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4-9Triangles4-9 IsoscelesIsoscelesandand EquilateralEquilateral TrianglesWarm UpLesson PresentationLesson QuizHoltMcDougalGeometryGeometryHolt
4-9 Isosceles and Equilateral TrianglesWarm Up1. Find each angle measure.60 ; 60 ; 60 True or False. If false explain.2. Every equilateral triangle is isosceles.True3. Every isosceles triangle is equilateral.False; an isosceles triangle can haveonly two congruent sides.Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesObjectivesProve theorems about isosceles andequilateral triangles.Apply properties of isosceles andequilateral triangles.Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesVocabularylegs of an isosceles trianglevertex anglebasebase anglesHolt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesRecall that an isosceles triangle has at least twocongruent sides. The congruent sides are called thelegs. The vertex angle is the angle formed by thelegs. The side opposite the vertex angle is called thebase, and the base angles are the two angles thathave the base as a side.3 is the vertex angle.1 and2 are the base angles.Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesHolt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesReading MathThe Isosceles Triangle Theorem issometimes stated as “Base angles of anisosceles triangle are congruent.”Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesExample 1: Astronomy ApplicationThe length of YX is 20 feet.Explain why the length of YZ is the same.The m YZX 180 – 140,so m YZX 40 .Since YZXX, XYZ isisosceles by the Converseof the Isosceles TriangleTheorem.Thus YZ YX 20 ft.Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesExample 2A: Finding the Measure of an AngleFind m F.m F m D x Isosc. Thm.m F m D m A 180 Sum Thm.Substitute thex x 22 180 given values.Simplify and subtract2x 158 22 from both sides.x 79Thus m F 79 Holt McDougal GeometryDivide bothsides by 2.
4-9 Isosceles and Equilateral TrianglesExample 2B: Finding the Measure of an AngleFind m G.m J m G Isosc. Thm.(x 44) 3x44 2xSubstitute thegiven values.Simplify x fromboth sides.Divide bothsides by 2.Thus m G 22 44 66 .x 22Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesCheck It Out! Example 2AFind m H.m H m G x Isosc. Thm.m H m G m F 180 Sum Thm.Substitute thex x 48 180 given values.Simplify and subtract2x 132 48 from both sides.Divide bothsides by 2.Thus m H 66 x 66Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesCheck It Out! Example 2BFind m N.m P m N Isosc. Thm.(8y – 16) 6y2y 16y 8Substitute thegiven values.Subtract 6y andadd 16 to bothsides.Divide bothsides by 2.Thus m N 6(8) 48 .Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesThe following corollary and its converse show theconnection between equilateral triangles andequiangular triangles.Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesHolt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesExample 3A: Using Properties of EquilateralTrianglesFind the value of x. LKM is equilateral.Equilateral equiangular (2x 32) 602x 28x 14Holt McDougal GeometryThe measure of eachequiangular is 60 .Subtract 32 both sides.Divide both sides by 2.of an
4-9 Isosceles and Equilateral TrianglesExample 3B: Using Properties of EquilateralTrianglesFind the value of y. NPO is equiangular.Equiangular equilateral 5y – 6 4y 12y 18Holt McDougal GeometryDefinition ofequilateral .Subtract 4y and add 6 toboth sides.
4-9 Isosceles and Equilateral TrianglesCheck It Out! Example 3Find the value of JL. JKL is equiangular.Equiangular equilateral 4t – 8 2t 12t 9t 4.5Definition ofequilateral .Subtract 4y and add 6 toboth sides.Divide both sides by 2.Thus JL 2(4.5) 1 10.Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesRemember!A coordinate proof may be easier if youplace one side of the triangle along thex-axis and locate a vertex at the origin oron the y-axis.Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesExample 4: Using Coordinate ProofProve that the segment joining the midpointsof two sides of an isosceles triangle is half thebase.Given: In isosceles ABC, X is the mdpt. of AB, andY is the mdpt. of AC.Prove: XY 1AC.2Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesExample 4 ContinuedProof:Draw a diagram and place the coordinates as shown.By the Midpoint Formula,the coordinates of X are(a, b), and Y are (3a, b).By the Distance Formula,XY 4a2 2a, and AC 4a.1Therefore XY AC.2Holt McDougal Geometry
4-9 Isosceles and Equilateral TrianglesCheck It Out! Example 4What if.? The coordinates of isosceles ABC areA(0, 2b), B(-2a, 0), and C(2a, 0). X is the midpointof AB, and Y is the midpoint of AC. Prove XYZ isisosceles.yProof:A(0, 2b)Draw a diagram and place thecoordinates as shown.XYZB(–2a, 0)Holt McDougal GeometryxC(2a, 0)
4-9 Isosceles and Equilateral TrianglesCheck It Out! Example 4 ContinuedBy the Midpoint Formula, the coordinates. of X are(–a, b), the coordinates. of Y are (a, b), and thecoordinates of Z are (0, 0) . By the DistanceyFormula, XZ YZ a2 b2 .So XZA(0, 2b)YZ and XYZ is isosceles.XYZB(–2a, 0)Holt McDougal GeometryxC(2a, 0)
4-9 Isosceles and Equilateral TrianglesLesson Quiz: Part IFind each angle measure.1. m R28 2. m P124 Find each value.3. x5. xHolt McDougal Geometry204. y26 6
4-9 Isosceles and Equilateral TrianglesLesson Quiz: Part II6. The vertex angle of an isosceles trianglemeasures (a 15) , and one of the baseangles measures 7a . Find a and each anglemeasure.a 11; 26 ; 77 ; 77 Holt McDougal Geometry
Holt McDougal Geometry 4-9 Isosceles and Equilateral Triangles Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the bas
