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## Class 8

### Course: Class 8>Unit 6

Lesson 2: Properties of a parallelogram

# Proof: Opposite angles of a parallelogram

Sal proves that opposite angles of a parallelogram are congruent. Created by Sal Khan.

## Want to join the conversation?

• When comparing angles of equal measure in this video Sal uses a congruent symbol. Would it be equally acceptable to use a plain equals sign instead?
If not, when would you use a congruent symbol and when would you use an equals sign to compare angles?
(15 votes)
• No, it would not. If you were talking specifically about the measures of the angles, then you could, but when referring to the angle itself, you should use the congruence symbol.
(2 votes)
• Is point D a right angle at ?
(3 votes)
• Technically, point D is not an angle :)

Since Sal is talking about parallelograms, the angle BDC (the angle defined by the line segments BD and CD) isn't necessarily a right angle, but it's not impossible for it to be a right angle.

You can't assume that BDC is a right angle, because if you did, you'd only prove that the opposite angles of a parallelogram are equal if one of the angles is a right angle (from which would follow that the other three angles are right angles too).

When proving things, you want to make as little assumptions as possible so that your proof applies to as many scenarios as possible.
(8 votes)
• Math really is another language. And I am not fluent.

How does one name angles? It seems like it's random mostly, but I'm not sure.

Just wondering. :D
(6 votes)
• In all reality, you can name an angle anything, as long as it is a variable. But sometimes people name them as numbers, but the y have to be in order and they all have to in numbers. Hope this helped.
(2 votes)
• Aren't opposite angles of a parallelogram corresponding angles or alternate interior angles?
(5 votes)
• No they are not corresponding or alternate interior angles. They are just called opposite angles and they are equal to each other.
(2 votes)
• What about a trapezoid?
(3 votes)
• If 𝐴𝐵𝐶𝐷 is a trapezoid, where side 𝐴𝐵 is parallel to side 𝐶𝐷,
then angles 𝐴 and 𝐷 are supplementary, as are angles 𝐵 and 𝐶.
(3 votes)
• why is the ~ place top on the =?
(0 votes)
• That means that the two figures are congruent. Just ~ means similar. Hope I helped!
(8 votes)
• What is the difference between Congruence and Equivalence?
(2 votes)
• `The fact is that they are not the same. Congruence is a relationship of shapes and sizes, such as segments, triangles, and geometrical figures, while equality is a relationship of sizes, such as lengths, widths, and heights. Congruence deals with objects while equality deals with numbers.`
(3 votes)
• define congruent, does it mean equal
(1 vote)
• In geometry two shapes are congruent if they have the same shape or size. So yes, congruent means "Equal in size or shape."
(5 votes)
• I am confused with the naming. our teacher says, it should be named in a correct way like ABCD but here it is named ABDC or CDBA but now that I've come to it, is it necessary to name it in order? or we can name however we like?.
(2 votes)
• Wait, but couldn't you also find the vertical angle of <BDC, because that would also correspond to <CAB?
(2 votes)
• You are correct that they are congruent as vertical angles, but what purpose does it serve?
(3 votes)

## Video transcript

What I want to do in this video is prove that the opposite angles of a parallelogram are congruent. So for example, we want to prove that CAB is congruent to BDC, so that that angle is equal to that angle, and that ABD, which is this angle, is congruent to DCA, which is this angle over here. And to do that, we just have to realize that we have some parallel lines, and we have some transversals. And the parallel lines and the transversals actually switch roles. So let's just continue these so it looks a little bit more like transversals intersecting parallel lines. And really, you could just pause it for yourself and try to prove it, because it really just comes out of alternate interior angles and corresponding angles of transversals intersecting parallel lines. So let's say that this angle right over here-- let me do it in a new color since I've already used that yellow. So let's start right here with angle BDC. And I'm just going to mark this up here. Angle BDC, right over here-- it is an alternate interior angle with this angle right over here. And actually, we could extend this point over here. I could call this point E, if I want. So I could say angle CDB is congruent to angle EBD by alternate interior angles. This is a transversal. These two lines are parallel. AB or AE is parallel to CD. Fair enough. Now, if we kind of change our thinking a little bit and instead, we now view BD and AC as the parallel lines and now view AB as the transversal, then we see that angle EBD is going to be congruent to angle BAC, because they are corresponding angles. So angle EBD is going to be congruent to angle BAC, or I could say CAB. They are corresponding angles. And so if this angle is congruent to that angle and that angle is congruent to that angle, then they are congruent to each other. So angle-- let me make sure I get this right-- CDB, or we could say BDC, is congruent to angle CAB. So we've proven this first part right over here. And then to prove that these two are congruent, we use the exact same logic. So first of all, we view this as a transversal. We view AC as a transversal of AB and CD. And let me go here and let me create another point here. Let me call this point F right over here. So we know that angle ACD is going to be congruent to angle FAC because they are alternate interior angles. And then we change our thinking a little bit. And we view AC and BD as the parallel lines and AB as a transversal. And then angle FAC is going to be congruent to angle ABD, because they're corresponding angles. Angle F to angle ABD, and they are corresponding angles. So in the first time, we viewed this as the transversal, AC as a transversal of AB and CD, which are parallel lines. Now AB is the transversal and BD and AC are the parallel lines. And obviously, if this is congruent to that, and that is congruent to that, then these two have to be congruent to each other. So we see that if we have opposite angles are congruent-- or if we have a parallelogram, then the opposite angles are going to be congruent.