GEOMETRY
Prove Theorems about Parallelograms
OBJECTIVE #: G.CO.11
OBJECTIVE
• 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.
BIG IDEA (Why is this included in the curriculum?)
• A natural extension to triangle properties and relationships is the study of quadrilaterals. This is
because every quadrilateral can be divided into triangles.
PREVIOUS KNOWLEDGE (What skills do they need to have to succeed?)
• The student must have a strong grasp on the isometric transformations and their definitions.
• The student must know the criteria for triangle congruence.
• The student must know the angle relationships relating to parallel lines and a transversal.
• The student must know triangle side and angle relationships.
VOCABULARY USED IN THIS OBJECTIVE (What terms will be essential to understand?)
PREVIOUS VOCABULARY (Terms used but defined earlier)
• Bisect: To divide into two congruent parts.
• Consecutive Angles: Two angles that share a side of a polygon.
• Equiangular Polygon: A polygon with all angles congruent.
• Equilateral Polygon: A polygon with all sides congruent.
• Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
 Rectangle: A parallelogram with four right angles.
 Rhombus: A parallelogram with four congruent sides.
 Square: A parallelogram with four right angles and four congruent sides.
• Perpendicular: Two lines/segments/rays that intersect to form right angles.
• Quadrilateral: A four sided polygon.
• Reflection: A rigid transformation in which the image is a mirror image of the pre-image, thus
ensuring the pre-image and the image are equidistant from the line of reflection.
• Rotation: A rigid transformation that turns a figure about a fixed point, thus ensuring the preimage and image are congruent.
• Translation: A type of transformation that maps every two points P and Q in the plane to
points P ' and Q ' , so that the following two properties are true. (1) PP ' = QQ ' . (2) PP ' QQ ' or
PP ' and QQ ' are collinear.
NEW VOCABULARY (New Terms and definitions introduced in this objective)
• Diagonal: A segment that joins two non-adjacent vertices of a polygon.
SKILLS (What will they be able to do after this objective?)
• The student will be able to prove properties of parallelograms and then apply them.
• The student will be able to prove the properties of rectangles and then apply them.
• The student will be able to prove the properties of rhombi and then apply them.
• The student will be able to prove the properties of squares and then apply them.
• The student will be able to classify a quadrilateral by its properties.
SHORT NOTES
(1) Prove that opposite sides of a parallelogram are congruent.
Suggested Proofs:
Regular Geometry 1a & 1b
/ Honors Geometry 1a & 1b
a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)
Given: Parallelogram ABCD
C
Prove that opposite sides of the parallelogram are congruent.
B
Established in G.CO.3, a parallelogram has rotational symmetry,
when rotated about its center 180°.
Rotational symmetry would map AB onto CD and AD onto CB ,
thus making them congruent.
This can be tested using patty paper. Create a parallelogram ABCD
on a piece of paper. Draw in the two diagonals AC and BD to
determine the intersection E. Copy the parallelogram onto the patty
paper. Line up the two parallelograms and then pin the patty paper
with your pencil at E and begin to rotate the patty paper. At a 180°
rotation, you will see that AB ≅ CD and AD ≅ CB .
E
D
A
A
D
E
B
C
b) Proof by Congruent Triangles (Formal – Classic Approach)
Given: Parallelogram ABCD
Prove: AB ≅ CD , AD ≅ CB
Construct an auxiliary line that is the diagonal BD.
STATEMENT
REASON
ABCD is a parallelogram
Given
Definition of Parallelogram
AB || CD , AD || CB
∠ABD ≅ ∠CDB
If ||, then alternate interior ∠’s ≅
∠ADB ≅ ∠CBD
If ||, then alternate interior ∠’s ≅
Reflexive Property (Common Side)
BD ≅ BD ,
ASA
∆ABD ≅ ∆CDB
CPCTC
AB ≅ CD , AD ≅ CB
C
B
D
A
(2) Prove that opposite angles of a parallelogram are congruent.
Suggested Proofs:
Regular Geometry 1a & 1b
/ Honors Geometry 1a & 1b
a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)
Given: Parallelogram ABCD
C
Prove that opposite angles of the parallelogram are congruent.
