Geometry CCSS: G.CO 11.
PROVE theorems about parallelograms. Theorems
INCLUDE: opposite sides ARE congruent, opposite angles
ARE congruent, the diagonals of a parallelogram BISECT
each other, and conversely, rectangles ARE parallelograms
with congruent diagonals.
 What
are the properties of different
quadrilaterals? How do we use the
formulas of areas of different
quadrilaterals to solve real-life
problems?
 Now
a little review:
11 PROVE theorems about parallelograms. Theorems INCLUDE: opposite sides ARE congruent, opposite angles ARE congruent, the d








1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
60 120
120
6
6
50°
130°
Test your prior knowledge and try to fill in the chart
with properties of the following quadrilaterals:
Rectangle
Rhombus
Square
Properties
Properties
Properties
 Use
properties of sides and angles of
rhombuses, rectangles, and squares.
 Use properties of diagonals of
rhombuses, rectangles and squares.
 What
are properties of sides and
angles of rhombuses, rectangles, and
squares?
 In
this lesson, you will study three special
types of parallelograms: rhombuses,
rectangles and squares.
A rectangle is a parallelogram
with four right angles.
A rhombus is a parallelogram
with four congruent sides
A square is a parallelogram with
four congruent sides and four right
angles.

Each shape has the properties of every group that
it belongs to. For instance, a square is a rectangle,
a rhombus and a parallelogram; so it has all of the
properties of those shapes.
parallelograms
rhombuses
rectangles
squares
Rhombuses
Rectangles
Some examples of a rhombus

a.
b.
Decide whether the statement is always, sometimes, or
never true.
A rhombus is a rectangle.
A parallelogram is a rectangle.
parallelograms
rhombuses
rectangles
squares
Rhombuses
Rectangles

a.
Decide whether the statement is always, sometimes, or never true.
A rhombus is a rectangle.
The statement is sometimes true. In the Venn diagram, the regions
for rhombuses and rectangles overlap. IF the rhombus is a square, it
is a rectangle.
parallelograms
rhombuses
rectangles
squares
Rhombuses
Rectangles

b.
Decide whether the statement is always, sometimes, or never true.
A parallelogram is a rectangle.
The statement is sometimes true. Some parallelograms are
rectangles. In the Venn diagram, you can see that some of the
shapes in the parallelogram box are in the area for rectangles, but
many aren’t.
parallelograms
rhombuses
rectangles
squares
Rhombuses
Rectangles

ABCD is a rectangle. What
else do you know about
ABCD?

Because ABCD is a
rectangle, it has four right
angles by definition. The
definition also states that
rectangles are
parallelograms, so ABCD
has all the properties of a
parallelogram:
• Opposite sides are parallel
and congruent.
• Opposite angles are
congruent and consecutive
angles are supplementary.
• Diagonals bisect each
other.
A rectangle is defined as a parallelogram with four
right angles. But any quadrilateral with four right
angles is a rectangle because any quadrilateral
with four right angles is a parallelogram.
 Corollaries about special quadrilaterals:

• Rhombus Corollary: A quadrilateral is a rhombus if and
only if it has four congruent sides.
• Rectangle Corollary: A quadrilateral is a rectangle if
and only if it has four right angles.
• Square Corollary: A quadrilateral is a square if and only
if it is a rhombus and a rectangle.
• You can use these to prove that a quadrilateral is a
rhombus, rectangle or square without proving first that
the quadrilateral is a parallelogram.
Characteristics
Both pairs of opposite sides parallel
Diagonals are congruent
Both pairs of opposite sides congruent
At least one right angle
Both pairs of opposite angles congruent
Exactly one pair of opposite sides parallel
Diagonals are perpendicular
All sides are congruent
Consecutive angles congruent
Diagonals bisect each other
Diagonals bisect opposite angles
Consecutive angles supplementary
Parallelogram
Rectangle
Rhombus
Square
 In
the diagram at the right,
PQRS is a rhombus. What
is the value of y?
P
Q
2y + 3
S
5y - 6
All four sides of a rhombus are ≅, so RS = PS.
5y – 6 = 2y + 3 Equate lengths of ≅ sides.
5y = 2y + 9 Add 6 to each side.
3y = 9
Subtract 2y from each side.
y=3
Divide each side by 3.
R
The following theorems
are about diagonals of
rhombuses and
rectangles.
 Theorem 6.11: A
parallelogram is a
rhombus if and only if
its diagonals are
perpendicular.
 ABCD is a rhombus if
and only if AC BD.

