1. What kind of
Quadrilateral is it
really?

2. Review Types of Quadrilaterals
• Can you name these figures?
Trapezoid
Quadrilateral Parallelogram
Rectangle
Square
Rhombus

3. Common Properties
• What do all these
shapes have in
common?
• They all have 4 sides.
• They all have 4
vertices, or points.
(It’s ok if you didn’t
get this one!)
Trapezoid
Parallelogram
Rectangle
Square
Rhombus
Quadrilateral

4. Different but Similar
• What characteristics make each shape
unique?
Trapezoid
Parallelogram
Rectangle Square
Rhombus
Quadrilateral

5. Different but Similar (con’t)
• The Quadrilateral has
✓ 4 sides.
• All 4 sided polygons are called
Quadrilaterals but sometimes
they have a more descriptive
name.
• The Trapezoid has
✓ 4 sides.
✓ 1 pair of parallel sides.
Trapezoid
Quadrilateral

6. Different but Similar (con’t)
• The Parallelogram has
✓ 4 sides.
✓ opposite sides are parallel.
✓ opposite sides are congruent.
• The Rectangle has
✓ 4 sides.
✓ opposite sides are parallel.
✓ opposite sides are congruent.
✓ 4 right angles.
Parallelogram
Rectangle

7. Different but Similar (con’t)
• The Rhombus has
✓ 4 sides.
✓ opposite sides are parallel.
✓ all sides are congruent.
• The Square has
✓ 4 sides.
✓ opposite sides are parallel.
✓ all sides are congruent.
✓ 4 right angles.
Square
Rhombus

8. Relationship Between Quadrilaterals
1 pair parallel sides4 sides
2 pair parallel sides
Opposite sides congruent
4 right angles
Square
All sides congruent

9. What figure is this?
• How do you know ABCD is a Quadrilateral?
• There are 4 sides but
does the figure fit in
a more specific
category?
• How can we check?
A
B
C
D

10. Think Back
• When determining if a Quadrilateral is a
parallelogram, rectangle, square, or rhombus, each
polygon has at least 1 pair of opposite sides with
the same measure.
• What have you learned that can be used to
determine the length of a segment on a coordinate
plane?
• The distance formula!

11. Use the Distance Formula
• Let’s check if the polygon has opposite sides with the same
measure.
• Pick any side such as side AB.
• Identify the coordinates of
the endpoints.
• A(-6, 0) and B(-3, 5)
• Use the distance formula
to find the length of
this side.
A
B
C
D

12. • Label the points with 1s & 2s.
• Substitute into the distance formula
and simplify to find the length of AB.
Use the Distance Formula (con’t)
d = y2 − y1( )2
+ x2 − x1( )2
A −6,0( ) B −3,5( )
x1,y1( ) x2,y2( )
d = 5 − 0( )2
+ −3− −6( )( )
2
= 5( )2
+ −3+ 6( )2
= 25 + 3( )2
= 25 + 9 = 34
A
B
C
D

13. • Now find the length of the opposite
side, CD, following the same steps.
Use the Distance Formula (con’t)
d = y2 − y1( )2
+ x2 − x1( )2
C 6,1( ) D 3,−5( )
x1,y1( ) x2,y2( )
d = −5 −1( )2
+ 3− 6( )2
d = −6( )2
+ −3( )2
d = 36 + 9
A
B
C
D= 45

14. What do we have?
• The length of AB is
• The length of CD is
• Are they the same?
• NO!
• Because they are different,
this polygon cannot be a
parallelogram, rectangle,
rhombus or square.
A
B
C
D
34.
45.
34
45

15. Could it be a Trapezoid?
• What makes a Quadrilateral a Trapezoid?
• One (and only one) pair of parallel sides.
• How can we determine
if sides are parallel?
• Slope! If the slope is
the same, the sides are
parallel.
A
B
C
D

16. Slope Formula
A
B
C
D
m =
y2 − y1
x2 − x1
.
• Find the slope of opposites side to determine if
Quad ABCD is a Trapezoid.
• Recall the formula to find
the slope between two
points is

17. Finding the slope
• Find the slope of each side and then compare.
A −6,0( ) B −3,5( ) C 6,1( ) D 3,−5( )
mAB =
y2 − y1
x2 − x1
x1,y1( ) x2,y2( )
mAB =
5 − 0
−3− −6( )
mAB =
5
−3+ 6
mAB =
5
3
mBC =
y2 − y1
x2 − x1
x1,y1( ) x2,y2( )
mBC =
1− 5
6 − −3( )
mBC =
−4
6 + 3
mBC =
−4
9
mCD =
y2 − y1
x2 − x1
mCD =
−5 −1
3− 6
mCD =
−6
−3
mCD = 2
mAD =
y2 − y1
x2 − x1
mAD =
−5 − 0
3− −6( )
mAD =
−5
3+ 6
mAD =
−5
9
x1,y1( ) x2,y2( )x1,y1( ) x2,y2( )

