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Slides: 34
Duration: 00:10:40
Filename: C:\Users\jpage\Documents\NCVPS Learning Objects\Geometry Proving Parallelograms Navigation to PPT W\Proving
Parallelograms.pptx
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Slide 1
Proving Parallelograms
Duration: 00:00:26
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Slide 2
In this Lesson...
Duration: 00:00:10
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Slide 3
Method 1
Duration: 00:00:18
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Notes:
Proving parallelograms
It’s important that we’re able to verify, or prove,
things in life. Is the bridge being built safe? Does
the car I’m buying really get good gas mileage?
Is something what they say it is?
The same is true in geometry. Just because a
quadrilateral looks like a parallelogram doesn’t
mean that it is one. How can we prove that it is
or it isn’t?
Notes:
In this lesson we will:
 Learn the six ways to prove a
quadrilateral is a parallelogram.
 Apply these methods to problems and
proofs.
Notes:
Method 1:
If both pairs of opposite sides in a quadrilateral
are parallel, then the quadrilateral is a
parallelogram.
Notice this is the definition of a parallelogram.
When we use this method we may write it out or
state, “definition of parallelogram.”
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Slide 4
Method 2
Duration: 00:00:09
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Slide 5
Method 3
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Slide 6
Method 4
Duration: 00:00:11
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Notes:
Method 2:
If both pairs of opposite sides in a quadrilateral
are congruent, then the quadrilateral is a
parallelogram.
Notes:
Method 3:
If both pairs of opposite angles in a quadrilateral
are congruent, then the quadrilateral is a
parallelogram.
Notes:
Method 4:
If one angle in a quadrilateral is supplementary
to both of its consecutive angles, then the
quadrilateral is a parallelogram.
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Slide 7
Method 5
Duration: 00:00:09
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Slide 8
Method 6
Duration: 00:00:11
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Slide 9
Examples
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Notes:
Method 5:
If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
Notes:
Method 6:
If one pair of opposite sides are both parallel and
congruent, then the quadrilateral is a
parallelogram.
Notes:
Examples:
Determine whether you have enough information
to determine the quadrilateral is a parallelogram.
Explain your reasoning.
Please pause the presentation to copy and
complete examples 1 – 3. Resume the
presentation when you’re done to check your
work.
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Slide 10
Examples
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Slide 11
Examples
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Slide 12
Examples
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Notes:
Example 1:
This is a parallelogram using the reasoning, “If
both pairs of opposite angles in a quadrilateral
are congruent, then it is a parallelogram.”
Notes:
Example 2:
Yes, this is a parallelogram. If one angle in a
quadrilateral is supplementary to both of its
consecutive angles, then it is a parallelogram.
Notes:
Example 3:
Using the definition of a parallelogram which
states, “if both pairs of opposite sides of a
quadrilateral are parallel, then it is a
parallelogram.”
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Slide 13
Examples
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Slide 14
Examples
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Slide 15
Examples
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Notes:
More Examples:
Determine whether you have enough information
to determine the quadrilateral is a parallelogram.
Explain your reasoning.
Please pause the presentation to copy and
complete examples 4 – 6. Resume the
presentation when you’re done to check your
work.
Notes:
Example 4:
Yes, this is a parallelogram. If both pairs of
opposite sides of a quadrilateral are congruent,
then it is a parallelogram.
Notes:
Example 5:
Yes, this is a parallelogram. One of our methods
states, “if the diagonals of a quadrilateral bisect
each other, then it is a parallelogram.”
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Slide 16
Examples
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Slide 17
Example 7
Duration: 00:00:23
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Slide 18
Example 7
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Notes:
Example 6:
While the vertical angles are marked to be
congruent, and the picture looks like a
parallelogram, there is not enough information
provided to prove this quadrilateral to be a
parallelogram.
Notes:
Writing a proof.
Example 7:
Given: ∆PQT ≅ ∆RST
Prove: PQRS is a parallelogram
Please pause the presentation as you copy
example 7. Resume the presentation when you
are ready to begin the proof.
Notes:
We’re given ∆PQT ≅ ∆RST. How can we use
parts of our congruent triangles to prove that the
quadrilateral is a parallelogram? What if we use
the sides of the triangles that make up the
diagonals? If we can state that the diagonals
bisect each other, then we can use, “if the
diagonals of a quadrilateral bisect each other,
then it is a parallelogram.”
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Slide 19
Example 7
Duration: 00:00:17
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Slide 20
Example 7
Duration: 00:00:17
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Slide 21
Example 7
Duration: 00:00:24
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Notes:
We start by stating ̅̅̅̅
QT ≅ ̅̅̅
ST, ̅̅̅̅
PT ≅ ̅̅̅̅
TR using
CPCTC or corresponding parts of congruent
triangles are congruent.
Notes:
The definition of “bisect” addresses length, not
congruence. Therefore we must state QT = ST,
PT = TR by the definition of congruence.
