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Math 1312 Section 4. The Trapezoid
Definition:
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases of the trapezoid. The nonparallel sides are referred to as the legs of the trapezoid.
Example 1:
Definition:
An altitude of a trapezoid is a segment drawn from any point on one of the parallel sides (base) perpendicular to the opposite side (the other base).
An infinite number of altitudes may be drawn in a trapezoid.
Example 2:
Definition:
A median of a trapezoid is the segment that joins the midpoints of the nonparallel sides (legs).
Example 3:
Base
Base
Leg Leg
Base Base
Leg
Leg
Median Median
Theorem:
The median of a trapezoid is parallel to each base and the length of the median equals one-half the sum of the lengths of the two bases.
Example 4:
Definition:
An isosceles trapezoid is a trapezoid in which the legs (nonparallel sides) are congruent.
An isosceles trapezoid features some special properties not found in all trapezoids.
Properties of Isosceles Trapezoid
- The legs are congruent.
- The bases are parallel.
- The lower base angles of an isosceles trapezoid are congruent.
- The upper base angles of an isosceles trapezoid are congruent.
- The lower base angle is supplementary to any upper base angle.
- The diagonals of an isosceles trapezoid are congruent.
- The median is parallel to the base.
- The length of the median equals one-half the sum of the lengths of the two bases.
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Example 7: Given an isosceles trapezoid ABCD with legs AD and BC.
a. m ∠ DAB = 75 °. Find the m ∠ ADC.
b. AC = 40. Find BD.
c. If m ∠ A = 6 x + 25 and m ∠ B = 8 x + 15. Find the measures of angles C and D.
Example 8: Given an isosceles trapezoid HJKL with bases HJ , LK , and median RS.
a. If LK = 30 and HJ = 42 find RS.
b. If RS = 17 and HJ = 14 find LK.
c. If RS = x + 5 and HJ + LK = 4 x + 6 find RS.
SUMMARY CHARTS:
Special Diagonals Are Always Diagonals Always Bisect Quadrilateral Congruent Perpendicular Each Other Angles
Parallelogram No No Yes No Rectangle Yes No Yes No Rhombus No Yes Yes Yes Square Yes Yes Yes Yes
Trapezoid No No No No Isosceles Trapezoid
Yes No No No
Property Rectangle Rhombus Square
- All the properties of a parallelogram? Yes Yes Yes
- Equiangular (4 right angles?) Yes No Yes
- Equilateral (4 congruent sides?) No Yes Yes
- Diagonals bisect angles? No Yes Yes
- Diagonals congruent? Yes No Yes
- Diagonals perpendicular? No Yes Yes
