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Geometry

Geometry. 5.4 Isosceles and Equilateral Triangles. What conjectures can you make about congruent angles and sides?. Topic/Objective. Use properties of isosceles triangles. Use properties of equilateral triangles. EF is opposite D. E is opposite side DF. Opposite Angles and Sides. E. D.

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Geometry

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  1. Geometry 5.4 Isosceles and Equilateral Triangles What conjectures can you make about congruent angles and sides?

  2. Topic/Objective • Use properties of isosceles triangles. • Use properties of equilateral triangles. Geometry 5.4 Isosceles, Equilateral Triangles

  3. EF is opposite D. E is opposite side DF. Opposite Angles and Sides E D F Geometry 5.4 Isosceles, Equilateral Triangles

  4. Isosceles Triangles Vertex Angle Leg Leg Base Angles Base Geometry 5.4 Isosceles, Equilateral Triangles

  5. A construction. Begin with an isosceles triangle, ABC. Draw the angle bisector from the vertex angle. The angle bisector intersects the base at M. ACM  BCM. Why? SAS A  B. Why? CPCTC C A B M Geometry 5.4 Isosceles, Equilateral Triangles

  6. Theorem 5.6 Base Angles Theorem. • If two sides of a triangle are congruent, then the angles opposite them are congruent. • (Easy form) The base angles of an isosceles triangle are congruent. Geometry 5.4 Isosceles, Equilateral Triangles

  7. This: Means this: Visually: The base angles of an isosceles triangle are congruent. Geometry 5.4 Isosceles, Equilateral Triangles

  8. Example Problem Solve for x. x + x + 52 = 180 2x + 52 = 180 2x = 128 x = 64 52° x° x° Geometry 5.4 Isosceles, Equilateral Triangles

  9. Example 2 Solve for x and y. In an isosceles triangle, base angles are congruent. So y is… 42° Now use the triangle angle sum theorem: x + 42 + 42 = 180 x + 84 = 180 x = 96° 42° x° y° Geometry 5.4 Isosceles, Equilateral Triangles

  10. 50° y° x° 50° y° Example 3. You try it. x = 65° y = 32.5° Find x and y. 2y + 115 = 180 2y = 65 y = 32.5° y° 32.5°` 115° x° 65° 65° x° 2x + 50 = 180 2x = 130 x = 65 180 – 65 = 115 Geometry 5.4 Isosceles, Equilateral Triangles

  11. (2x)° (3x – 25)° Example 4 Solve for x. 3x – 25 = 2x x = 25 Geometry 5.4 Isosceles, Equilateral Triangles

  12. Theorem 5.7 • Converse of the Base Angles Theorem. • If two angles of a triangle are congruent, then the sides opposite them are congruent. Geometry 5.4 Isosceles, Equilateral Triangles

  13. Example 5 Since base angles are equal, opposite sides are equal. 4x + 52 = 2x + 68 2x + 52 = 68 2x = 16 x = 8 4x + 52 2x + 68 4(8) + 52 = 84 Solve for x, then find the length of the legs. Geometry 5.4 Isosceles, Equilateral Triangles

  14. Example 6 You do it. Find the length of each side. 5x = 3x + 16 2x = 16 x = 8 40 4x – 2 5x 30 3x + 16 40 Geometry 5.4 Isosceles, Equilateral Triangles

  15. Equilateral TrianglesCorollaries to Base Angles Theorem • If a triangle is equilateral, then it is also equiangular. • If a triangle is equiangular, then it is also equilateral. Geometry 5.4 Isosceles, Equilateral Triangles

  16. Geometry 5.4 Isosceles, Equilateral Triangles

  17. 3x – 10 x + 10 2x Example 7 Solve for x. All sides are congruent. 3x – 10 = x + 10 2x = 20 x = 10 2x = x + 10 x = 10 3x – 10 = 2x x – 10 = 0 x = 10 Geometry 5.4 Isosceles, Equilateral Triangles

  18. One last problem. Solve for x and y. x° y° 50° Solution… Geometry 5.4 Isosceles, Equilateral Triangles

  19. This triangle is equilateral. Each angle is? These angles form straight angle. The missing angle is? Solution… 40° x° 70° 80° y° 60° 70° ? 50° 50° 60° 60° Geometry 5.4 Isosceles, Equilateral Triangles

  20. Summarize what you have learned today • The base angles of an isosceles triangle are congruent. • Equilateral triangles are Equiangular. Geometry 5.4 Isosceles, Equilateral Triangles

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