B
Established in G.CO.3, a parallelogram has rotational symmetry,
when rotated about its center 180°.
Rotational symmetry would map ∠A onto ∠C and ∠B onto ∠D, thus
making them congruent.
This can be tested using patty paper. Create a parallelogram ABCD
on a piece of paper. Draw in the two diagonals AC and BD to
determine the intersection E. Copy the parallelogram onto the patty
paper. Line up the two parallelograms and then pin the patty paper
with your pencil at E and begin to rotate the patty paper. At a 180°
rotation, you will see that ∠A ≅∠C and ∠B ≅ ∠D.
E
D
A
A
D
E
B
C
b) Proof by Congruent Triangles (Formal – Classic Approach)
Given: Parallelogram ABCD
Prove: ∠A ≅∠C
Draw an auxiliary line that is the diagonal BD.
STATEMENT
REASON
ABCD is a parallelogram
Given
Definition of Parallelogram
AB || CD , AD || CB
C
B
D
A
∠ABD ≅ ∠CDB
If ||, then alternate interior ∠’s ≅
∠ADB ≅ ∠CBD
If ||, then alternate interior ∠’s ≅
Reflexive Property (Common Side)
BD ≅ BD
ASA
∆ABD ≅ ∆CDB
CPCTC
∠A ≅∠C
Using diagonal AC we could follow a symmetrical argument to
prove that ∠B ≅ ∠D. I will demonstrate it below.
Given: Parallelogram ABCD
Prove: ∠B ≅ ∠D
Draw an auxiliary line that is the diagonal AC
STATEMENT
REASON
ABCD is a parallelogram
Given
Definition of Parallelogram
AB || CD , AD || CB
∠BAC ≅ ∠DCA
∠BCA ≅ ∠DAC
AC ≅ AC
∆BAC ≅ ∆DCA
∠B ≅ ∠D
If ||, then alternate interior ∠’s ≅
If ||, then alternate interior ∠’s ≅
Reflexive Property (Common Side)
ASA
CPCTC
C
B
D
A
(3) Prove that diagonals bisect each other in a parallelogram.
Suggested Proofs:
Regular Geometry 1a & 1b
/ Honors Geometry 1a & 1b
a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)
Given: Parallelogram ABCD, with diagonals AC and BD
C
Prove that diagonals bisect each other.
B
Established in G.CO.3, a parallelogram has rotational symmetry, when
rotated about its center 180°.
Rotational symmetry would map DE onto BE and CE onto AE , thus
making them congruent.
E
D
A
This can be tested using patty paper. Create a parallelogram ABCD on a
piece of paper. Draw in the two diagonals AC and BD to determine
the intersection E. Copy the ∆CED onto the patty paper. Line up the
patty paper with the original parallelogram and then pin the patty paper
with your pencil at E and begin to rotate the patty paper. At a 180°
rotation, you will see that DE ≅ BE and
CE ≅ AE.
b) Proof by Congruent Triangles (Formal – Classic Approach)
Given: Parallelogram ABCD, with diagonals AC and BD
C
Prove: DE ≅ BE , CE ≅ AE
STATEMENT
ABCD is a parallelogram
AB || CD , AD || CB
∠CDE ≅ ∠ABE
∠DCE ≅ ∠BAE
CD ≅ AB
∆ABD ≅ ∆CDB
DE ≅ BE , CE ≅ AE
B
REASON
Given
Definition of Parallelogram
If ||, then alternate interior ∠’s ≅
If ||, then alternate interior ∠’s ≅
Opposite Sides of Parallelogram ≅
ASA
CPCTC
E
A
D
(4) Prove that diagonals are congruent in a rectangle.
Suggested Proofs:
Regular Geometry 1a & 1b
/ Honors Geometry 1a & 1b
a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)
Given: Rectangle ABCD
Prove that diagonals are congruent.
C
B
Established in G.CO.3, a rectangle has reflectional symmetry, when
reflected over the line through the midpoints of opposite sides. This
would map B onto C and A onto D, thus the isometric properties of
reflection would preserve the distances of BD and CA .
