9

Mathematics
Quarter 3 – Module 1
Week 1-2

Parallelogram and Its Family

(Rectangle, Square and Rhombus)
Reminders to Learners
The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on any part of
the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in this module.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.

If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.

We hope that through this material, you will experience meaningful learning
and gain deep understanding of the relevant competencies. You can do it!

About the Module

This module was designed and written with you in mind. It is here to help you master
about Parallelogram and its Family. The scope of this module permits it to be used
in many different learning situations. The language used recognizes the diverse
vocabulary level of students. The lessons are arranged to follow the standard sequence
of the course. But the order in which you read them can be changed to correspond
with the textbook you are now using.

The module is divided into 4 lessons, namely:

Lesson 1 – Introduction to Parallelogram: Its Properties
Lesson 2 – Special Parallelogram: Rectangle and Its Properties
Lesson 3 – Special Parallelogram: Square and Its Properties
Lesson 4 – Special Parallelogram: Rhombus and Its Properties

After going through this module, you are expected to:

• determine the conditions that make a quadrilateral a parallelogram;

• use properties to find measures of angles, sides and other quantities involving
parallelograms; and
• prove theorems on the different kinds of parallelograms (rectangle, square and
rhombus).

1
 What I Know (Pre-Test)

Instructions: Read each item carefully and choose only the letter of the correct
answer. Write your chosen answer on a separate sheet of paper.
1. How do you call a parallelogram that has 4 congruent sides and 4 right angles?
a. square c. rectangle
b. trapezoid d. rhombus

2. Which of the following statement is FALSE?

a. The diagonals of a rectangle are congruent.
b. The diagonals of an isosceles trapezoid are congruent.
c. The diagonals of a square are perpendicular and bisect each other.
d. The diagonals of a rhombus are congruent and perpendicular to each other.
P A
3. What is the value of 𝑏 if the rectangle PALE
1
has diagonals 𝐴𝐸 = 3𝑏 + 6 and 𝑃𝐿 = 6𝑏 − 12 ?
3
a. 16 c. 8
b. 12 d. 6 2
E L
4. Which of the following should be the measure of angle 3 on rectangle PALE when
the measure of angle 1 is 55°?
a. 70° b. 80° c. 110° d. 130°
P E
5. Square PEAR has the measures 𝑃𝐸 = 6𝑥 − 2 and
𝐴𝑅 = 4𝑥 + 6, find the exact measure of side PE.
a. 16 c. 22
b. 20 d. 28
R A
6. The perimeter of the square PEAR is 36 cm. Find the length of diagonal ER.
a. 6√2 b. 9√2 c. 12√2 d. 14√2
M A
7. What is the measure of angle MNE in rhombus
MANE with center R if 𝑚∡AMN = 34°? R
a. 34° c. 68°
b. 46° d. 86°
E N
8. What is the measure of diagonal MN of rhombus MANE if 𝑀𝑅 = 13?
a. 18 b. 22 c. 26 d. 28

9. What kind of angles creates perpendicular lines?

a. vertical angles b. acute angles c. obtuse angles d. right angles
F A
10. Which of the following is the exact measure of the diagonal
2𝑥 + 7
TA in parallelogram FAST?
a. 8 c. 35 3𝑥 − 1
b. 31 d. 46
T S

2
 Lesson
Introduction to Parallelogram:
1 Its Properties

What I Need to Know

At the end of this lesson, you are expected to:
o determine the conditions that make a quadrilateral a parallelogram; and
o use properties to find measures of angles, sides and other quantities
involving parallelograms.

What’s In
❖ Flashback
To recall a lesson in grade 8 during fourth quarter, you have
learned how to prove properties of parallel lines cut by a
transversal through various examples given by your teacher.

Try to re-examine your learning on transversal with a follow-up

activity with the help of the concept that is shown on the box
below.

Parallel Lines Cut by a Transversal

A diagonal line that passes through a b
two or more parallel lines is called b a
transversal line. And this transversal line
a b
creates pairs of congruent alternate interior
and exterior angles. b a

Missing Diagram and Parts Explanation

Values
✓ 𝒂 = 𝟏𝟑𝟎° since vertical angles,
𝑎 𝑎𝑛𝑑 130°, are congruent
1. a = _______ 𝟏𝟑𝟎° ✓ 𝒃 = 𝟓𝟎° since adjacent angles
b = _______ are supplementary to each
b a other
c = _______ ✓ 𝒄 = 𝟏𝟑𝟎° since alternate angles,
c e 𝑎 𝑎𝑛𝑑 𝑐, are congruent
d = _______
d ✓ 𝒅 = 𝟏𝟑𝟎° since vertical angles,
e = _______ 𝑐 𝑎𝑛𝑑 𝑑, are congruent
✓ 𝒆 = 𝟓𝟎° since alternate angles,
𝑏 𝑎𝑛𝑑 𝑒, are congruent

3
 What’s New

❖ Level Up!

So, how was it answering the activity above? Were you able to
get the same answer as provided on the table above? Were the
solutions able to guide you in recalling the past lesson on
transversal line? Or do you have any way of solving the
missing values in the given problem above?

If you were able to get it right, I guess you have perfectly

understood that lesson way back in Grade 8.

Based on the activity above, kindly give your idea on these following questions.
1. How can you relate the concept on transversal to our new lesson on
quadrilaterals which is a close geometrical figure?
2. What are the conditions that will prove a quadrilateral to be a parallelogram?
3. How are angles, sides and other quantities relating to parallelograms
measured?

What Is It

❖ What should we know about quadrilaterals?

