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5.6 – Trapezoids

Key Terms

  • Base Angles – The two angles formed by the base of a trapezoid and the two adjacent sides.
  • Bases – The parallel sides of a trapezoid.
  • Isosceles Trapezoid – A trapezoid with two congruent legs.
    • The base angles of an isosceles trapezoid are also congruent.
  • Legs – The nonparallel sides of a trapezoid.
  • Median – The line segment that joins the midpoints of the legs of a trapezoid.
    • The median is parallel to the bases, and its length equals the mean of their lengths.
  • Trapezoid – A quadrilateral with exactly one pair of parallel sides.

 


Review

  • Isosceles
    • Two congruent sides
    • Congruent base angles

 

  • Consecutive Interior Angles
    • Angles between two parallel lines, along the same side (transversal)
    • Supplementary (adds up to 180°)
    • Adjacent (side by side)

 

  • Diagonals
    • Segments that connect opposite vertices.

 


Notes

  • Quadrilateral Family Tree and Diagrams

GeoB 5.6 - Quad Family Trap Tree GeoB 5.6 - Quad Family Trap

 


  • Diagram of a Trapezoid

GeoB 5.6 - Trap Base Angles GeoB 5.6 - Trap Base GeoB 5.6 - Trap Legs GeoB 5.6 - Trapezoid

  • Trapezoid Properties
    • A trapezoid is a quadrilateral with exactly one pair of parallel sides.
    • Trapezoids are NOT parallelograms
      • In a parallelogram, both pairs of opposite sides are parallel.
      • A trapezoid is not a parallelogram because only one pair of its sides is parallel.
    • Trapezoids are NEVER rectangles, because they can only have ONE pair of parallel sides (not two).
    • Trapezoids can have two right angles, but not four.

GeoB 5.6 - Trapezoid Prop

  • Isosceles Trapezoids
    • No right angles, not even one.

GeoB 5.6 - Trapezoid Isos Prop

GeoB 5.6 - Trapezoid IsosGeoB 5.6 - Congruent Base Angles GeoB 5.6 - Trapezoid Isos Cong Legs

  • Median
    • … is a segment that connects the midpoints of the legs of a trapezoid.
    • … is always parallel to both bases of a trapezoid.
    • … Connects the midpoints of the legs.
    • The length of the median is half the sum of the lengths of the bases.

GeoB 5.6 - Trapezoid Median

GeoB 5.6 - Median Ex GeoB 5.6 - Median Ex2

  • Median of a Trapezoid Formula
    • median=\frac{1}{2}\cdot (B_1+B_2)

 


Examples

  • Ex 1. If TRAP is an isosceles trapezoid, what is the value of x?
    • Since \overline{TR} and \overline{PA} are parallel, \angle T and \angle P are consecutive interior angles (which are supplementary).
    • Setup: 4x + 12 + 2x = 180
    • Combine like terms: 6x + 12 = 180
    • Subtract 12 on both sides: 6x = 168
    • Divide by 6 on both sides: x = 28°

GeoB 5.6 - Ex2c

 

  • Ex 2. If SLDG is an isosceles trapezoid, what is the value of x?
    • Since the diagonals of an isosceles trapezoid are congruent, they are equal.
    • Setup: 8x – 5 = 6x + 21
    • Subtract 6x on both sides: 2x – 5 = 21
    • Add 5 to both sides: 2x = 26
    • Divide both sides by 2: x = 13°

GeoB 5.6 - Ex3c

 

  •  Ex 3. Given the median \overline{QR} and trapezoid MOPN, what is the value of x?
    • The formula for finding the median is median=\frac{1}{2}\cdot (B_1+B_2)
    • Setup: \frac{1}{2}\cdot [(5x-7)+(6x+6)]=27
    • Combine like terms: \frac{1}{2}\cdot (11x-1)=27
    • Multiply by 2 on both sides: 11x – 1 = 54
    • Add 1 to both sides: 11x = 55
    • Divide by 11 on both sides: x = 5

GeoB 5.6 - Ex4c

 

  • Ex 4. DEFG is an isosceles trapezoid. Find the measure of \angle G.
    • Isosceles trapezoids have congruent base angles. So, if \angle D=121 then \angle G=121
    • Also, \angle G and \angle F are supplementary (add up to 180°); so, 59 + G = 180.  Solve for G.
    • Answer: 121°

GeoB 5.6 - Ex8c

  • Ex 5. Jaime wants to tile his floor using tiles in the shape of a trapezoid.
    • To make the pattern a little more interesting he has decided to cut the tiles in half along the median.
    • The top base of each tile is 15 inches in length and the bottom base is 21 inches.
    • How long of a cut will Jaime need to make so that he cuts the tiles along the median?
      • Formula: median=\frac{1}{2}\cdot (B_1+B_2)
      • Add the bases: median=\frac{1}{2}\cdot (15+21)
      • What is one-half of 36?  Divide 36 by 2 to find out: median=\frac{1}{2}\cdot (36)
      • The median = 18 inches

 


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