The trapezoid

Definition: a trapezoid is a quadrilateral with exactly one pair of parallel sides.

• The parallel sides are the bases of the trapezoid;
• the other two sides are its legs.

Theorem: in a trapezoid, the two angles beside each leg add up to $\displaystyle 180^{\circ}$.

• The leg $\displaystyle AD$ cuts both parallel bases, so $\displaystyle \angle A$ and $\displaystyle \angle D$ are co-interior angles;
• so are $\displaystyle \angle B$ and $\displaystyle \angle C$, beside the leg $\displaystyle BC$;
• for example, if $\displaystyle {\angle A}={110^{\circ}}$, then $\displaystyle {\angle D}={180^{\circ}-110^{\circ}}={70^{\circ}}$.

Given: $\displaystyle AB\parallel DC$.
Prove: $\displaystyle {\angle A+\angle D}={180^{\circ}}$ and $\displaystyle {\angle B+\angle C}={180^{\circ}}$.

Definition: a height of a trapezoid is a perpendicular from a point on one base to the other base.

Theorem: all the heights of a trapezoid are equal.

Given: $\displaystyle AB\parallel DC$.
The segments $\displaystyle AE$ and $\displaystyle BF$ are perpendicular to $\displaystyle DC$.
Prove: $\displaystyle {AE}={BF}$.

The proof below draws the segment $\displaystyle AF$ and shows that triangles $\displaystyle ABF$ and $\displaystyle FEA$ are congruent:

All the heights are equal, so the height of a trapezoid, $\displaystyle h$, is one number: the distance between the bases.

The diagonal $\displaystyle AC$ cuts the trapezoid into two triangles, and both have the same height $\displaystyle h$:

• in triangle $\displaystyle ACD$ the base is $\displaystyle {DC}={b}$, and the area is $\displaystyle \frac{b\cdot h}{2}$;
• in triangle $\displaystyle ABC$ the base is $\displaystyle {AB}={a}$, its height from the vertex $\displaystyle C$ is $\displaystyle h$, and the area is $\displaystyle \frac{a\cdot h}{2}$.

Together, the area of the trapezoid:

In words: the sum of the bases times the height, divided by $\displaystyle 2$.

In trapezoid $\displaystyle ABCD$ the bases are $\displaystyle {AB}={5}$ and $\displaystyle {DC}={26}$, the legs are $\displaystyle {AD}={10}$ and $\displaystyle {BC}={17}$, and the height is $\displaystyle 8$.

• The area: $\displaystyle {S}={\frac{(5+26)\cdot8}{2}}={\frac{31\cdot8}{2}}={124}$;
• the perimeter: $\displaystyle {5+26+10+17}={58}$.

The quadrilateral $\displaystyle ABCD$ in the coordinate system is a right trapezoid:

• $\displaystyle A$ and $\displaystyle B$ have the same $\displaystyle y$, so $\displaystyle AB$ is parallel to the $\displaystyle x$ axis. So is $\displaystyle DC$, so $\displaystyle AB\parallel DC$;
• $\displaystyle BC$ is perpendicular to the $\displaystyle x$ axis and $\displaystyle AD$ is not, so they are not parallel;
• $\displaystyle BC$ is perpendicular to the bases, so it is also the height: $\displaystyle {h}={5-1}={4}$;
• the bases: $\displaystyle {AB}={6-3}={3}$ and $\displaystyle {DC}={6-0}={6}$;
• the leg $\displaystyle AD$, by Pythagoras: the difference in $\displaystyle x$ is $\displaystyle 3$, the difference in $\displaystyle y$ is $\displaystyle 4$, so $\displaystyle {AD}={\sqrt{3^2+4^2}}={5}$;
• the area: $\displaystyle {S}={\frac{(3+6)\cdot4}{2}}={18}$, and the perimeter: $\displaystyle {3+6+4+5}={18}$.

In trapezoid $\displaystyle ABCD$ the bases are $\displaystyle AB$ and $\displaystyle DC$.
Find its other angles, and give the reasons.

$\displaystyle {\angle A}={115^{\circ}}$ and $\displaystyle {\angle C}={50^{\circ}}$.

Angle $\displaystyle B$ is twice angle $\displaystyle C$, and angle $\displaystyle D$ is $\displaystyle 72^{\circ}$.

Work out:

The bases of a trapezoid are $\displaystyle 7$ and $\displaystyle 13$, and its height is $\displaystyle 6$.
What is its area?

The area of a trapezoid is $\displaystyle 45$, and its bases are $\displaystyle 4$ and $\displaystyle 11$.
What is its height?

The perimeter of a trapezoid is $\displaystyle 40$.
Its bases are $\displaystyle 8$ and $\displaystyle 14$, and one of its legs is $\displaystyle 7$.
How long is the other leg?

The quadrilateral $\displaystyle KLMN$ is drawn in the coordinate system.

Show that $\displaystyle KLMN$ is a trapezoid: which sides are parallel, and why?

Find the area of the trapezoid.

Find the length of the leg $\displaystyle KN$.

In trapezoid $\displaystyle ABCD$ the bases are $\displaystyle AB$ and $\displaystyle DC$.
The diagonal $\displaystyle DB$ bisects angle $\displaystyle D$.
Prove: $\displaystyle {AD}={AB}$.