Subject: Geometry

Parallelograms

What is a parallelogram?

A parallelogram is a quadrilateral whose opposite sides are parallel. The converse of this definition is “any quadrilateral that has both opposite sides parallel is a parallelogram”.

This is a parallelogram:


Figure 1: A parallelogram: Widths and lengths are the same.

You can imagine if one – and only one – of these sides was longer, the two sides next to it could no longer be parallel (or congruent):


Figure 2: Not a parallelogram: There are no sides same lengths or width.

None of these are parallelograms:

Let’s have a look at a few principles involving parallelograms.

Principles

Principle 1.

Opposite sides of a parallelogram are parallel (this is the above definition).

Principle 2.

Opposite sides of a parallelogram are congruent.

Obviously they must be congruent in order to be parallel. This is the principle of parallel lines, that states “through a given point not on a given line, one and only one line can be drawn parallel to a given line”.

Principle 3.

Opposite angles of a parallelogram are congruent.

These first three principles are demonstrated in the following diagram (not to scale).


Figure 4: PARALLELOGRAM WITH MARKED SIDES LENGTHS AND INTERIOR ANGLES.

Principle 4.

A diagonal of a parallelogram creates two congruent triangles.

The triangles made will be congruent, regardless of which diagonal the line is drawn from.


Figure 5: PARALLELOGRAM CUT IN TWO.

Principle 5.

The two diagonals of a parallelogram bisect each other.

The diagonals bisect at each diagonals midpoint. Thus, four lines are created radiating from the central point (E in the following diagram). A mathematical way of expressing this is \bar{AE} = \bar{CE} and \bar{BE} = \bar{DE}.


Figure 6: PARALLELOGRAM CUT IN FOUR.

The diagonals bisect at each diagonals midpoint. Thus, four lines are created radiating from the central point (E in the following diagram). A mathematical way of expressing this is \bar{AE} = \bar{CE} and \bar{BE} = \bar{DE}.

Question:

What does it mean if:

1. All four lines from the central point are equal in length?

2. None of the lines are equal in length?

Answers are at the end of this article.

Many of these principles tie in with the basic principles of parallel lines. For instance, in any parallelogram, two consecutive angles (in geometry, consecutive means next to each other) will be supplementary. This is demonstrated in the fourth diagram.

Proving that a quadrilateral is a parallelogram

Question:

Have a look at the following diagrams. Use mathematical reasoning, not your eyesight. What can you say about the following shapes? Can they be parallelograms with the lengths and angles they have been given? Are the statements written about them true or false? Again, answers are at the end of the article.

3.


Figure 7: PARALLELOGRAM WITH SAME ANGLES (a=d=e=h, b=c=f=g).

4.


Figure 8: QUADRILATERAL WITH DIFFERENT LENGHTS.

5.


Figure 9: QUADRILATERAL WITH X AT THE MIDDLE (a \neq b, c \neq d).

Answers are at the end of this article.

Areas of Parallelograms

To find the area of a square or rectangle, we use the formula base multiplied by height. We can use the same formula for parallelograms. To find the area of this parallelogram, we would multiply its base by its height (11 X 4).


Figure 10

It’s important to note here that the term ‘base’ applies to the length of the bottom side or edge, it does not include the amount of overhang. The term height refers to the distance between the bases, NOT the lengths of the sides.

The following series of diagrams shows why this formula works. The base of this parallelogram is 11 units. The length of each side is 5. The vertical line at the right indicates the height value, 4 units, and the horizontal line at the left indicates the amount of ‘overhang’ in the shape, 3 units. When we draw lines perpendicular to the bases the overhang value will be equal to one side of our right angled triangles. So if we imagine cutting it up like this…


Figure 11: PARALLELOGRAM WITH EDGES CUT OFF AND MEASUREMENTS

We can put it together like this…


Figure 12: SAME PARALLELOGRAM AS ABOVE, MEASUREMENTS, TRIANGLES ARRANGED BETTER

4 X 11 = 44. Our new selection of shapes shows a rectangle of 8 units by 4 units (32 square units) and two triangles that can be put together to form one rectangle of 3 units by 4 units (12 square units). Adding the two totals together will give us an area of 44 square units.

Answers

1.

If all four lines are equal, the shape is a rectangle or square.

2.

If none of the lines are equal in length, the shape is not a parallelogram, square or rectangle. (There is a second possibility that the point is not in the exact centre of the shape. That was not being asked here, but is something to be aware of.)

3.

These statements (a=d=e=h, b=c=f=g) imply that each triangle created by the two intersecting lines in the middle is identical. They would be true if all four sides were equal, but they are not. This can be demonstrated practically. If we stand the shape upright so the longest line is vertical, we can see that while each triangle has an identical twin, the four of them are not identical. Look at the four angles in the middle: two are greater than 90°, two are less than 90°.


Figure 13: STANDING UP

The difference is very slight when viewed normally, especially in this shape where the

lengths are almost equal. It would be wise to point out a couple of things:

Firstly, the statements a=d=e=h and b=c=f=g are only true if all four sides of a parallelogram are of equal length. You can test this on paper, too: If you fold the shape in half twice, along the lines in the middle, you would emerge with one triangle.

Secondly, in a parallelogram with two sides longer than the others, lines that come from the corners to meet in the middle do not bisect the angles exactly. The greater the difference in the length of the sides, the greater the difference in the angles.

Thirdly, in this shape a=e, b=f, c=g, d=h.

4.

How can this shape be a parallelogram? The opposite sides are not congruent; this violates the second principle. The opposite angles are not congruent, this violates the third principle. It might look like one, but it’s not.

5.

For the statements made here (a≠b, c≠d) to be true, the four sides must not be of equal length, in other words the shape must be longer or shorter in one dimension. This shape is wider than it is tall, so it meets the criteria. The statement is true, and it is definitely a parallelogram because its opposite sides are equal. If four sides are all of equal length, the angles in the middle will be 90°. This statement is true of square as well, but obviously not rectangles.