Theorem 6.12: A
parallelogram is a
rhombus if and only if
each diagonal bisects a
pair of opposite
angles.
 ABCD is a rhombus if
and only if AC bisects
DAB and BCD and
BD bisects ADC and
CBA.

Theorem 6.13: A
parallelogram is a
rectangle if and only if
its diagonals are
congruent.
 ABCD is a rectangle if
and only if AC ≅ BD.
A
B

D
C
 You
can rewrite Theorem 6.11 as a
conditional statement and its converse.
 Conditional statement: If the diagonals
of a parallelogram are perpendicular,
then the parallelogram is a rhombus.
 Converse: If a parallelogram is a
rhombus, then its diagonals are
perpendicular.
A
B
X
Statements:
1. ABCD is a rhombus
2. AB ≅ CB
3. AX ≅ CX
4. BX ≅ DX
5. ∆AXB ≅ ∆CXB
6. AXB ≅ CXB
7. AC  BD
Reasons:
1. Given
D
C
A
B
X
D
Statements:
1. ABCD is a rhombus
2. AB ≅ CB
3. AX ≅ CX
4. BX ≅ DX
5. ∆AXB ≅ ∆CXB
6. AXB ≅ CXB
7. AC  BD
Reasons:
1. Given
2. Given
C
A
B
X
D
Statements:
1. ABCD is a rhombus
2. AB ≅ CB
3. AX ≅ CX
4. BX ≅ DX
5. ∆AXB ≅ ∆CXB
6. AXB ≅ CXB
7. AC  BD
C
Reasons:
1. Given
2. Given
3. Def. of . Diagonals
bisect each other.
A
B
X
D
Statements:
1. ABCD is a rhombus
2. AB ≅ CB
3. AX ≅ CX
4. BX ≅ DX
5. ∆AXB ≅ ∆CXB
6. AXB ≅ CXB
7. AC  BD
C
Reasons:
1. Given
2. Given
3. Def. of . Diagonals
bisect each other.
4. Def. of . Diagonals
bisect each other.
A
B
X
D
Statements:
1. ABCD is a rhombus
2. AB ≅ CB
3. AX ≅ CX
4. BX ≅ DX
5. ∆AXB ≅ ∆CXB
6. AXB ≅ CXB
7. AC  BD
C
Reasons:
1. Given
2. Given
3. Def. of . Diagonals
bisect each other.
4. Def. of . Diagonals
bisect each other.
5. SSS congruence post.
A
B
X
Statements:
1. ABCD is a rhombus
2. AB ≅ CB
3. AX ≅ CX
4. BX ≅ DX
5. ∆AXB ≅ ∆CXB
6. AXB ≅ CXB
7. AC  BD
D
C
Reasons:
1. Given
2. Given
3. Def. of . Diagonals
bisect each other.
4. Def. of . Diagonals
bisect each other.
5. SSS congruence post.
6. CPCTC
A
B
X
D
Statements:
1.
ABCD is a rhombus
2.
AB ≅ CB
3.
AX ≅ CX
4.
BX ≅ DX
5.
∆AXB ≅ ∆CXB
6.
AXB ≅ CXB
7.
AC  BD
C
Reasons:
1.
Given
2.
Given
3.
Def. of . Diagonals
bisect each other.
4.
Def. of . Diagonals
bisect each other.
5.
SSS congruence post.
6.
CPCTC
7.
Congruent Adjacent s



