Presentation on theme: "Triangle Fundamentals"— Presentation transcript:
1 Triangle Fundamentals Intro to G.10TriangleFundamentalsModified by Lisa Palen
2 Triangle What’s a polygon? Definition: A triangle is a three-sided polygon.What’s a polygon?
3 PolygonsDefinition: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
4 Definition 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.
5 Triangles can be classified by: Their sidesScaleneIsoscelesEquilateralTheir anglesAcuteRightObtuseEquiangular
6 Classifying 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
7 Classifying Triangles by Angles Obtuse:1084428BCAA triangle in which one angle is....obtuse.Right:A triangle in which one angle is...right.
8 Classifying Triangles by Angles Acute:574776GHIA triangle in which all three angles are....acute.Equiangular:A triangle in which all three angles are...congruent.
10 Classification by Sides polygonsPolygontrianglesTrianglescaleneisoscelesScaleneIsoscelesequilateralEquilateral
11 Classification by Angles polygonsPolygontrianglesTrianglerightacuteequiangularRightObtuseAcuteobtuseEquiangular
12 Naming Triangles We name a triangle using its vertices. For example, we can call this triangle:∆ABC∆ACBReview: What is ABC?∆BAC∆BCA∆CAB∆CBA
13 Parts of Triangles Every triangle has three sides and three angles. For example, ∆ABC hasSides: Angles: CAB ABC ACB
14 Opposite 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
15 Interior 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
16 Exterior 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
17 Interior 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
19 m<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
20 Third Angle CorollaryIf two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.
21 Third 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
22 Corollary Each angle in an equiangular triangle measures 60˚. 60 60
23 CorollaryThere can be at most one right or obtuse angle in a triangle.ExampleTriangles???
24 CorollaryAcute angles in a right triangle are complementary.Example
25 Exterior 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°
26 Exterior 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)
28 Introduction There are four segments associated with triangles: MediansAltitudesPerpendicular BisectorsAngle Bisectors
29 Median - 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.
30 Altitude - 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.
31 Perpendicular 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.