Subject: Geometry


The Breakdown

What are quadrilaterals?

A quadrilateral is a four-sided, two-dimensional enclosed figure (quad means four, lateral means side). Squares and rectangles are quadrilaterals, but quadrilaterals are not always square or rectangular. The sides do not have to be the same length, and can meet at any angle.

This section will look at angle relationships within quadrilaterals. Here are a few shapes that meet the requirements:

Figure 1: Examples of quadrilaterals

Here are a few that don’t. None of the following shapes are quadrilaterals. If you don’t understand why, slow down and have a think about each one. The first sentence in this article ought to help you out.

Figure 2: Examples of shapes that are not quadrilaterals

Sum of all angles in a quadrilateral

What do you think the angles of different quadrilaterals add up to? Why? Do you think quadrilaterals all have different angle sums, or does it depend on the length of their sides? (An angle sum is the total value of all interior angles in a shape.) In the following diagram (see Figure 3), does A have the same angle sum as B?

Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same

Think about cutting a square or rectangle in half diagonally…

If you already know that the angles of any and every triangle add to 180°, it should make sense to you that the angles of a quadrilateral sum to 360° (two times 180°). This is easy to do with square and rectangular quadrilaterals, but harder when the sides are not all equal.

Principles related to quadrilaterals

Later on in this article we’ll have a look at trapezoids. For now, let’s examine some principles related to quadrilaterals of a general nature.

Principle 1:

Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2:

If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

In the following diagram, angles a and b must add to 180° for the total enclosure to equal 360°.

Figure 4: Since quadrilaterals have sum of its angles not more than 360 degrees, angles a and b must add up to 180.

Principle 3:

Quadrilateral edges drawn to extend past corners will create supplementary interior/exterior angles.

In the following diagram, angles a and c are supplementary, and angles b and d are supplementary.

Figure 4: Quadrilaterals with extending sides.

What are trapezoids?

A trapezoid is a quadrilateral that has exactly two parallel sides. In other words, it has two parallel sides and two non-parallel sides. The parallel sides are called bases, the non-parallel sides connecting the bases are called legs. The median is the finite line segment connecting the two midpoints of the legs. In the following diagram showing Trapezoid Y, \bar{AC} and \bar{BD} are the legs, \bar{AB} and \bar{CD} are the bases, and \bar{EF} is the median.

Figure 6: A diagram of Trapeziod Y.

Trapezoid Y is an isosceles trapezoid because its legs are congruent (\bar{AC} = \bar{BD}). A trapezoid without congruent legs is simply called a trapezoid.

An isosceles trapezoid is symmetrical when cut at a right angle to the midpoint of the median.

The base angles of a trapezoid are the angles at the ends of its longer base, in Trapezoid Y’s case, \angle{C} and \angle{D} .

Principles related to trapezoids

Let’s look at some basic principles pertaining trapezoids.

Principle 1:

The base angles of an isosceles trapezoid are congruent.

Figure 7: Other form of trapezoid.

In the above diagram \angle{a} \propto \angle{b}.

Principle 2:

Any trapezoid with congruent base angles is an isosceles trapezoid.

Thus, in the diagram used above, if \angle{A} \propto \angle{B}, then \angle{A} \propto \angle{B} .

Applying Algebra to Trapezoid Figures

In the following diagram, use your knowledge of geometry to find values for all four angles of each trapezoid by first finding values for x and y. These shouldn’t be too hard for you to solve, just use a little logic and you’ll be fine.

Figure 8

Example #1

ABCD is an isosceles trapezoid.

Please note that a lower case x means ‘letter x’ which replaces a number, while upper case X denotes multiplication.

The statement that ABCD is isosceles has been given to us. From this we can deduce two truths:

1. \angle{C}= 5x + 60°. 2. 5x = 2x + y.

Therefore 5x – y = 2x. Therefore y = 3x. Working out the values for x and y is not too difficult from here. Because and are parallel, we know that = 180°. In other words 10x + 60 = 180°. Therefore 10x = 120°. (Deduct 60° from 180°.) Therefore x = 120°/10 x = 12°. Finding values for angles is as simple as substituting this value into x in the diagram. = 5x = 5 X 12 = 60° = 5x + 60 = (5 X 12) + 60 = 120° Because ABCD is isosceles, we know and .

Figure 9

Example #2

As before, work out the angle values for each corner.

Solving the second diagram is a little more interesting. The fact that alone tells us that ABCD is not an isosceles trapezoid, so we’ll have to work out at least three angles in order to deduce the fourth. We know that because and are parallel, we know that = 180°, and that = 180°. We’ll work with and first. To put this mathematically, 12x – 15 + 15 = 180°. Simplified, 12x = 180°. Therefore x = 180/12 = 15°. = 5 X 15 = 75°. = 7 X 15 = 105°. (A possibility to consider: These two values are equal to 180, therefore the lines are parallel, ABCD is indeed a trapezoid. If 180°, and are not parallel… and ABCD is not a trapezoid.) = 9 X 15 = 135°. There are two possible ways to find a value for . The first is by deductive reasoning: = 360 – ( + + ) = 360 – (75 + 105 + 135) = 45°. To put this in words, must equal 360 minus the sum of the other three angles. The second is by algebra: We know that there are 24 fifteens in 360 ( 360/15 = 24). Because the other three angles sum to 21x, we know = 3x. We’ve been given a value of x + 2y for . From this we can conclude x = y because x + 2y = 3x. = 3x = 3 X 15 = 45°.