The document provides a lesson on theorems for different types of parallelograms including rectangles, rhombuses, and squares. It begins with examples that prove various properties of these shapes using logical reasoning and theorems. It then summarizes the unique properties of each shape type, such as rectangles having four right angles and rhombuses having diagonals that bisect opposite angles. The lesson aims to help students understand and apply the theorems to solve problems involving different kinds of parallelograms.

1. 9
Mathematics
Quarter 3 - Module 2:
Theorems on Different Kinds of
Parallelogram
(Rectangle, Rhombus, and Square
9

2. Development Team of the Module
Writers: Rony C. Roncales, MAT
Erickson C. Mata
Editors: Donato B. Chico
Jerry M. Padigdig
Rolando C. Rullan Jr.
Mark V. Gatbunton
Joialie O. Gonzales
Levy B. Hernal
Reviewer: SDO Nueva Ecija
Layout Artist: Rony C. Roncales, MAT
Erickson C. Mata
Management Team: Jayne M. Garcia, EdD
Florentino O. Ramos, PhD
Beverly T. Mangulabnan, PhD
Eleanor A. Manibog, PhD
Mathematics – Grade 9
Alternative Delivery Mode
Quarter 3 – Module 2: Theorems on Different Kinds of Parallelogram (Rectangle,
Rhombus, and Square)
First Edition, 2020
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3. Introductory Message
This Self-Learning Module is prepared so that you, our dear learners,
can continue your studies and learn while at home. Activities,
questions, directions, exercises, and discussions are clearly stated
for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you
step-by-step as you discover and understand the lessons prepared
for you.
Pre-tests are provided to measure your prior knowledge on lessons
in each SLM. This will tell you if you need to proceed on completing
this module if you need to ask your facilitator or your teacher’s
assistance for better understanding of the lesson. At the end of
each module, you need to answer the post-test to self-check your
learning. Answer keys are provided for each activity and test. We
trust that you will be honest in using these.
In addition to the material in the main test, Notes to the teacher are
also provided to our facilitators and parents for strategies and
reminders on how they can best help you on your home-based
learning.
Please use this module with care. Do not put unnecessary marks on
any part of this SLM. Use a separate sheet of paper in answering the
exercises and tests. Read the instructions carefully before
performing each task.
If you have any question in using this SLM or any difficulty in
answering the tasks in this module, do not hesitate to consult your
teacher or facilitator.
Thank you.

4. 1
What I Need to Know
This module serve as a guide and tool for teaching theorems in different
kinds of parallelogram such as rectangle, rhombus, and square.
This module contains:
Module 2 – Theorems on Different Kinds of Parallelogram (Rectangle,
Rhombus and Square).
After accomplishing this module, you are expected to:
1. prove theorems of the different kinds of parallelograms (Rectangle,
Rhombus, and Square) (M9 GE-III-c-1).
2. apply the properties of a rectangle, rhombus, and square to solve
problems involving rectangle, rhombus, and square.
What I Know
Let us first review the properties of a parallelogram.
Activity 1: Match my properties!
Directions: Determine which property of parallelogram is being described in
the following figures. Match the figures of parallelogram in column A with the
corresponding description in column B. Write the letterof the correct answer
on a separate sheet of paper.
Column A Column B
1. A. A diagonal of a parallelogram
forms two congruent triangles.
10 m
10 m
8m 8 m

5. 2
B. In a parallelogram, any two
opposite angles are congruent.
2.
C.In a parallelogram, any two
consecutive angles are
supplementary.
3. D. In a parallelogram, any two
opposite sides are congruent.
E. The diagonals of a
parallelogram bisect each
other
4.
5.
Congratulations for the job well done. If you get a score of 4 or 5, this means
you are ready for the next lesson.
However, if you get a score of 3 and below, it is recommended to
reevaluate the work done to be able to learn the next lesson more.
1000
1000
3 cm.
3 cm.
4 cm. 4 cm.
6 m
8 m
6 m
8 m
1000
800

