Geometry / Unit 7: Polygons and Quadrilaterals
7.5 Trapezoids and Kites
Notes
- A trapezoid has exactly one pair of parallel sides, the bases. The other sides are the legs.
- Isosceles trapezoid: congruent legs, congruent base angles (each pair), and congruent diagonals.
- Trapezoid Midsegment Theorem: the midsegment is parallel to the bases and its length is the average: $\displaystyle \frac{b_1 + b_2}{2}$.
- A kite has two pairs of consecutive congruent sides. Its diagonals are perpendicular, and one pair of opposite angles (between the unequal sides) is congruent.
- Example: bases 10 and 18 give a midsegment of 14.
Practice
A trapezoid has bases of 9 and 21. How long is its midsegment?
$\displaystyle \frac{9 + 21}{2} = 15$.
In isosceles trapezoid $ABCD$ with bases $\overline{AB}$ and $\overline{CD}$, $m\angle A = 70^\circ$. Find $m\angle D$.
$\angle A$ and $\angle D$ are same-side interior angles along a leg between parallel bases, so they are supplementary.
Kite $ABCD$ has $AB = AD$ and $CB = CD$. $m\angle B = 115^\circ$ and $m\angle A = 50^\circ$. Find $m\angle C$.
$\angle B \cong \angle D = 115^\circ$, so $m\angle C = 360 - 50 - 115 - 115 = 80^\circ$.
Ready for more? Take the Unit 7 test.