This can be tested using patty paper. Create a rectangle ABCD on a
piece of paper. Draw in the two diagonals AC and BD . Copy the
rectangle onto the patty paper. Place only one of the diagonals on
the patty paper. Reflect the patty paper by reversing it to the other
side. Match up the two rectangles (A onto D, B onto C, C onto B, D
onto A). Notice that the diagonal on the patty paper matches the
other diagonal. BD ≅ CA .
D
A
C
B
D
A
B
C
A
D
b) Proof by Triangle Congruence (Formal – Classic Approach)
Given: Rectangle ABCD, with diagonals AC and BD
Prove: BD ≅ AC
STATEMENT
ABCD is a rectangle
A rectangle is a parallelogram
AB ≅ CD
AD ≅ AD
∠BAD and ∠CDA are right
angles
∠BAD ≅ ∠CDA
∆BAD ≅ ∆CDA
BD ≅ AC
C
B
REASON
Given
Definition of Rectangle
Opposite Sides ≅ in Parallelogram
Reflexive Property (Common Side)
Definition of Rectangle
All Right ∠ ' s are ≅
SAS
CPCTC
D
A
(5) Prove that the diagonals of a rhombus are angle bisectors.
Suggested Proofs:
Regular Geometry 1a & 1b
/ Honors Geometry 1a & 1b
a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)
Given: Rhombus ABCD, with diagonals AC and BD
B
Prove that diagonals are angle bisectors.
Established in G.CO.3, a rhombus has reflectional symmetry, when
reflected over either diagonal. If we use diagonal AC , this would
map B onto D, A onto A and C onto C. Thus the isometric properties
of a reflection would preserve the angle measures. Thus ∠BAC ≅
∠DAC and ∠BCA ≅ ∠DCA. Using the other diagonal, we could also
establish that ∠ABD ≅ ∠CBD and ∠ADB ≅ ∠CDB.
C
E
A
D
This can be tested using patty paper. Create a rhombus ABCD on a
piece of patty paper. Draw in the diagonals. Now fold the patty
paper so that A maps onto C, crease the patty paper. Notice that
∠ABD ≅ ∠CBD and ∠ADB ≅ ∠CDB. Now fold the patty paper so that
B maps onto D, crease the patty paper. Notice that ∠BAC ≅ ∠DAC
and ∠BCA ≅ ∠DCA.
b) Proof by Triangle Congruence (Formal – Classic Approach)
Given: Rhombus ABCD, with diagonals AC and BD
Prove: AC is an angle bisector
STATEMENT
REASON
ABCD is a rhombus
Given
A rhombus is a parallelogram
Definition of Rectangle
Opposite Sides ≅ in parallelogram
AB ≅ CD
Opposite Sides ≅ in parallelogram
AD ≅ BC
Reflexive Property (Common Side)
AC ≅ AC
SSS
∆BAD ≅ ∆CDA
CPCTC
∠BAC ≅ ∠DAC and
∠BCA ≅ ∠DCA
Congruent Adjacent Angles
AC is an angle bisector
Using symmetric argument we could prove that ∆ABD ≅ ∆CDB and
from there establish that BD , the other diagonal, is also an angle
bisector
B
C
A
E
D
(6) Prove that the diagonals of a rhombus are perpendicular.
a) Proof by Symmetry and Patty Paper (Informal – Transformational Approach)
Given: Rhombus ABCD, with diagonals AC and BD
B
Prove that diagonals are perpendicular.
C
E
A
Established in G.CO.3, a rhombus has reflectional symmetry, when
reflected over either diagonal. That would require AC to be the
perpendicular bisector of BD . Thus BD ⊥ AC .