6. 3
The previous module tackles the properties of the parallelogram. This
module focuses on different kinds of parallelogram. You are encouraged to
understand the following theorems or properties of different kinds of
parallelogram: Rectangle, Rhombus and Square. You are also advised to
identify the unique properties of each kind of parallelogram.
In addition, this module will help you solve problems related to different
kinds of parallelogram using each theorem and property.
What’s In
Activity 2: JUMBLED LETTERS
Directions: Arrange the following jumbled letters using the given hint below.
Write the correct word/s on a separate answer sheet.
1. It is a shape of Sponge Bob’s pants.
2. Its shape looks like a chalkboard.
3. Its shape looks like a diamond.
4. These are two line
segments meeting at
a point forming right
angle. It has a
symbol of .
.
5. An angle measuring 900.
Lesson
1
Theorems on Different
Kinds of Parallelogram

7. 4
What’s New
You are therefore encouraged to explore this lesson through the activity
below.
Activity 3: Do it in your Own!
Directions: Read and follow the procedures. Then complete the following
statements below.
A. Rectangle Probe
Materials: Graphing Paper
Compass
Ruler
STEP 1. Using the lines on a piece of graphing paper as a guide, draw a
rectangle.
STEP 2. Draw both diagonals. Using your ruler, compare the lengths of the
two diagonals.
STEP 3. Draw different rectangles and repeat STEP 2 to verify your
conjectures inductively.

8. 5
Complete the following conjecture:
1. The diagonals of a rectangle are ______________________________.
B. Rhombus Probe
Materials: four strips of cardboard of equal length
four roundhead fasteners
puncher and ruler/protractor
STEP 1. Punch holes on the four cardboards at both ends. Make sure that
you form 4 congruent-sides parallelogram.
STEP 2. Connect these four cardboards to form an equilateral quadrilateral.
STEP 3. The figure formed is a rhombus.
STEP 4. Draw the diagonals. Use a ruler and a protractor to measure
diagonals and angles, then complete the following conjectures:
1. Diagonals of a rhombus are___________________________.
2. The _________________ of a rhombus bisect the angles.
You must remember that the rhombus has four congruent sides.

9. 6
What is It
Rectangles, rhombuses, and squares are considered special types of
parallelogram.
Apart from properties inherent to parallelogram, each kind of
parallelogram possesses unique properties.
Theorems on Rectangle
Theorems on Rhombus
Properties of Square
This time you should study the following examples.
Example 1:
Given: RAIN is a parallelogram where
R is a right angle.
Prove: A, I, and N are right angles.
Theorem 1: If a parallelogram is rectangle, it has four right angles
Theorem 2: If a parallelogram has congruent diagonals, it is a rectangle.
Theorem 3: The diagonals of a rhombus are perpendicular.
Theorem 4: Each diagonal of a rhombus bisects opposite angles.
a. All properties of a parallelogram.
b. All properties of a rectangle.
c. All properties of a rhombus.
R
N I
A
Note: Remember that perpendicular lines are two lines that intersect
to form a right angle.
A
B
D
C

10. 7
Proof:Statements Reasons
1. RAIN is a parallelogram with R
is a right angle.
1.Given
2. m R = 90 2.Definition of right angle.
3. R ≅ I & A ≅ N 3.In a parallelogram. Opposite angles
are congruent.
4.m R = m I , m A = m N 4.Definition of congruent angles.
5. m R = 90 5. Substitution (SN 2 & 4)
6.m R + m A = 180 6.Consecutive angles are
supplementary.
7.900 + m R = 180 7.Substitution (SN 2 & 6)
8. 900 = 900 8. Reflexive Property
9.m A = 90 9.Subtraction Property (SN 7 & 8)
10.m N = 90 10. Substitution (SN 4 and 9 )
11. A , I , and N are right
angles.
11.Definition of right angle.
12. RAIN is a rectangle. 12. Definition of rectangle.
Note: SN: Statement Number
Example 2:
Given: Rectangle ABCD with diagonals AC and BD.
Prove: AC ≅ BD
D
B
A
C

11. 8
Proof:
Statements Reasons
1.Rectangle ABCD with diagonals AC
and DB.
1. Given
2. AB  DC 2.Opposite sides of a rectangle
are congruent.
3. BAD and CDA are right angles. 3. Definition of Rectangle
4. BAD ≅ CDA 4. Any two right angles are
congruent.
5. AD ≅ DA 5. Reflexive Property
6. BAD ≅ CDA 6. Side-Angle-Side Postulate
7. AC ≅ BD 7. Corresponding Parts of
Congruent Triangles are
Congruent
Example 3: In RENT answer the following:
a. Find ET if RS = 13 units
b. Find m TRN if m RNT =24
Solutions:
a. Diagonal ET ≅ Diagonal RN
RN = 2RS = 2(13) = 26 units
- Since the diagonals of a rectangle
bisect each other.
RN= ET, and RN = 26
Therefore, ET = 26 units
R
T N
E
S
R
T N
240
1.Identify the theorem that can be used
to solve the problem: (The diagonals of
a rectangle are congruent).
2.Substitute and solve.
Note: The two pairs of opposite sides RT and EN, RE and TN with pairs of a single
and double arrow heads respectively means they are both parallel.

