Properties of Isosceles Trapezoids

Here are some properties of isosceles trapezoids. A trapezoid is a cyclic quadrilateral (that is, can fit inside a circle) with one pair of parallel sides. An isosceles trapezoid has one pair of parallel sides and another, different pair of congruent sides. They look like this:

The trapezoid has one line of symmetry - the cyan line. One green and one orange angle sum to 180 degrees - they are supplementary. The green angles are congruent, and the orange angles are congruent.

Perimeter and Area

Say we call side AB c, side BC d, side CD a, and AD d (BC and AD are both called d because AD=BC).Then the perimeter is: $$P= a+c+2d$$

The area is: $$A= 1/2(b_1+b_2)h$$

Here's how to get the area.

Suppose

Then you would double the trapezoid, and place it like the two together were a parallelogram.

The area of this parallelogram is bh, or: $$A= (b_1+b_2)h$$

Of course, since we doubled the parallelogram, it must be half of this area, which is:

$$A= 1/2(b_1+b_2)h$$

Trapezoids are quite cool! Let's take a mini quiz on trapezoids!

Solution to the last question.

I hope you enjoyed! ;D ~ Abigale Kim (a+bi)

One way to prove that a quadrilateral is an isosceles trapezoid is to show:

1. The quadrilateral has two parallel sides.

2. The lower base angles are congruent and the upper base angles are congruent.

Let's look at an example to illustrate how one might go about proving that a quadrilateral is an isosceles trapezoid.

First, let's show that $\overline{AD}$ is parallel to $\overline{BC}.$ The line segment that contains $\overline{AB}$ is a transversal that intersects with $\overline{AD}$ and $\overline{BC}.$ Notice that $m\angle BAD$ AND its corresponding angle are both equal to $75°.$ Thus, $\overline{AD} || \overline{BC}.$ For more information on why this is true, here is an explanation on the relationship between transversals, parallel lines, and angles.

Since $\overline{AD} || \overline{BC},$ we know that ABCD is a trapezoid. Next, we have to show that the corresponding base angles are congruent.

If we look at the trapezoid, we can see that $\angle ADC$ is the supplementary angle to $105°.$ To calculate $m\angle ADC,$ we subtract $105°$ from $180°$ to get $75°.$ So: $$m\angle ADC = m\angle BAD = 75°$$ The lower base angles are congruent!

We can also check that the upper base angles are congruent. Because $\overline{AD} || \overline{BC},$ we can conclude that $m\angle BCD = 105°.$ We also know that $\angle ABC$ is supplementary to $75°,$ so: $$m\angle ABC = 180° - 75° = 105°$$ So: $$m\angle BCD = m\angle ABC = 105°$$ The upper base angles are congruent!

We have shown that this quadrilateral has two parallel sides and congruent corresponding base angles. Thus, ABCD is an isosceles trapezoid.