Isosceles & Equilateral Triangles

Transcription

1 Isosceles & Equilateral Triangles Objective: Use and apply properties of isosceles triangles. Refer to textbook pages Write NOTES on DNG pages

2 Vocabulary Isosceles is derived from the Greek isos for (NOTES DNG page 83) equal and skelos for leg. different forms of a chemical element with the same atomic number.

3 Isosceles & Equilateral Triangles Objective: Use and apply properties of isosceles triangles. (NOTES DNG page 82) If two sides of a triangle are congruent, then the angles opposites those sides are congruent. If two angles of a triangle are congruent, then the sides opposites those angles are congruent. The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

4 Isosceles & Equilateral Triangles Objective: Use and apply properties of isosceles triangles. Vocabulary A corollary is a statement that follows immediately from a theorem. (Isosceles Triangle Theorem) (NOTES DNG page 82) If a triangle is equilateral, then the triangle is equiangular. (Converse of Isosceles Theorem) If a triangle is equiangular, then the triangle is equilateral.

5 Isosceles & Equilateral Triangles Objective: Use and apply properties of isosceles triangles. Architecture The A-shaped roof has congruent legs and congruent base angles. Seventeen spires, pictured at the left, cover the Cadet Chapel at the Air Force Academy in Colorado Springs, Colorado. Each spire is an isosceles triangle with a 40 vertex angle.

6 Isosceles & Equilateral Triangles Objective: Use and apply properties of isosceles triangles. XY XZ 1 2 BX BX Given Definition of Angle Bisector Reflexive Prop. of Congruence BXY BXZ SAS Postulate Y Z CPCTC

7 Isosceles & Equilateral Triangles DNG page 83 Explain why ABC is isosceles. By the definition of an isosceles triangle, ABC is isosceles. Can you deduce that RUV is isosceles? Explain. No; neither RUV nor RVU can be shown to be to R.

8 Suppose that m L = y. Find the value of x and y. The bisector of the vertex angle of an Isosceles is the bisector of the base. x = 90 Definition of perpendicular. m N = m L Isosceles Theorem m L = y m N = y Given Transitive Property of Congruence y m N + m NMO + m MON = 180 Triangle-Sum Theorem y + y + 90 = 180 Substitute 2y + 90 = 180 Simplify y =90 y 2y = 90 Subtract 90 from each side. y = 45 Divide each side by 2. Therefore, x = 90 and y = 45.

9 Using Isosceles Triangles The garden shown at the right is in the shape of a regular hexagon. Suppose that a segment is drawn between the endpoints of the angle marked x. Find the angle measures of the triangle that is formed. The measure of x = 120. Because the garden is a regular hexagon, the sides have equal length, so the triangle is isosceles. By the Isosceles Triangle Theorem, the raised bed garden unknown angles are congruent. The measure of the angle marked x is 120, and the sum of the angle measures of a triangle is ( Label each unknown angle y : x + y + y = y = 180 railroad tie steps 2y = 60 y = 30 So the angle measures of the triangle are 120, 30, and 30. flagstone walk

10 Isosceles & Equilateral Triangles Theorem 4-5: The bisector of the vertex of an isosceles is the bisector of the base. Suppose m L = 43. Find the values of x and y. y + x + 43 = 180 y = 180 y = 180 y = 47 x = 90, y = 47 Isosceles Theorem raised bed garden railroad tie steps z z = 150 flagstone walk

11 Practice 4-5 page 323 Isosceles & Equilateral Triangles Find the values of the variables. Note: An equilateral is also equiangular. Thus each interior angle measures x + y = 180 x + y = 70 but x = y, x + x = 70 2x = 70 x = 35 y = 35 x = 180 x +100 = 180 x = 80 y = 90 t = 360 t +210 = 360 t = 150

12 Practice 4-5 page 323 Isosceles & Equilateral Triangles Find the values of the variables. r r + r + 90 = 180 2r = 90 r = 45 but s = r, s = 45 x +125 = 180 x = 55 y +110 =180 y = 70 =55 55 z = 125 mof an interior n = 5 n n m XYZ = b + 72 = 108 b = a = 360 a = 360 a = 132 m XYZ = ( c = 60

13 Practice 4-5 page 323 Isosceles & Equilateral Triangles Find the values of the variables. c 60 c c c c 2x 6 = x x = 6 2c + 30 = 180 2c = 150 c = 75 2c + a = 180 2(75)+ a = a = 180 a = 30 z = z = 120 2c + b = 180 b = 30

14 ( ( Practice 4-5 page 323 Isosceles & Equilateral Triangles Complete each statement. Explain why it is true. AD, GA, KJ, DC, BA, CB, F ACG KJI CED AJB BHC D AGC KIJ CDE ABJ BCH Given m D = 25, find the measure of each angle = 180 ( ) = = 130 = ½ (130) = 65 = 130 = 90

15 Practice 4-5 page 323 Isosceles & Equilateral Triangles Find the values of x and y. x x = 180 x = 180 x = 70 y + y + 70 = 180 2y = 110 y = 55 x = x = 70 y + x + 90 = 180 y = 180 y = 180 y = 20 (Isosceles right triangle) x + x + 90 = 180 2x = 90 x = 45 y = 45