Assign coordinates.
Because AC BD, place
ABCD in the coordinate
plane so AC and BD lie on
the axes and their
intersection is at the origin.
Let (0, a) be the coordinates
of A, and let (b, 0) be the
coordinates of B.
Because ABCD is a
parallelogram, the
diagonals bisect each other
and OA = OC. So, the
coordinates of C are (0, - a).
Similarly the coordinates of
D are (- b, 0).
A(0, a)
D(- b, 0)
B(b, 0)
C(0, - a)

Find the lengths of the
sides of ABCD. Use the
distance formula (See –
you’re never going to get
rid of this)

AB=√(b – 0)2 + (0 – a)2 = √b2 + a2
BC= √(0 - b)2 + (– a - 0)2 = √b2 + a2

CD= √(- b – 0)2 + [0 - (– a)]2 = √b2 + a2

DA= √[(0 – (-
b)]2
+ (a –
0)2
=
√b2
+
A(0, a)
D(- b, 0)
B(b, 0)
C(0, - a)
a2
All the side lengths are equal,
so ABCD is a rhombus.
4 feet

a.
b.
CARPENTRY. You are
building a rectangular
frame for a theater set.
First, you nail four pieces
of wood together as
shown at the right. What
is the shape of the frame?
To make sure the frame is
a rectangle, you measure
the diagonals. One is 7
feet 4 inches. The other is
7 feet 2 inches. Is the
frame a rectangle?
Explain.
6 feet
6 feet
4 feet
4 feet
a.
First, you nail four
pieces of wood
together as shown at
the right. What is the
shape of the frame?
6 feet
6 feet
Opposite sides are
congruent, so the
frame is a
parallelogram.
4 feet
4 feet
To make sure the
frame is a rectangle,
you measure the
diagonals. One is 7
feet 4 inches. The
other is 7 feet 2
inches. Is the frame a
rectangle? Explain.
The parallelogram is
NOT a rectangle. If it
were a rectangle, the
diagonals would be
congruent.
b.
6 feet
6 feet
4 feet
You’ve just had a new door installed, but it
doesn’t seem to fit into the door jamb
properly. What could you do to determine if
your new door is rectangular?
1. Take out a piece
of notebook paper
and make a hot
dog fold over from
the right side over
to the pink line.
The fold crease
2. Now, divide the right
hand section into 5
sections by drawing 4
evenly spaced lines.
3. Use scissors to cut
along your drawn line,
but ONLY to the crease!
4. Write
QUADRILATERALS
down the left hand
side
The fold
crease
5. Fold over the top
cut section and write
PARALLELOGRAM
on the outside.
6. Reopen the fold.
The fold
crease
7. On the left hand
section, draw a
parallelogram.
8. On the right hand
side, list all of the
properties of a
parallelogram.
1. Opposite angles are congruent.
2. Consecutive angles are
supplementary.
3. Opposite sides are congruent.
4. Diagonals bisect each other.
5. Opposite sides are parallel
* Fold over the
second cut section
and write
RECTANGLE on the
outside.
* Reopen the fold.
1. Opposite angles are congruent.
2. Consecutive angles are
supplementary.
3. Opposite sides are congruent.
4. Diagonals bisect each other.
5. Opposite sides are parallel
* On the left hand
section, draw a
rectangle.
1. Opposite angles are congruent.
2. Consecutive angles are
supplementary.
3. Opposite sides are congruent.
4. Diagonals bisect each other.
5. Opposite sides are parallel
1. Special parallelogram.
* On the right hand
side, list all of the
properties of a
rectangle.
2. Has 4 right angles
3. Diagonals are congruent.
* Fold over the third
cut section and write
RHOMBUS on the
outside.
1. Opposite angles are congruent.
2. Consecutive angles are
supplementary.
3. Opposite sides are congruent.
4. Diagonals bisect each other.
5. Opposite sides are parallel
1. Special parallelogram.
2. Has 4 right angles
3. Diagonals are congruent.
* Reopen the fold.
* On the left hand
section, draw a
rhombus.
1. Opposite angles are congruent.
2. Consecutive angles are
supplementary.
3. Opposite sides are congruent.
4. Diagonals bisect each other.
5. Opposite sides are parallel
1. Special parallelogram.
* On the right hand
side, list all of the
properties of a
rhombus.
2. Has 4 right angles
3. Diagonals are congruent.
1. Special Parallelogram
2. Has 4 Congruent sides
3. Diagonals are perpendicular.
4. Diagonals bisect opposite angles
* Fold over the third
cut section and write
SQUARE on the
outside.
1. Opposite angles are congruent.
2. Consecutive angles are
supplementary.
3. Opposite sides are congruent.
4. Diagonals bisect each other.
5. Opposite sides are parallel
1. Special parallelogram.
2. Has 4 right angles
3. Diagonals are congruent.
* Reopen the fold.
1. Special Parallelogram
2. Has 4 Congruent sides
3. Diagonals are perpendicular.
4. Diagonals bisect opposite angles
* On the left hand
section, draw a
square.
1. Opposite angles are congruent.
2. Consecutive angles are
supplementary.
3. Opposite sides are congruent.
4. Diagonals bisect each other.
5. Opposite sides are parallel
1. Special parallelogram.
* On the right hand
side, list all of the
properties of a
square.
* Place in your
notebook and save
for tomorrow.
2. Has 4 right angles
3. Diagonals are congruent.
1. Special Parallelogram
2. Has 4 Congruent sides
3. Diagonals are perpendicular.
4. Diagonals bisect opposite angles
1. All the properties of parallelogram,
rectangle, and rhombus
2. 4 congruent sides and 4 right
angles
Name the figure described.
1.
2.
3.
A quadrilateral that is both a rhombus
and a rectangle.
A quadrilateral with exactly one pair of
parallel sides.
A parallelogram with perpendicular
diagonals
 What
are the properties of different
quadrilaterals? How do we use the
formulas of areas of different
quadrilaterals to solve real-life
problems?
 What
are some properties of
trapezoids and kits?
 Use
properties of trapezoids.
 Use properties of kites.

