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Chapter 4

Chapter 4. 4-9 Isosceles and Equilateral Triangles. Objectives. Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Isosceles triangles.

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Chapter 4

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  1. Chapter 4 4-9 Isosceles and Equilateral Triangles

  2. Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles.

  3. Isosceles 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 base angles are the two angles that have the base as a side. • 3 is the vertex angle. • 1 and 2 are the base angles

  4. Theorems for isosceles triangles

  5. Example#1 • The length of YX is 20 feet. • Explain why the length of YZ is the same. • Since YZXX,∆XYZ is isosceles by the Converse of the IsoscelesTriangle Theorem. • Thus YZ = YX = 20 ft. The mYZX = 180 – 140, so mYZX = 40°.

  6. Example 2: Finding the Measure of an Angle • Find mF.

  7. Example#3 • Find mG

  8. Example#4 • Find mN

  9. Student guided practice • Do problems 3-6 in your book pg.288

  10. Equilateral triangles • The following corollary and its converse show the connection between equilateral triangles and equiangular triangles.

  11. Equilateral Triangle

  12. Example#5 • Find the value of x.

  13. Example#6 • Find the value of y.

  14. Example#7 • Find the value of JL

  15. Student guided practice • Do problems 7-10 in your book 288

  16. Example#8 Using coordinate proofs • Prove that the segment joining the midpoints of two sides of an isosceles triangle is half the base. • Given: In isosceles ∆ABC, X is the mdpt. of AB, and Y is the mdpt. of AC. • Prove:XY = 1/2 AC.

  17. Solution • Proof: • Draw a diagram and place the coordinates as shown.

  18. Solution • 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. • Therefore XY = 1/2 AC.

  19. y A(0, 2b) X Y x Z B(–2a, 0) C(2a, 0) Example #9 • What if...? The coordinates of isosceles ∆ABC are A(0, 2b), B(-2a, 0), and C(2a, 0). X is the midpoint of AB, and Y is the midpoint of AC. Prove ∆XYZ is isosceles.

  20. Solution • By the Midpoint Formula, the coordinates. of X are (–a, b), the coordinates. of Y are (a, b), and the coordinates of Z are (0, 0) . By the Distance Formula, XZ = YZ = √a2+b2 . • So XZ YZ and ∆XYZ is isosceles.

  21. Homework • Do problems 13-20 in your book page 289

  22. Closure • Today we learned about isosceles and equilateral triangles • Next class we are going to learned about Perpendicular and angle bisector

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