12. 9
b. T is a right angle.
- Since RENT is a rectangle. So, RNT
is a right triangle. The acute angles
TRN and RNT are complementary.
m TRN = 90 – m RNT
= 90 – 24
m TRN = 66
Example 4
Given: Rhombus ROSE
Prove: RS ⊥ OE
Proof:
Statements Reasons
1.Rhombus ROSE. 1.Given
2.OS ≅ OE 2.Definition of rhombus.
3.RS and EO bisect each other 3.The diagonals of parallelogram
bisect each other.
4.H is the midpoint of RS 4.EO bisects RS at H.
5.RH ≅ HS 5.Definition of Midpoint
6.OH ≅ OH 6.Reflexive property
7. RHO ≅ SHO 7.Side-Side-Side Congruence
Postulate
8. RHO ≅ SHO 8.Corresponding Parts of
Congruent Triangles are
Congruent.
9. RHO and SHO are right angles 9. RHO and SHO from a
linear pair and are congruent.
10. RS ⊥ OE 10.Definition of perpendicular
lines.
1.Identify the theorem that
can be used to solve the
problem.
(If a parallelogram has one right angle,
then it has four right angles and the
parallelogram is a rectangle.)
2.Substitute and solve.
R O
H
E S

13. 10
Example 5:
Given: Rhombus VWXY
Prove: 1 ≅ 2 ; 3 ≅ 4
Proof:
Statements Reasons
1. Rhombus VWXY 1.Given
2.YV ≅ VW; WX ≅ XY 2.Definition of
rhombus
3. WY ≅ YW 3.Reflexive Property
4. YVW ≅ WXY 4.Side-Side-Side
Congruence
Postulate
5. 1 ≅ 2 ; 3 ≅ 4 5.Corresponding
Parts of Congruent
Triangles are
Congruent
Example 6: Quadrilateral SAME is a rhombus. Find the measures of each of
the following:
a. ASM d. SM
b. SMA e. SPE
c. PM f. AS if AM =(4x – 7)
and SE = (x + 5)
Solutions
a. ASM≅ ESM
Since m ESM = 50,
Then m ASM = 50
b. ESM ≅ SMA
V
1
W
Y X
2
4
3
S
M
A
E
P
6
500
1.Identify the theorem that can
be used to solve the problem:
(Each diagonal bisects the
opposite angles of the
rhombus.)
2.Substitute and solve.
1.Identify the theorem that can
be used to solve the problem:
(Alternate interior angles
are congruent.)

14. 11
1.Identify the theorem that can
be used to solve the
problem:(The diagonals of a
rhombus are perpendicular.)
1.Identify the theorem or property
that can be used to solve the
problem: (All sides in a rhombus
are congruent.)
2. Equate SE and AM and solve.
for x.
4. Use multiplication property of
equality.
3. Apply addition property of
equality.
m ESM = m SMA = 50
c. PM ≅ PS
PM = PS = 6 units
d. PM ≅ PS
SM = PS + PM
= 6 + 6
SM = 12 units
e. SPE is a
right angle
m SPE = 90 Since ∠SPE is an angle formed by
perpendicular lines, therefore, ∠SPE is a
right angle.
f. AS ≅ SE ≅ AM
AM = x + 5
SE = 4x – 7
4x – 7 = x + 5
3x – 7+7 = 5 + 7
(
1
3
) 3x = 12(
1
3
)
x = 4
2.Substitute and solve.
1.Identify the theorem that can
be used to solve the problem:
(Diagonals of the rhombus
bisect each other.)
2.Substitute and solve.
1.Identify the theorem that can
be used to solve the problem:
(The diagonals of a rhombus
bisect each other.)
2.Substitute and solve.
2. Perpendicular lines are lines that
intersect at right angles.

