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Trapezoid

A trapezoid is a quadrilateral with one pair of parallel lines

Bases - The two parallel lines are called the bases.

The Legs - The two non parallel lines are the legs.

Diagram 1 Trapezoid Picture , bases and legs laballed
Diagram 2 base and legs of trapezoid

Properties

  • Property #1) The angles on the same side of a leg are called adjacent angles and are supplementary( more )
  • Property #2) Area of a Trapezoid = $$ Area = height \cdot \left( \frac{ \text{sum bases} }{ 2 } \right) $$ ( more )
  • Property #3) Trapezoids have a midsegment which connects the mipoints of the legs( more )

Adjacent Angles of Trapezoid

The angles on the same side of a leg are called adjacent angles such as $$\angle A $$ and $$ \angle D $$ are supplementary. For the same reason, $$ \angle B $$ and $$ \angle C $$ are supplementary.

Adjacent Angles of a Trapezoid are supplementary

Practice Problems

Problem 1

Use the adjacent angles theorem to determine m $$ \angle ZWX $$.

Adjacent angles of trapezoid

$$ \angle ZWX = 180 − 44 = 136° $$

Problem 2

Use adjacent angles theorem to calculate m $$ \angle MLO $$.

Base angles of trapezoid

$$ \angle MLO = 180-124 = 56° $$

Problem 3

Find the value of x in the trapezoid below, then determine the measure of angles $$ \angle WXY $$ and $$ \angle XYZ $$.

Same Side interior angles of trapezoid
angles
Problem 4

What is wrong with trapezoid LMNO pictured below? (Explain why LMNO cannot be a trapezoid based on the information provided).

Base angles of trapezoid

If LMNO is a trapezoid and its bases LO and MN are parallel then, $$ \angle MNO $$ and $$ \angle NOL $$ which must be supplementary however, the sum of these angles is not 180 111 + 68 ≠ 180.

Area of Trapezoid

Trapezoid Area formulla
Problem 5
trapezoid area

$ Area = height \cdot \left( \frac{ \text{sum bases} }{ 2 } \right) \\ = 7 \cdot \left( \frac{ 4 + 8 }{ 2 } \right) \\ =7 \cdot \left( \frac{ 12 }{ 2 } \right) \\ = 7 \cdot 6 \\ = \fbox {42 } ft^2 $

Midsegment of Trapezoid

Problem 6

Use the midsegment theorem to determine the length of midsegment ON.

Parallelogram Sides Picture

To calculate the length of the midsegment find the average of the bases length of midsegment = (6 + 4) / 2 = 5.

Midsegment of Trapezoid

The most important thing to remember is that a midpoint bisects a line (cuts a line into two equal halves).

Parallelogram Sides Picture

The midpoint of the red segment pictured below is the point $$(A, 2b)$$ (click button below to see).

The midsegment of a trapezoid is the segment that joins the midpoints of the nonparallel sides of a trapezoid.

In the trapezoid below, the midpoints of the non-parallel sides are points S and V. The midsegment is the red line segment from S to V.

Midsegment of Trapezoid, Picture
Example Midsegment
Trapezoid #10 Trapezoid picture
Step 1
Calculating the length of the bases . Upper Base:
$$ 35 - 16 = 9 $$
Step 2
Calculating Low Base:
$$ 45 - 0 = 45 $$
Step 3
Calculating the sum of the bases
$$ 9 + 45 = 54 $$
Step 4
Divide the sum by 2
$$ \frac {54}{2} = \boxed{27}$$
Problem 8

What is the length of midsegment SV in the trapezoid below?

Midsegment of trapezoid diagram and problem
Step 1
Calculating the length of the bases . Upper Base:
$$ 17 - 8 = 9 $$
Step 2
Calculating Low Base:
$$ 20 - 0 = 20 $$
Step 3
Calculating the sum of the bases
$$ 9 + 20 = 29 $$
Step 4
Divide the sum by 2
$$ \frac {29}{2} = \boxed{14.5}$$
Problem 9

Is the red segment below a midsegment?

Trapezoid Brain Teaser

It is not a true midsegment because its length does not equal half the sum of the lengths of the bases.