This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I’m gong to talk about the properties trapezoids and show how to prove
those properties. So let’s start with the definition of a
trapezoid. A trapezoid is a quadrilateral — so it’s a four-sided figure — and it’s
got one pair of parallel sides. Now I’ve drawn a quadrilateral,
a four-sided figure that I’ve labeled ABCD and I’m going to mark side BC as being parallel to side AD So this is a
trapezoid. I’ve got a property here that says consecutive angles between parallel
sides are supplementary. Well angle A and angle B are consecutive
angles. If I go around the trapezoid, angle A
and angle B come after one another and they are between the two parallel sides, BC and AD. I want to show that they’re supplementary.
So what I’m going to do is extend side AB so that I form an exterior angle at B. Now there’s that exterior angle and
right next to it is the interior angle. These two angles
together share a straight line, a 180 degree
angle, which means that they’re supplementary
angles. Now side AB is a transversal that intersects parallel lines BC and AD. So that means the angle at A has to be congruent with the exterior angle at B because
they’re corresponding angles along a transversal. The exterior angle at B is supplementary to the interior angle
at B, so that means angle A must also be supplementary to it. So therefore, these two interior angles,
consecutive angles between parallel sides
are supplementary. Now in the same way I could go
over to angles C and D and show that they’re supplementary.
So that’s our property for all trapezoids. Consecutive
angles between the parallel sides are supplementary to each other.
Now we have a special kind trapezoid, a specific kind which is called
an isosceles trapezoid. An isosceles trapezoid is a trapezoid with a pair of congruent base angles.
Let’s understand what that means. I’ve got a trapezoid that I’ve drawn and I’ll label the parallel sides and AD. Now when talk about trapezoids, the parallel sides are very often called
the bases, and this definition says that there’s a
pair of congruent base angles. So angle A and angle D would be
congruent base angles,
they would have the same measure. And now we have this property.
The property says each pair of base angles is congruent. So
in other words, A and D are congruent, but the other pair,
B and C, would also have to be congruent.
Here’s how we’ll prove that. Angle A and angle B are consecutive angles between the
parallel sides, between the bases, So they must be supplementary to each
other. Now angle B would also have to be
supplementary to angle D, since A and D are congruent.
Then we’ve got angle C, and angle C has to be supplementary
to D, because they’re consecutive angles between the parallel sides. So if angle D and angle B are supplementary, and angle D and angle C are supplementary,
angle B and and C must equal each other. So that’s our property: each pair of base angles are congruent. Now we’ve got another property. It says a pair of opposite sides are congruent.
So we have to show that side AB is congruent with side CD. I’ll get rid of some of the notation here
so we can see this clearly. What I’m going to do is I’m going to draw a perpendicular
line from B down to the base, AD. When I draw that perpendicular line, it’s going to form a right angle with the
base. I’m gong to draw another perpendicular line from C down to the base, and that will form a
right angle. So I’ve made two triangles. I can show
those triangles are congruent. Angle A and angle D are congruent
angles. The two right angles are congruent. And now I just have to show that those two
perpendiculars are congruent so I can have angle-angle-side congruency. those perpendiculars that I drew
are the distance, each one measures the distance, between
the two parallel sides, between the bases. Now when you have parallel lines, the distance
is constant, the distance is always the same.
So these perpendicular lines are going to be congruent with each,
they’ll be the same length. Therefore I have two congruent triangles, and that means that side AB is going to be congruent with the
corresponding side, CD. So I’ve got a pair of opposite sides, AB and CD, that are congruent. And my last property says the diagonals
are congruent. So once again I’ll clean this up and show that the diagonals are gonna be
congruent. So I’ll draw the diagonals. I have a line connecting B and D and another diagonal from A to C. Let’s show that those two diagonals are congruent with each other. So I’ve formed two triangles.
One triangle would be BAD and the other triangle would be CDA. If I can show that those two triangles
are congruent, then it would mean that those diagonals, which are
corresponding sides of the two triangles, will also be congruent. So let’s see. I know that BA you is congruent with CD. So I’ve got two sides that are
congruent. I know that angle A is congruent with
the angle D, so I’ve got two angles that are congruent.
And both of the triangles share the same base AD. So I’ve got side-angle-side congruency for the two triangles. That means that their corresponding
sides, AB and DC… I’m sorry we’re doing the diagonals… That means their corresponding sides AC and DB are go to be congruent. Those are the
diagonals and therefore the diagonals are congruent. So those are your three properties for
isosceles triangles: both pairs of base angles are congruent — so A and D were congruent
and B and C were congruent; there’s a pair of congruent opposite sides; and the diagonals are congruent. Okay,take care. I’ll see you next time.