15. 12
AS = SE
AS = x + 5
AS = (4) + 5
AS = 9 units
Let us summarize the properties of parallelogram, rectangle,
rhombus, and square based on graphic organizer below.
Properties
1.Two pairs of opposite sides are parallel and
congruent
2.Opposite angles are congruent.
3.Consecutive angles are supplementary.
4.Each diagonal divides the quadrilateral into
two congruent triangles.
5.Diagonals bisect each other
Parallelogram
Properties
1.All angles are congruent.
2. All angles are right angles.
3. Diagonals are congruent.
4.All properties of a parallelogram.
Properties
1. Each diagonal bisects
quadrilateral’s angles.
2. All sides are
congruent.
3. Diagonals are
perpendicular.
4. All properties of a
parallelogram.
Property
1. All properties of
parallelogram,
rectangle, and
rhombus.
Rectangle Rhombus
Square
5. Substitute and simplify.

16. 13
What’s More
Activity 4: Linking!
Directions: Consider the figure at the right and answer the following.
Write your answers on separate sheet/s of paper.
TIME is a square;
1. what is m∠1?
2. what is m∠MET?
If TM = 12 ft.,
3. what is IE?
4. what is IX?
5. What is TE if
TI = 8 ft.?
LOVE is a rhombus;
6. what is m∠LYO?
If m∠2 = 37,
7. what is m∠3?
8. what is m∠4?
9. What is OV if LO = 13 ft.?
10. What is LY if LV = 21 ft.?
What I Have Learned
Let us summarize what you have learned!
Activity 5: Fill Me In!
Directions: Fill in the blank. Give the correct information on each
blank. Write your answers on a separate sheet of paper.
1. If a parallelogram has four _________, it is rectangle.
2. The diagonals of a rectangle are ______________.
3. The diagonals of a rhombus are ______________.
4. Each diagonal of a rhombus bisects ___________.
E
I
M
T
X
1
2
Y
E V
L O
3
4

17. 14
5. All the properties of parallelograms and the theorems on
rectangles and rhombuses are true to all _____________.
What I Can Do
Activity 6: Let’s Apply It!
Directions: Answer the following problems. Write your answers on a separate
sheet/s of paper.
Consider the figure of a flat screen
television at the right.
If in the given rectangle CARE, AE = 49 in.,
and CE = 25 in., what is the measure
of the following?
1. CT = _____
2. AR =_____
3. CR =_____
4. ∠ACE = ____
5. ∠TRE= _______ if m∠ART=56.
In the given picture frame,
the diagonals of square EASY meet at P.
6. What is the value of x if EA = (3x – 11)
and AS = (x + 3)?
7. What is ES if EP = (x + 2)in.
and AY = (4x – 6)in.?
In the given logo at the right, the diagonals of
rhombus HOPE meet at J. If m∠OJP = (2y+10)
and m∠2 = (y+20),
8. what is m∠3?
9. what is m∠JEP?
If EH = 2x – 3 and HO = x + 1,
10. find PE?
R
T
E
A
C
P
S
Y
E A
O
J
E P
H
2
3

18. 15
Assessment
Directions: Read each item carefully. Write the letter of the correct
answer on a separate sheet of paper.
1. Which statement best differentiates squares from the rectangles?
A. Squares must have four 90° angles, rectangles do not have
all 90° angles.
B. Squares have two sets of equal sides, rectangles have only
one pair of equal sides.
C. Squares have four equal sides, rectangles have two pairs
of equal opposite sides.
D. Squares have the diagonals that bisect each other,
rectangles have diagonals that are perpendicular.
2. What parallelogram has diagonals bisect both pairs of opposite
angles?
A. Rectangle
B. Rhombus
C. Square
D. Rhombus or square
3. What is m∠PTS if PQRS is a rhombus and the two diagonals
intersect at T?
A. 45°
B. 60°
C. 90°
D. 120°
4. STAR is a rhombus. The two diagonals intersect at X. How will
you prove that its diagonals are perpendicular?
A. Show∠SXT ≅ ∠AXT, Since SA
̅̅̅
̅ and TR
̅̅̅̅ intersect to form congruent
adjacent angles, SA
̅̅̅
̅ ⊥ TR
̅̅̅̅.
B. Show ΔSXT ≅ ΔAXT, Since congruent parts of congruent triangles
are congruent, then SA
̅̅̅
̅ ⊥ TR
̅̅̅̅
C. Show ΔTSR ≅ ΔTAR, Since congruent parts of congruent triangles
are congruent, then ∠SXT ≅ ∠AXT then SA
̅̅̅
̅ ⊥ TR
̅̅̅̅.
D. Show ΔSXT ≅ ΔAXR, since ∠SXT ≅ ∠AXR, then SA
̅̅̅
̅ ⊥ TR
̅̅̅̅.