D
This can be tested using patty paper. Create a rhombus on a piece of
patty paper. Draw in the diagonals. Now fold the patty paper so that
A maps onto C, crease the patty paper. Notice that the crease is the
diagonal. Thus the diagonal is a perpendicular bisector of the other
two vertices making them perpendicular to each other.
b) Proof by Triangle Congruence (Formal – Classic Approach)
Given: Rhombus ABCD
B
C
Prove: BD ⊥ AC
STATEMENT
ABCD is a rhombus
A rhombus is a parallelogram
AB ≅ BC ≅ CD ≅ DA
AE ≅ EC
BE ≅ DE
∆BAE ≅ ∆BCE ≅ ∆DAE ≅ ∆DCE
∠DEA ≅ ∠DEC ≅ ∠BEA ≅ ∠BEC
m∠DEA = m∠DEC = m∠BEA = m∠BEC
m∠DEA + m∠DEC + m∠BEA + m∠BEC =360°
m∠DEA + m∠DEA + m∠DEA + m∠DEA =360°
4m∠DEA =360°, m∠DEA =90°
BD ⊥ AC
A
E
D
REASON
Given
Definition of a Rhombus
Definition of a Rhombus
Diagonals bisect each other in
parallelogram
Diagonals bisect each other in
parallelogram
SSS
CPCTC
Definition of Congruence
Angle Sum about a Point = 360°
Substitution Property
Simplify & Division Property
Intersect at Right Angles
FUTURE CONNECTIONS (What will they use these skills for later?)
• Learning the properties of these quadrilaterals will help us investigate them through similarity,
coordinate analysis and circle geometry.
• At this time, we have examined properties of quadrilaterals using congruency. In future objectives, we
will re-examine these properties using coordinate geometry proof.
ADDITIONAL EXTENSIONS OR EXPLANATIONS (What needs greater explanation?)
• QUADRILATERAL PROPERTY FLOWCHART PROVIDED
• The objective specifically addresses to parallelograms and rectangles. However, in future objectives,
students will need to understand and apply the properties of all special quadrilaterals. Therefore, it is
important to also address the following objectives:
• The student will be able to prove the properties of trapezoids and then apply them.
• The student will be able to prove the properties of kites and then apply them.
• Rectangle: An equiangular parallelogram.
• Rhombus: An equilateral parallelogram.
• Square: A rhombus and a rectangle.
ASSESSMENT ITEMS (What questions would evaluate these skills?)
1) Circle whether the following are (T)rue or (F)alse. (1 point each)
•
•
•
•
•
•
•
Diagonals bisect angles in a parallelogram.
A quadrilateral is a 4 sided polygon.
Consecutive sides are congruent in a rectangle.
If a parallelogram has 4 ≅ angles, then it must be a square.
A square is both a rhombus & a rectangle
All rectangles are parallelograms.
All rhombi are rectangles.
T
T
T
T
T
T
T
or
or
or
or
or
or
or
F
F
F
F
F
F
F
2) Circle whether the relationship is (A)lways, (S)ometimes, or (N)ever True.
a) A rhombus is a quadrilateral
b) A rectangle is a rhombus.
c) Adjacent sides of a rectangle are congruent.
d) Diagonals of a square are perpendicular
e) Diagonals of a parallelogram are congruent.
f) If ABCD is a parallelogram, then AB = CD
g) Square is a rectangle
h) A parallelogram is a rhombus
i) A rhombus is a square
A
A
A
A
A
A
A
A
A
S
S
S
S
S
S
S
S
S
N
N
N
N
N
N
N
N
N
3) Making quadrilaterals through transformations:
a) Make a quadrilateral by rotating
b) Make a quadrilateral by rotating
∆RST 180° about U.
∆RST 180° about V.
S
U
R
S
o
U
V
x
T
R
o
V
x
T
What type of quadrilateral is formed?
What type of quadrilateral is formed?
Rectangle
Parallelogram
What did you see in the shape to conclude
the type of quadrilateral?
What did you see in the shape to conclude the
type of quadrilateral?
Opposite Sides are Parallel
4 congruent right angles
Opposite Sides are Parallel
Opposite Angles are Congruent
4) Complete the Venn diagram by correctly
labeling the parallelograms, the rhombi, the
rectangles and the squares locations in the
diagram.
Squares are the intersection of the rectangles and rhombi. Parallelograms are the largest boundary.
5) You know that in parallelogram ABCD the m∠A = 90°. What is the most specific name for this
parallelogram and why?
Rectangle. Opposite angles are equal and consecutive angles are supplement – so all 4 angles are right
angles.