A trapezoid is a
quadrilateral with
exactly one pair of
parallel sides. The
parallel sides are the
bases. A trapezoid has
two pairs of base angles.
For instance in trapezoid
ABCD D and C are
one pair of base angles.
The other pair is A and
B. The nonparallel
sides are the legs of the
trapezoid.
A
base
leg
leg
D
B
base
C
 If
the legs of a
trapezoid are
congruent, then the
trapezoid is an
isosceles trapezoid.
Theorem 6.14
 If a trapezoid is
isosceles, then each
pair of base angles is
congruent.
 A ≅ B, C ≅ D
A
D
B
C
Theorem 6.15
 If a trapezoid has a
pair of congruent
base angles, then it is
an isosceles
trapezoid.
D
 ABCD is an isosceles
trapezoid
A
B
C
Theorem 6.16
 A trapezoid is
isosceles if and only
if its diagonals are
congruent.
 ABCD is isosceles if
and only if AC ≅ BD.
A
D
B
C
 The
midsegment of a
trapezoid is the
B
segment that
connects the
midpoints of its legs.
Theorem 6.17 is
A
similar to the
Midsegment
Theorem for
triangles.
C
midsegment
D
 The
midsegment of a
trapezoid is parallel
to each base and its
length is one half the
sums of the lengths of
A
the bases.
 MN║AD, MN║BC
 MN = ½ (AD + BC)
B
M
C
N
D
 LAYER
CAKE A
baker is making a
cake like the one at
the right. The top
layer has a diameter
of 8 inches and the
bottom layer has a
diameter of 20
inches. How big
should the middle
layer be?
the midsegment
theorem for
trapezoids.
 DG = ½(EF + CH)=
½ (8 + 20) = 14”
E
 Use
D
C
F
G
D
A
kite is a
quadrilateral that has
two pairs of
consecutive
congruent sides, but
opposite sides are
not congruent.
Theorem 6.18
 If a quadrilateral is a
kite, then its
diagonals are
perpendicular.
 AC  BD
C
B
D
A
Theorem 6.19
 If a quadrilateral is a
kite, then exactly one
pair of opposite
B
angles is congruent.
 A ≅ C, B ≅ D
C
D
A
 WXYZ
is a kite so the
diagonals are
perpendicular. You
can use the
W
Pythagorean
Theorem to find the
side lengths.