19. 16
5. Which is not a property of a square?
A. All sides are congruent.
B. All angles are right angles.
C. Consecutive angles are supplementary but not congruent
D. Diagonals are perpendicular and congruent.
6. JADE is a square. What is the value of x if JE = 4x -5 and
AD = x + 4?
A. 1
B. 3
C. 6
D. 7
7. What is the length of diagonal IE in rectangle FIRE and OR=7m?
A. 7 m
B. 14 m
C. 21 m
D. 28 m
8. What kind of parallelogram is it if its diagonals bisect both pairs
of opposite angles and each bisected angle measures 45°?
A. Rectangle
B. Rhombus
C. Square
D. A parallelogram with a 90° angle
9. What condition will make parallelogram MNOP a rectangle?
A. ∠N is a right angle
B. MN
̅̅̅̅̅ ≅ OP
̅̅̅̅
C. MN
̅̅̅̅̅ ∥ OP
̅̅̅̅
D. MN
̅̅̅̅̅ and OP
̅̅̅̅ bisect each other
10.What is the m∠1 if m∠HMO=34 in rhombus RHOM?
A. 34°
B. 56°
C. 68°
D. 90°
F
O
E R
I
7 m
R
34°
M O
H
1

20. 17
Additional Activities
Parallelogram Analysis
Directions: Tell whether each statement is always true, sometimes true, or
never true. Write your answers on a separate sheet of paper.
1. A square is a rhombus.
2. A rectangle is a square.
3. A rectangle has congruent diagonals.
4. The diagonals of a square bisect its angles.
5. A square has four congruent angles.
6. A rhombus has four congruent angles.
7. A rectangle has perpendicular diagonals.
8. A parallelogram is a rhombus.
9. A rhombus is a square.
10. A rectangle has unequal interior angles.

21. 18
Answer Key
What’s
More
Activity
4
1.
45°
2.
90°
3.12
ft.
4.6
ft.
5.8
ft.
6.
90°
7.
37°
8.
37°
9.
13
ft.
10.
10.5
ft.
What
I
Have
Learned
Activity
5
1.
right
angle
2.
congruent
3.
perpendicular
4.
opposite
angles
5.
square
What
I
Can
Do
Activity
6
1.
24.5
in.
2.
25
in.
3.
49
in.
4.
°
°
ASSESSMENT
1.
C
6.
B
2.
D
7.
B
3.
C
8.
C
4.
A
9.
A
5.
C
10.
A
Additional
Activities
1.
always
true
6.
sometimes
true
2.
sometimes
true
7.
sometimes
true
3.
always
true
8.
sometimes
true
4.
always
true
9.
sometimes
true
5.
always
true
10.
never
true
What
I
know
Activity
1
1.D
2.B
3.E
4.A
5.C
What’s
in
Activity
2
1.SQUARE
2.RECTANGLE
3.RHOMBUS
4.PERPENDICULAR
5.RIGHT
ANGLE
What
is
it
Activity
3
A.
1.Equal/Congruent
B
1.Perpendicular
2.Diagonals

22. 19
References
Books:
Bryant, M., Bulalayao, L., Callanta, M., Cruz, J., De Vera, R., Garcia., Javier,
S., Lazaro., Mesterio, B., & Saladino RH., DepEd (2014). Mathematics
Grade 9 Learner’s Material. First Edition. ISBN:978-971-9601-71-5
Diaz, Z. B., Mojica M. P., Suzara, J. L., Mercado, J. P., Esparrago M. S., Reyes,
Jr., N. V.,(2014), Next Century Mathematics, Phoenix Publishing
House, Inc. ISBN 978-971-06-3548-1
Leonor, E.F., Timajo, R.P., (2014) Integrated Math for Grade 9, New Horizon
Publication ISBN 978-971-8593—88-2
Nivera, G. C. & Lapinid, M R. C., (2013), Grade 9 Mathematics Patterns and
Practicalities,Don Bosco Press, Inc., ISBN 978-971-9978-40
Nivera, G. C.,Dioquino, A. D., Buzon, O. N., Abalajon, T. J., (2003), Making
Connections in Mathematics, Vicarish Publication Trading Inc., ISBN
971-689-185-7