Presentation on theme: "Triangle Fundamentals"— Presentation transcript:
1Triangle Fundamentals Intro to G.10TriangleFundamentalsModified by Lisa Palen
2Triangle What’s a polygon? Definition: A triangle is a three-sided polygon.What’s a polygon?
3PolygonsDefinition:A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints.These figures are not polygonsThese figures are polygons
4Definition of a Polygon A polygon is a closed figure in a plane formed by a finite number of segments that intersect only at their endpoints.
5Triangles can be classified by: Their sidesScaleneIsoscelesEquilateralTheir anglesAcuteRightObtuseEquiangular
6Classifying Triangles by Sides Scalene:A triangle in which no sides are congruent.BC=5.16cmBCABC=3.55cmABCAB = 3.47 cmAC = 3.47 cmAB = 3.02 cmAC = 3.15 cmIsosceles:A triangle in which at least 2 sides are congruent.HI=3.70cmGHIEquilateral:A triangle in which all 3 sides are congruent.GI = 3.70 cmGH = 3.70 cm
7Classifying Triangles by Angles Obtuse:1084428BCAA triangle in which one angle is....obtuse.Right:A triangle in which one angle is...right.
8Classifying Triangles by Angles Acute:574776GHIA triangle in which all three angles are....acute.Equiangular:A triangle in which all three angles are...congruent.
9Classification of Triangles Flow Charts Venn Diagrams withFlow ChartsandVenn Diagrams
10Classification by Sides polygonsPolygontrianglesTrianglescaleneisoscelesScaleneIsoscelesequilateralEquilateral
11Classification by Angles polygonsPolygontrianglesTrianglerightacuteequiangularRightObtuseAcuteobtuseEquiangular
12Naming Triangles We name a triangle using its vertices. For example, we can call this triangle:∆ABC∆ACBReview: What is ABC?∆BAC∆BCA∆CAB∆CBA
13Parts of Triangles Every triangle has three sides and three angles. For example, ∆ABC hasSides: Angles: CAB ABC ACB
14Opposite Sides and Angles Side opposite of BAC :Side opposite of ABC :Side opposite of ACB :Opposite Angles:Angle opposite of : BACAngle opposite of : ABCAngle opposite of : ACB
15Interior Angle of a Triangle An interior angle of a triangle (or any polygon) is an angle inside the triangle (or polygon), formed by two adjacent sides.For example, ∆ABC has interior angles: ABC, BAC, BCA
16Exterior AngleAn exterior angle of a triangle (or any polygon) is an angle that forms a linear pair with an interior angle. They are the angles outside the polygon formed by extending a side of the triangle (or polygon) into a ray.Exterior AngleInterior AnglesAFor example, ∆ABC has exterior angle ACD, because ACD forms a linear pair with ACB.DBC
17Interior and Exterior Angles The remote interior angles of a triangle (or any polygon) are the two interior angles that are “far away from” a given exterior angle. They are the angles that do not form a linear pair with a given exterior angle.For example, ∆ABC has exterior angle:ACD andremote interior angles A and BExterior AngleRemote Interior AnglesADBC
19m<A + m<B + m<C = 180 Triangle Sum TheoremThe sum of the measures of the interior angles in a triangle is 180˚.m<A + m<B + m<C = 180IGO GeoGebra Applet
20Third Angle CorollaryIf two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.
21Third Angle Corollary Proof Given:The diagramProve:C Fstatementsreasons1. A D, B E2. mA = mD, mB = mE3. mA + mB + m C = 180ºmD + mE + m F = 180º4. m C = 180º – m A – mBm F = 180º – m D – mE5. m C = 180º – m D – mE6. mC = mF7. C F1. Given2. Definition: congruence3. Triangle Sum TheoremSubtraction Property of EqualityProperty: SubstitutionDefinition: congruenceQED
22Corollary Each angle in an equiangular triangle measures 60˚. 60 60
23CorollaryThere can be at most one right or obtuse angle in a triangle.ExampleTriangles???
24CorollaryAcute angles in a right triangle are complementary.Example
25Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.Remote Interior AnglesAExterior AngleDExample:Find the mA.BC3x - 22 = x + 803x – x =2x = 102x = 51mA = x = 51°
26Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.GeoGebra Applet (Theorem 1)
28Introduction There are four segments associated with triangles: MediansAltitudesPerpendicular BisectorsAngle Bisectors
29Median - Special Segment of Triangle Definition:A segment from the vertex of the triangle to the midpoint of the opposite side.BADECFSince there are three vertices, there are three medians.In the figure C, E and F are the midpoints of the sides of the triangle.
30Altitude - Special Segment of Triangle The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side.Definition:BADFIn a right triangle, two of the altitudes are the legs of the triangle.BADFIKIn an obtuse triangle, two of the altitudes are outside of the triangle.
31Perpendicular Bisector – Special Segment of a triangle A line (or ray or segment) that is perpendicular to a segment at its midpoint.Definition:The perpendicular bisector does not have to start from a vertex!ROQPExample:MLNCDAEABBIn the isosceles ∆POQ, is the perpendicular bisector.In the scalene ∆CDE, is the perpendicular bisector.In the right ∆MLN, is the perpendicular bisector.