Quadrilaterals
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Quadrilateral Overview
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Quadrilateral Properties
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Quadrilateral types
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Proof - Opposite Sides of Parallelogram Congruent
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Proof - Diagonals of a Parallelogram Bisect Each Other
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Proof - Opposite Angles of Parallelogram Congruent
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Quadrilateral angles
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Proof - Rhombus Diagonals are Perpendicular Bisectors
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Proof - Rhombus Area Half Product of Diagonal Length
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Area of a Parallelogram
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Whether a Special Quadrilateral Can Exist
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Quadrilaterals challenge
Proof - Diagonals of a Parallelogram Bisect Each Other Proving that a quadrilateral is a parallelogram if and only if its diagonals bisect each other
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- So, we have a parallelogram right over here
- So, what we wanna prove is that it's diagonals bisect each other
- So, the first thing we can think about; these aren't just diagonals,
- these are lines that are intersecting parallel lines
- So, you can also view them as transversals
- And if we focus on DB right over here, we see that it intersects DC
- and AB and it sits there
- Those we know are parallelograms
- We know that they are parallel
- This is a parallelogram
- Alternate interior angles must be congruent
- So, that angle must be equal to that angle there
- Let me make a label here
- Let me call that middle point E
- So, we know that angle ABE must be congruent to angle CDE
- by alternate interior angles of
- a transversal intersecting parallel line
- Alternate interior angles
- If we look at diagonal AC or we should call it transversal AC
- we can make the same argument
- It intersects here and here
- These two lines are parallel
- So, alternate interior angles must congruent
- So, angle DEC must be --- let me write this down ---
- angle DEC must be congruent to angle BAE
- by for the exact same reason
- Now we have something interesting
- If we look this top triangle over here and this bottom triangle,
- we have one set of corresponding angles that are congruent
- We have a side in between that's going to be congruent
- Actually, let me write that down explicitly
- We know and we've proved this to ourselves in the previous video
- that parallelograms not only are opposite sides are parallel they're
- also congruent
- So, we know from the previous video that that side is equal
- to that side
- So, let me go back to what I was saying
- We have two sets of corresponding angles that are congruent
- We have a side in between that's congruent
- And then we have another set of corresponding angles
- that are congruent
- So, we know that this triangle is congruent to that triangle
- by angle-side-angle
- So, we know that triangle --- I'm gonna go from the blue
- to the orange to the last one
- Triangle ABE is congruent to triangle blue, orange
- and the last one, CDE by angle-side-angle congruency
- Now what is that do for us
- What we know if two triangles are congruent, all of their
- corresponding features especially all of the corresponding
- sides are congruent
- So, we know that side EC corresponds to EA
- Or I could say side AE, we could say side AE,
- corresponds to side CE
- They're corresponding sides of congruent triangle
- So, their measures or their lengths must be the same
- So, AE must be equal to CE
- Let me put two slashes since I already used one slash over here
- let me focus on this -- we know that BE must be equal to DE
- Once again they're corresponding sides of two congruent triangles
- so they must have the same length
- So, this is corresponding sides of congruent triangles
- So, BE is equal to DE
- And we've done our proof
- We've showed that, look, diagonal DB is splitting AC into two
- segments of equal length and vice versa
- AC is splitting DB into two segments of equal lengths
- So, they are bisecting each other
- Now, let's go the other way around
- Let's prove to ourselves that if we have two diagonals
- of a quadrilateral that are bisecting each other that we're
- dealing with a parallelogram
- So, let me see
- So, we're gonna assume that the two diagonals
- are bisecting each other
- So, we're assuming that that is equal to that
- And that that, right over there, is equal to that
- Given that we wanna prove that this is a parallelogram
- And to do that we just have to remind ourselves
- We just have to remind ourselves that this angle is going
- to be equal to that angle
- One of the first things we learn because they're vertical angles
- So, let me write this down
- C -- label this point -- angle CED is going to be equal to
- or is congruent to angle, so I started is BEA, angle BEA
- And that, what is that, well that shows us that these
- two triangles are congruent 'cause we have a corresponding sides
- of a congruent and angle in between and on the other side
- So, we now know that the triangle, I'll keep this in yellow,
- triangle AEB is congruent to triangle DEC by side-angle-side
- congruency, by SAS congruent triangles
- Fair enough
- Now, if we know that two triangles congruent we know that all
- corresponding sides and angles are congruent
- So, for example, we know that angle CDE is going to be congruent
- to angle BAE
- And this is just corresponding angles of congruent triangles
- And now we have this kind of transversal of these two lines that
- could be parallel if the alternate interior angles are congruent
- And we see that they are
- These two are kind of candidate alternate interior angles and
- they are congruent
- So, AB must be parallel to CD
- So, AB, let's just draw one arrow, AB must be parallel to CD
- by alternate interior angles congruent of parallel lines
- I'm just writing in some short hand, forgive the cryptic nature
- of it although I'm saying it out
- And so we can then do the exact same -- while we just shown
- that these two sides are parallel -- we can do that exact same
- logic to show that these two sides are parallel
- I won't necessarily write it all out
- It's exact same proof to show that these two
- So, first of all, we know that this angle is congruent to that angle
- right over there
- And then we know, actually let me write it out, we know
- that angle AEC is congruent to angle DEB, I should say
- They are vertical angles
- And that is the reason up here as well
- Vertical angles
- And then we see that triangle AEC must be congruent
- to triangle DEB by side-angle-side
- So, then we have triangle AEC must be congruent to triangle
- DEB by SAScongruency
- Now, we know that corresponding angles must be congruent
- So, that we know that angle, so, for example angle CAE
- must be congruent to angle BDE and this is the corresponding
- angles of congruent triangles
- So, CAE, let me use a new color
- CAE must be congruent to BDE
- And now we have a transversal
- The alternate interior angles are congruent
- So, the two lines that the transversals are intersecting
- must be parallel
- So, this must be parallel to that
- So, then we have AC must be parallel to BD
- by alternate interior angles
- And we're done
- We've just proven that if the diagonals bisect each other,
- if we start that as a given then we end at a point where we say,
- "Hey, the opposite sides of this quadrilateral must be parallel
- or that ABCD is a parallelogram "
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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