Properties of Parallelogram Video Lecture | Advance Learner Course: Mathematics (Maths) Class 7

Solving problems involving
parallelograms. First, the definition of a
parallelogram. It's a quadrilateral with
opposite sides parallel. The notation
that we use to show that sides are
parallel are these little arrows.
Using just this definition, which of
these polygons can be classified as parallelograms?
Turns out four of them are also parallelograms because they fit the definition
that opposite sides are parallel. So we'll look at some properties of parallelograms
The first property is opposite angles are congruent
We can measure each angle. This
obtuse angle measures 125 degrees and
the angle opposite it, also measures 125
degrees. Now let's measure the acute angles.
This one here is 55 degrees and
the angle opposite it is also 55 degrees.
So we verified that property. Now we can
use the property opposite angles are congruent
to solve some problems. Given
parallelogram WXYZ, solve for x and y.
Let's pay attention to angles W and Y.
They are opposite angles so they are congruent.
We can set them equal. x equals 28.
Let's look at angles x and z ,the obtuse angles. they're opposite each other
We know that y plus 42 equals 96 so y equals 54. We'll look at another property of parallelograms
that also has to do with the angles.
Notice that the two angles along the
base, 55 degrees plus 125 degrees, they
add together to equal 180. Now that makes sense based on what I know about
parallel lines. And 125 degrees plus 55
degrees equals 180. Wo what we find out
in parallelograms, are consecutive angles
are supplementary. That comes in handy
when asked to solve a problem like this:
Given YFED is a parallelogram, find x and y.
Let's take a look at angle Y, which by the angle addition postulate, we know
has a measure of 115 degrees. The measures of angles Y and D are supplementary.
So we can say 5y + 115 = 180.
Same thing with angles F and Y. when we
add their measures together we get 180
because they are supplementary. We can
solve these simple equations for y.
The equation on the bottom we can
rewrite as 7x + 110 = 180.
Let's subtract 110 from both
sides. Let's divide through by 7 and
we get that x = 10. Another
property of parallelograms has to do
with the opposite sides. Turns out opposite sides are congruent.
They are not only parallel but
they're congruent. This particular
side measures 16 units and the side
opposite it is also 16 units.
The slanted sides are9.5 units and
9.5 units. So let's solve a problem
where we apply the property that
opposite sides are congruent.
We know that the two slanted sides here ,2x +3 and 9, they are congruent.
We can set them equal and we can
solve this simple equation for x.
Similarly 8y - 7 is congruent to 7 so we can set that equal.
We get that y = 1. The last property of parallelograms that we're going to talk about
has to do with the diagonals. If I were to measure this longer diagonal,
it looks like it's 22 units and the shorter diagonal is 13 units.
If I were to draw the two diagonals, and measure the little pieces that now make up
each one I can see that they bisect each other. So the 22 unit diagonal
is broken up into two 11 unit pieces and the 13 unit diagonal is broken into 6.5 unit pieces.
So let's use that property. Here I have a
parallelogram RSNU. We are asked to solve for x and y which lie along the diagonals
I know that the diagonals bisect each other so therefore y +7 should equal 20.
I can solve for y pretty simply. x + y = 16
but I have to substitute. y = 13 and solve for x.
Let's summarize what we talked about in this video. We started with the definition
of a parallelogram, that it's a
quadrilateral with opposite sides parallel.
Then we talked about the
properties that the opposite angles are
congruent. Consecutive angles are
supplementary. The opposite sides are
congruent and the diagonals bisect each
other. All of these properties hold
true for any figure that can be
classified as a parallelogram.
Therefore these figures here which we
typically might think of as rectangle,
rhombus, square and parallelogram. All of
the properties of a parallelogram hold
true for these figures.