calledcalled RHS RHS (Right(Right--hypotenusehypotenuse--side)side).. GivenGiven BelowBelow::--


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Inequalities in a triangleInequalities in a triangle

Theorem Theorem 11::-- IfIf twotwo sidessides ofof aa triangletriangle areareunequal,unequal, thethe largerlarger sideside hashas thethe greater greater angleangle

oppositeopposite toto it it.. For For exampleexample::-- ab= ab=44cm,cm,ac=ac=66cm,bc=cm,bc=55cmcm.. TheThe GreaterGreater angleangle isis angleangle BB..


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Theorem 2Theorem 2ndnd

TheThe sumsum ofof anyany twotwo sidessides ofof triangletriangle isis greatergreaterthanthan thethe thirdthird sideside.. ForFor exampleexample::--

abab ++ bcbc >> acac ((22++44 >>55)).. bcbc ++ caca >> ab(ab(44++55 >>22))..

caca ++ abab >> bc(bc(55++22 >>44))..


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Theorem 3Theorem 3rdrd

TheThe DifferenceDifference ofof anyany twotwo sidessides ofof triangletriangle isislesserlesser thanthan thethe thirdthird sideside.. ForFor exampleexample::--

bcbc -- abab


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Heron FormulaHeron Formula

ItIt waswas discovereddiscovered byby GreekGreek MathematicianMathematicianHero Hero ofof AlexandriaAlexandria.. He He waswas bornborn inin egyptegypt..HeronHeron FormulaFormula isis asas followsfollows::--

SemiSemi perimeterperimeter == (a(a ++ bb ++ c)c)..

ByBy Heron Heron FormulaFormula areaarea ofof triangletriangle ==

s(ss(s--a)a) (s(s--b)b) (s(s--c)c)

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TRIANGLES 8