WX = √202 + 122 ≈ 23.32
XY = √122 + 122 ≈ 16.97
Because WXYZ is a kite, WZ =
WX ≈ 23.32, and ZY = XY ≈
16.97
X
12
20
U 12
12
Z
Y
J
 Find
mG and mJ
in the diagram at the
H 132°
60°
right.
SOLUTION:
GHJK is a kite, so G ≅ J and mG
= mJ.
G
K
2(mG) + 132° + 60° = 360°Sum of measures of int. s of a quad. is 360°
2(mG) = 168°Simplify
mG = 84° Divide each side by 2.
So, mJ = mG = 84°
Quadrilaterals
4-sided polygons
Trapezoids
1. one pair of opposite  sides
Parallelograms
Kites
1. 2 pairs of opposite  sides
1. two pairs of
2. 2 pairs of opposite  sides
consecutive  sides
3. 2 pairs of opposite  angles
2. diagonals are 
4. consecutive angles are supplementary
5. diagonals bis ect each other
3. one pair of opposite
 angles
2. midsegment p arallel to both b ases
and average length of the bases
Isosceles Trapezoids
1. legs are congruent
Rhombus
1. 4  sides
2. base angles are congruent
2. diagonals are 
3. diagonals are congruent
Rectangle
1. 4 right angles
2. diagonals are congruent
3. diagonals bisect a pair of
opposite angles
Square
1. combination of rectangle and rhombus
Geometry
11 PROVE theorems about parallelograms. Theorems INCLUDE: opposite sides ARE congruent, opposite angles ARE congruent, the d








1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
PROVE theorems about parallelograms. Theorems
INCLUDE: opposite sides ARE congruent, opposite
angles ARE congruent, the diagonals of a parallelogram
BISECT each other, and conversely, rectangles ARE
parallelograms with congruent diagonals.
Name the figure:
 1. a quadrilateral with exactly one pair of
opposite angles congruent and
perpendicular diagonals
 2. a quadrilateral that is both a rhombus
and a rectangle
 3. a quadrilateral with exactly one pair of ll
sides.
 4. any ll’ogram with perpendicular
diagonals.

 Identify
special quadrilaterals based on
limited information.
 Prove that a quadrilateral is a special
type of quadrilateral, such as a rhombus
or trapezoid.
 How
do we simplify real life tasks,
checking if an object is rectangular,
rhombus, trapezoid, kite or other
quadrilateral?

Quadrilateral
In this chapter, you have
studied the seven special
Trapezoid
Kite
Parallelogram
types of quadrilaterals
shown at the right. Notice
Isosceles
Rhombus
Rectangle
that each shape has all the
trapezoid
Square
properties of the shapes
linked above it. For
instance, squares have the
properties of rhombuses,
rectangles, parallelograms,
and quadrilaterals.
 Quadrilateral
ABCD has at least one pair
of opposite sides congruent. What kinds
of quadrilaterals meet this condition?
Parallelogram
Rhombus
Opposites
sides are ≅.
Opposite sides
are congruent.
All sides are
congruent.
All sides are
congruent.
Legs are
congruent.