Notes:
Once we have stated QT = ST, PT = TR, we are
able to state ̅̅̅̅
QS and ̅̅̅̅
PR bisect each other by the
definition of bisect.
Since we have proved that the diagonals bisect
each other, we can now state that PQRS is a
parallelogram.
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Slide 22
Example 7
Duration: 00:00:21
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Slide 23
Example 8
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Slide 24
Example 8
Duration: 00:00:39
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Notes:
Our final step reads, “PQRS is a parallelogram”
because “if the diagonals of a quadrilateral bisect
each other, then it is a parallelogram.”
This isn’t the only method we could have used to
prove PQRS to be a parallelogram. Let’s look at
another way we could have constructed this
proof.
Notes:
Example 8:
Given: ∆PQT ≅ ∆RST
Prove: PQRS is a parallelogram
Please pause the presentation as you copy
example 8. Resume the presentation when you
are ready to complete the proof.
Notes:
Again, we begin with our given information.
Step 1: ∆PQT ≅ ∆RST by Given.
To use a different method of proving a
quadrilateral is a parallelogram, we will need to
use different corresponding parts of our triangles.
̅̅̅, is there a way we could also
If we use ̅̅̅̅
PQ ≅ ̅RS
prove these sides to be parallel? If so, we can
use the method, “if one pair of opposite sides of
a quadrilateral are both congruent and parallel,
then it is a parallelogram.”
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Slide 25
Example 8
Duration: 00:00:12
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Slide 26
Example 8
Duration: 00:00:19
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Slide 27
Example 8
Duration: 00:00:11
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Notes:
̅̅̅ by CPCTC.
Step 2: ̅̅̅̅
PQ ≅ ̅RS
To prove our sides parallel, we need an angle
pair.
Notes:
In step 3, I have chosen the angle pair <QPT ≅
<RST which I know to be congruent by CPCTC.
These two angles are alternate interior angles for
̅̅̅̅
̅̅̅.
PQ and ̅RS
Notes:
̅̅̅ by “if alternate interior angles
Therefore, ̅̅̅̅
PQ ll ̅RS
are congruent, then the lines are parallel.”
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Slide 28
Example 8
Duration: 00:00:19
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Slide 29
Example 9
Duration: 00:00:30
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Slide 30
Example 9
Duration: 00:00:36
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Notes:
̅̅̅ are both parallel
As we have shown ̅̅̅̅
PQ and ̅RS
and congruent, we can now state that PQRS is a
parallelogram because, “if one pair of opposite
sides of a quadrilateral are both congruent and
parallel, then it is a parallelogram.”
Notes:
Our last example deals with a figure created in
the coordinate plane.
Example 9:
Show that A(-1, 2), B(3, 2), C(1,
-2), and D(-3, -2) are vertices of a parallelogram.
Please pause the presentation to copy the
example. Resume the presentation when you are
ready to complete the proof.
Notes:
We begin by locating our vertices and connecting
them to form a quadrilateral.
To prove this figure is a parallelogram, we will
need the distance formula and/or the slope
depending on which method we would like to
use. The slope will tell us if sides are parallel
while the distance formula allows us to see that
they have equal lengths and are therefore
congruent.
In order to demonstrate using both the distance
formula and slope, today we will be using, “if one
pair of opposite sides of a quadrilateral are both
congruent and parallel, it is a parallelogram.”
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Slide 31
Example 9
Duration: 00:00:38
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Notes:
We start by choosing which sides we would like
to use. I have chosen ̅̅̅̅
AD and ̅̅̅̅
BC.
First we will use the distance formula d =
√(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 . Since both
̅̅̅̅ and BC
̅̅̅̅ have lengths about equal to 4.47, we
AD
̅̅̅̅ ≅ BC
̅̅̅̅.
can state that AD = BC and therefore AD
Slide 32
Example 9
Duration: 00:00:19
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Notes:
After determining the slope using
𝑦2 −𝑦1
𝑥2 −𝑥1
that both ̅̅̅̅
AD and ̅̅̅̅
BC have the slope of 2.
̅̅̅̅ is parallel to BC
̅̅̅̅.
Therefore, AD
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we see
Slide 33
Example 9
Duration: 00:00:13
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Slide 34
Summary
Duration: 00:00:30
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Notes:
We have shown that quadrilateral ABCD has one
pair of opposite sides that are both congruent
and parallel, therefore ABCD is a parallelogram.
Notes:
In this lesson you learned and applied:
 The six ways to prove a quadrilateral is a
parallelogram.
 Prove both pairs of opposite
sides are parallel.
 Prove both pairs of opposite
sides are congruent.
 Prove both pairs of opposite
angles are congruent.
 Prove the diagonals bisect each
other.
 Prove one angle is
supplementary to both of its
consecutive angles.
 Prove one pair of opposite sides
are both congruent and parallel.
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