Geometry Reference

Trapezoids

A trapezoid is a quadrilateral with one pair of parallel sides. Most trapezoid problems use the bases, height, median, or the special angle and diagonal facts of isosceles trapezoids.

Summary Table

Feature	Meaning	Rule
Bases	The parallel sides.	Usually called $b_1$ and $b_2$.
Legs	The non-parallel sides.	Congruent legs make an isosceles trapezoid.
Height	Perpendicular distance between the bases.	Use perpendicular height, not a slanted leg.
Median or midsegment	Segment joining the midpoints of the legs.	Parallel to the bases and equal to their average.
Same-leg interior angles	Angles along one leg between parallel bases.	They are supplementary.
Isosceles trapezoid	Legs are congruent.	Base angles are congruent and diagonals are congruent.

Content Formulas

Area

$$A=\frac12(b_1+b_2)h$$

Median / Midsegment

$$m=\frac{b_1+b_2}{2}$$

Area with Median

$$A=mh$$

Supplementary Same-Leg Angles

$$\angle A+\angle D=180^\circ$$ $$\angle B+\angle C=180^\circ$$

The area formula can be read as average of the bases times height. That average is exactly the median length.

Classic Mastery Problems

Area from Bases and Height

A trapezoid has bases $8$ and $14$ and height $5$. Find its area.

Trapezoid MoveAverage the bases, then multiply by the perpendicular height.

$$A=\frac12(b_1+b_2)h$$ $$=\frac12(8+14)(5)$$ $$=55$$

Find a Missing Base

A trapezoid has area $96$, height $8$, and one base $10$. Find the other base.

Trapezoid MoveSubstitute what is known into the area formula and solve for the missing base.

$$96=\frac12(10+b_2)(8)$$ $$96=4(10+b_2)$$ $$24=10+b_2$$ $$b_2=14$$

Median Length

The bases of a trapezoid are $11$ and $25$. Find the median length.

Trapezoid MoveThe median is the average of the bases.

$$m=\frac{11+25}{2}$$ $$=18$$

Use Median to Find Area

A trapezoid has median $17$ and height $9$. Find its area.

Trapezoid MoveThe median is the average of the bases, so area is median times height.

$$A=mh$$ $$=17(9)$$ $$=153$$

Same-Leg Angles

In a trapezoid, one angle along a leg is $68^\circ$. Find the other angle along the same leg.

Trapezoid MoveSame-leg interior angles are supplementary because the bases are parallel.

$$68^\circ+x=180^\circ$$ $$x=112^\circ$$

Isosceles Trapezoid Angles

An isosceles trapezoid has one base angle of $74^\circ$. Find the other three angles.

Trapezoid MoveBase angles match, and same-leg interior angles are supplementary.

$$\text{same base angle: }74^\circ$$ $$180^\circ-74^\circ=106^\circ$$ $$\text{other angles: }74^\circ,\ 106^\circ,\ 106^\circ$$

Isosceles Trapezoid Diagonals

In an isosceles trapezoid, the diagonals are labeled $3x+2$ and $5x-10$. Find $x$ and the diagonal length.

Trapezoid MoveDiagonals of an isosceles trapezoid are congruent.

$$3x+2=5x-10$$ $$12=2x$$ $$x=6$$ $$3(6)+2=20$$

Trapezoid Checklist

Question	Check
Which sides are the bases?	Find the parallel sides.
Is the height given?	Use perpendicular height, not a leg.
Is a median or midsegment given?	Treat it as the average of the bases.
Is the trapezoid isosceles?	Use congruent legs, congruent base angles, and congruent diagonals.
Are angles on the same leg?	They add to $180^\circ$.