When you join the
midpoints of the
sides of any
quadrilateral, what
special
quadrilateral is
formed? Why?
A
F
E
B
D
G
H
C
A
Solution: Let E, F, G, and H
be the midpoints of the
sides of any quadrilateral, E
ABCD as shown.
D
 If you draw AC, the
Midsegment Theorem for
H
triangles says that FG║AC
and EG║AC, so FG║EH.
C
Similar reasoning shows that
EF║HG.
 So by definition, EFGH is a
parallelogram.

F
B
G
 When
you want to prove that a
quadrilateral has a specific shape, you
can use either the definition of the shape
as in example 2 or you can use a
theorem.
You have learned 3 ways to prove that a quadrilateral is a
rhombus.

You can use the definition and show that the quadrilateral is a
parallelogram that has four congruent sides. It is easier,
however, to use the Rhombus Corollary and simply show that
all four sides of the quadrilateral are congruent.

Show that the quadrilateral is a parallelogram and that the
diagonals are perpendicular (Thm. 6.11)

Show that the quadrilateral is a parallelogram and that each
diagonal bisects a pair of opposite angles. (Thm 6.12)
8




K(2, 5)N(6, 3) = 4.47 cm
Show KLMN is a rhombus
Solution: You can use any of the M(2, 1)N(6, 3) = 4.47 cm
L(-2, 3)M(2, 1) = 4.47 cm
three ways described in the
6
L(-2, 3)K(2, 5) = 4.47 cm
concept summary above. For
instance, you could show that
opposite sides have the same
slope and that the diagonals are
4
perpendicular. Another way
L(-2, 3)
shown in the next slide is to prove
that all four sides have the same
2
length.
AHA – DISTANCE FORMULA If
you want, look on pg. 365 for the
whole explanation of the distance
formula
So, because LM=NK=MN=KL,
KLMN is a rhombus.
-2
K(2, 5)
N(6, 3)
M(2, 1)
5
A

What type of quadrilateral
is ABCD? Explain your
reasoning.
D
60°
120°
120°
60°
B
C
A

What type of quadrilateral is
ABCD? Explain your reasoning.
D
60°
120°
120°

Solution: A and D are
supplementary, but A and B
are not. So, AB║DC, but AD is not
parallel to BC. By definition, ABCD
is a trapezoid. Because base angles
are congruent, ABCD is an isosceles
trapezoid
60°
B
C
The diagonals of quadrilateral ABCD intersect at
point N to produce four congruent segments: AN ≅
BN ≅ CN ≅ DN. What type of quadrilateral is
ABCD? Prove that your answer is correct.
 First Step: Draw a diagram. Draw the diagonals as
described. Then connect the endpoints to draw
quadrilateral ABCD.

B


First Step: Draw a diagram.
Draw the diagonals as
described. Then connect the
endpoints to draw
quadrilateral ABCD.
2nd Step: Make a conjecture:
• Quadrilateral ABCD looks
like a rectangle.

3rd step: Prove your
conjecture
• Given: AN ≅ BN ≅ CN ≅ DN
• Prove ABCD is a rectangle.
C
N
A
D
Because you are given information about diagonals, show
that ABCD is a parallelogram with congruent diagonals.
 First prove that ABCD is a parallelogram.

• Because BN ≅ DN and AN ≅ CN, BD and AC bisect each
other. Because the diagonals of ABCD bisect each other,
ABCD is a parallelogram.

Then prove that the diagonals of ABCD are congruent.
• From the given you can write BN = AN and DN = CN so, by
the addition property of Equality, BN + DN = AN + CN. By
the Segment Addition Postulate, BD = BN + DN and AC =
AN + CN so, by substitution, BD = AC.
• So, BD ≅ AC.
ABCD is a parallelogram with congruent diagonals, so
ABCD is a rectangle.