18. • Compare the slopes of AB and CD. Are they
parallel?
• No! They are not the same
so they are not parallel.
• Compare the slopes of
BC and AD. Are they
parallel?
• They are different so they
are not parallel.
Compare the Slopes
mAB =
5
3
mBC =
−4
9
mCD = 2
A
B
C
D
mAD =
−5
9

19. • Because the polygon does not have at least 1 pair
of opposite sides with the same measure and no
opposites sides are parallel, this
polygon is just a Quadrilateral.
• Phew, that was a lot but
the steps are necessary
to prove the polygon is
only a Quadrilateral.
So what is the polygon?
A
B
C
D

20. Parallelogram vs Rectangle

21. Perpendicular Lines
• Perpendicular Lines meet at a right angle.
• Recall that Perpendicular Lines have slopes that
are negative reciprocals.
• Such as -2 and 1/2 are negative reciprocals
because their product is -1.
• If consecutive sides have slopes that are negative
reciprocals then the sides form a right angle.

22. Proof Made Easy
• When justifying or proving the type of Quadrilateral
on a coordinate plane, you need to show the polygon
cannot be another.
• Each problem is a little different but there is a way to
help minimize the steps to prove the type of polygon.
• Follow the flow chart on the next page when proving
the type of figure to help minimize the steps.

23. Check if opposite
sides are parallel.
Check if other pair of
sides are parallel.
Check if consecutive
sides are congruent.
Check if consecutive
sides are congruent.
Figure is a
Rectangle.
Figure is a
Square.
Figure is a Quadrilateral.
Figure is a
Trapezoid.
Figure is a Parallelogram.
Figure is a
Rhombus.
no
no
yes
yes
yes
noCheck if consecutive
sides are perpendicular.
yes
no
no
yes

24. What type of Quadrilateral do the points form?
• A(-4, -2), B(-2, 4), C(4, 2), D(2, -4)
• We see it looks like a square but we need to
justify or prove it is.
• First, check if a pair of
opposites sides is parallel.
• Let’s check AB and CD. A
B
C
D

25. • A(-4, -2), B(-2, 4), C(4, 2), D(2, -4)
• Find the slope of AB.
• Find the slope of CD.
• Slopes are the same.
Next step is to check if
BC and AD are also parallel.
A
B
C
D
mAB =
y2 − y1
x2 − x1
x1,y1( ) x2,y2( )
=
4 − −2( )
−2 − −4( )
=
4 + 2
−2 + 4
=
6
2
= 3
mCD =
y2 − y1
x2 − x1
=
−4 − 2
2 − 4
=
−6
−2
= 3
x1,y1( ) x2,y2( )

26. • A(-4, -2), B(-2, 4), C(4, 2), D(2, -4)
• Find the slope of BC.
• Find the slope of AD.
• Slopes are the same.
BC and AD are parallel.
A
B
C
D
mBC =
y2 − y1
x2 − x1
x1,y1( ) x2,y2( )
=
2 − 4
4 − −2( )
=
−2
4 + 2
=
−2
6
=
−1
3
mAD =
y2 − y1
x2 − x1
=
−4 − −2( )
2 − −4( )
=
−4 + 2
2 + 4
=
−2
6
x1,y1( ) x2,y2( )
=
−1
3

27. • Now check if consecutive sides are perpendicular.
• How do we know if 2 slopes are perpendicular?
• They are negative reciprocals, which means their product is -1.
• Here are the slopes of the 4 sides again:
• Any consecutive sides can be checked
but let’s check AB and BC.
• Their product is -1 so AB and BC are
perpendicular.
A
B
C
D
mAB = 3 mBC =
−1
3
mCD = 3 mAD =
−1
3
mAB ×mBC = 3×
−1
3
= −1

28. • Almost there! Check if consecutive sides are congruent.
• Why don’t we check if opposite sides are congruent?
• We’ve already verified opposites sides are parallel so we
know the figure is a type of parallelogram.
• We also verified the consecutive sides are perpendicular,
which means the figure must be either a rectangle or
square.
• By determining if consecutive sides are congruent or not,
we can then determine if the polygon is rectangle or a
square.

29. • A(-4, -2), B(-2, 4), C(4, 2), D(2, -4)
• Check if AB and BC are congruent using the
distance formula.
A
B
C
D
dAB = y2 − y1( )2
+ x2 − x1( )2
x1,y1( ) x2,y2( )
= 4 − −2( )( )
2
+ −2 − −4( )( )
2
= 4 + 2( )2
+ −2 + 4( )2
= 6( )2
+ 2( )2
= 36 + 4 = 40

30. • A(-4, -2), B(-2, 4), C(4, 2), D(2, -4)
• Now find the length of BC.
• AB and BC are congruent.
• Therefore, Quad ABCD is
a square. A
B
C
D
dBC = y2 − y1( )2
+ x2 − x1( )2
= 2 − 4( )2
+ 4 − −2( )( )
2
= −2( )2
+ 4 + 2( )2
= 4 + 6( )2
x1,y1( ) x2,y2( )
= 4 + 36 = 40
40
40