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Exploring Isosceles and Equilateral Triangles

Discover the properties of isosceles and equilateral triangles, including Base Angles Theorem, Equilateral Triangle Theorem, and practical constructions. Learn through examples and investigations.

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Exploring Isosceles and Equilateral Triangles

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  1. Warm-Up 1 If the lengths of two sides of an isosceles triangle are 3 and 7, what is the perimeter of the triangle?

  2. Warm-Up 2 Use your straightedge to draw an acute angle. How could you use your compass to turn this angle into an isosceles triangle?

  3. Definition: Review A triangle is an isosceles triangle if and only if it has at least two congruent sides.

  4. 4.7 Use Isosceles and Equilateral Triangles Objectives: • To complete and use the Base Angles Theorem and its converse • To deduce the Equilateral Triangle Theorem from the Base Angles Theorem

  5. Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent.

  6. Example 1 • AC = AB Given • BM = CM Def. of Median • AM = AM Reflex • AMC = AMB SSS • <B = <C CPCTC Median

  7. Example 2 Try this one in your notebook Altitude

  8. Example 3 Find the value of x. 35

  9. Example 4: SAT If AB = BC, what is y in terms of x? y=2x-4

  10. Converse of the Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent.

  11. Investigation 3 In this Investigation, we will use some more compass and straightedge wizardry to construct a perfect equilateral triangle.

  12. Investigation 3 • Draw AB.

  13. Investigation 3 • Using A as center and AB as the radius, draw a quarter-circle arc through B.

  14. Investigation 3 • Using B as center and AB as the radius, draw a quarter-circle arc through A.

  15. Investigation 3 • Label the point of intersection C.

  16. Investigation 3 • Draw AC and BC.

  17. Investigation 3 Click button below to watch a construction video

  18. Investigation 3 In ΔABC, measure each angle with the protractor. What do you notice about the angle measurements? Use deductive reasoning to show that these measurements must be true based on the Base Angles Theorem.

  19. Equilateral Triangle Theorem A triangle is equilateral if and only if it is equiangular.

  20. Example 5 Find the values of x and y in the diagram. Answer in your notebook

  21. Example 6: SAT Find the value of x. 30

  22. Example 7: SAT In the figure shown, if O is the center of the circle, what is the value of x? 55

  23. Example 8: SAT In the figure shown, if AB = AC and AC||BD, what is the value of x? 65

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