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Understanding Trapezoids: Definitions, Theorems, and Examples

Learn about trapezoids, isosceles trapezoids, area calculations, and how to prove a quadrilateral is a trapezoid. Explore practical examples and applications in geometry.

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Understanding Trapezoids: Definitions, Theorems, and Examples

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  1. Notes 8-6 and 11-4 Trapezoids and Area of Irregular Shapes

  2. What is a Trapezoid? • A trapezoidis a quadrilateral with exactly one pair of parallel sides. • Parallel sides, base • Nonparallel sides, legs • Base angles, two consecutive angles whose common side is a base

  3. What is an Isosceles Trapezoid? • Definition: Trapezoid with congruent legs. • Theorem: Each pair of base angles are congruent. • Theorem: The diagonals are congruent.

  4. Example: Find mF. mF = 131°

  5. Example: JN = 10.6, and NL = 14.8. Find KM. KM = 10.6 + 14.8 = 25.4

  6. Example: Find the value of a so that PQRS is isosceles. a = 9 or a = –9

  7. Example: AD = 12x – 11, and BC = 9x – 2. Find the value of x so that ABCD is isosceles. x = 3

  8. Example: Finding Measurements of Trapezoids Find the area of a trapezoid in which b1 = 8 in., b2 = 5 in., and h = 6.2 in. A = 40.3 in2

  9. Example: Finding Measurements of Trapezoids Find b2 of the trapezoid, in which A = 231 mm2. b2 = 19 mm

  10. Example: Find the area of the triangle. A = 96 m2

  11. To Prove a Quadrilateral is a Trapezoid: • If given vertices on coordinate plane: • Prove exactly one pair of opposite sides are parallel (Slope Formula). • Prove it is Isosceles by showing both legs are congruent (Distance Formula). • Example: Is Quadrilateral ABCD a Trapezoid? Isosceles Trapezoid? A(-5, -3), B(-4, 2), C(-1, 4), D(1, 1)

  12. Median of a Trapezoid: • The Median, or midsegment, of a trapezoidis the segment whose endpoints are the midpoints of the legs. • The Median is parallel to the bases. • The median’s measure is half the sum of the bases. (Median)

  13. Example: Finding Lengths Using Midsegments Find EF. EF = 10.75

  14. Example: Find EH. 8= EH

  15. Lesson Quiz: Use the diagram for items 1 and 2. 1. mWZY = 61°. Find mWXY. 2.XV = 4.6, and WY = 14.2. Find VZ. 3. Find LP. 119° 9.6 18

  16. Example: Finding the Areas of Composite Figures Find the shaded area. Round to the nearest tenth, if necessary.

  17. Example: Finding the Areas of Composite Figures Find the shaded area. Round to the nearest tenth, if necessary. shaded area: 40 + 25 = 65 ft2

  18. Example: Find the shaded area. Round to the nearest tenth, if necessary. Total shaded area is about 1781.3 m2.

  19. Example: Finding the Areas of Composite Figures Find the shaded area. Round to the nearest tenth, if necessary. area of figure: 234 – 10.125 ≈ 202.2 ft2

  20. Example: Finding the Areas of Composite Figures Find the shaded area. Round to the nearest tenth, if necessary. area of figure: 100 –128  186.2 cm2

  21. Example: Find the shaded area. Round to the nearest tenth, if necessary. area of figure: 28.3 – 18 = 10.3 in2

  22. Example: Fabric Application A company receives an order for 65 pieces of fabric in the given shape. Each piece is to be dyed red. To dye 6 in2 of fabric, 2 oz of dye is needed. How much dye is needed for the entire order?

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