A quadrilateral is a four-sided closed geometrical figure that is defined by its

different properties involving its sides and angles. It can also be categorized into 3 sub-
families namely; parallelograms, trapezoids and kites.

So, the first family of quadrilaterals that you will learn and discover is the family
of parallelograms.

❖ What should I know then?

For you to determine if the given quadrilateral is a parallelogram, look at these

conditions or properties pertaining to PARALLELOGRAM.

Properties of Parallelogram
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary. c a
4. Diagonals are not congruent, but it bisects d b
7 f
each other and not the opposite angles.
e e
5. Each diagonal creates four pairs of alternate
f
interior angles (𝑎 𝑎𝑛𝑑 𝑏)𝑎𝑛𝑑 (𝑐 𝑎𝑛𝑑 𝑑) b d
6. Diagonals create two pairs of congruent a c
triangles.
7. Diagonals create two pairs of vertical angles
in the middle (𝑒 𝑎𝑛𝑑 𝑓).

4
To clearly visualize the said properties of a PARALLELOGRAM, study a few samples to
help you measure the angles, sides and other quantities pertaining to it.

1. Find the indicated values of this

parallelogram. a ✓ m∡𝒚 = 𝟔𝟖° since consecutive angles are
𝑥° supplementary to each other
✓ m∡𝒙 = 𝟏𝟏𝟐° since opposite angles are congruent
7 b to each other
✓ 𝒂 = 𝟏𝟐 since opposite sides are congruent
𝑦° 112° ✓ 𝒃 = 𝟕 since opposite sides are congruent
12
2. Find the indicated values of this
parallelogram. ✓ m∡𝒚 = 𝟑𝟑° since vertical angles are congruent to
each other
✓ m∡𝒛 = 𝟏𝟒𝟕° since adjacent angles are
supplementary to each other
10 c
✓ 𝒂 = 𝟏𝟎 since bisected diagonal have equal measure
7 𝑦° in length
33° b
15 ✓ 𝒃 = 𝟕 since opposite sides are congruent
𝑧° a
✓ 𝒄 = 𝟏𝟓 since bisected diagonal have equal measure
in length
✓
3. Find the indicated values of this
✓ m∡𝒛 = 𝟕𝟎° since alternate interior angles are
parallelogram.
congruent
42° 4𝑥° ✓ 𝒚 = 𝟏𝟒 since 42° = 3𝑦° because alternate interior
70° angles are congruent
4𝑎 𝑎 + 12 42 3𝑦
=
3 3
𝑧° 14 = 𝑦
𝑤° 3𝑦° ✓ 𝒙 = 𝟏𝟕 since the sum of the angles in a triangle is
180° and by solving for 𝑥,
✓ 𝒂 = 𝟒 since opposite sides are congruent 42° + 𝑧° + 4𝑥° = 180°
wherein by solving, 42 + 70 + 4𝑥 = 180
4𝑎 = 𝑎 + 12 112 + 4𝑥 = 180
4𝑎 − 𝑎 = 12 4𝑥 = 180 − 112
3𝑎 12 4𝑥 68
= =
3 3 4 4
𝒂=𝟒 𝑥 = 17
And to get the exact measurement of its ✓ m∡𝒘 = 𝟔𝟖° since opposite angles are congruent
side, you will plug-in the value of 𝑎 = 4 to wherein by solving,
4𝑎 = 𝑎 + 12 4𝑥° = 𝑤°
4(4) = 4 + 12 4(17) = 𝑤°
𝟏𝟔 = 𝟏𝟔 68° = 𝑤°

4. Find the indicated values of this

✓ 𝒚 = 𝟒𝟏 since opposite angles are congruent
parallelogram.
wherein by solving,
3𝑦° 3𝑦 123
= 𝑡ℎ𝑒𝑛, 𝑦 = 41
3 3
✓ 𝒙 = 𝟑𝟏 since consecutive angles are
supplementary wherein by solving,
2𝑥 − 5 + 123 = 180
(2𝑥 − 5)° 123° 2𝑥 = 180 − 118 𝑡ℎ𝑒𝑛, 𝑥 = 31

5
Are the examples clear enough to illustrate the properties of a parallelogram? How are
missing angles, sides and other quantities being solved? If there are no questions, you
can now answer the next activity.

What’s More

Activity 1: NOW IT’S YOUR TURN!

Instruction: Refer to the given figure below and use the properties to measure the
angles, sides and other quantities pertaining to parallelogram. Show your complete
solution on your activity notebook.

1. Refer to the figure below to find the 2. Refer to the figure below to find the
indicated values. indicated values.

3𝑥° 5𝑧°
85°
7𝑎 4𝑎 + 21 𝑎° 42°
𝑏°
𝑦°
𝑤° 45°

3. Refer to the figure below to find the 4. Refer to the figure below to find the
indicated values. indicated values.

5𝑦° 105° 31°

𝑦°

44°
35° (14𝑥 + 5)° 𝑧° 𝑥°

What I Need to Remember

You can use the properties of a parallelogram to measure the angles, sides and
other quantities relating to it.

Properties of Parallelogram
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary.
4. Diagonals are not congruent, but it bisects each other and not the opposite
angles.
5. Each diagonal creates four pairs of alternate interior angles
(𝑎 𝑎𝑛𝑑 𝑏)𝑎𝑛𝑑 (𝑐 𝑎𝑛𝑑 𝑑)
6. Diagonals create two pairs of congruent triangles.
7. Diagonals create two pairs of vertical angles in the middle (𝑒 𝑎𝑛𝑑 𝑓).

6
 Lesson
Special Parallelograms:
2 Rectangle and Its Properties

What I Need to Know

At the end of this lesson, you are expected to:

o prove theorems on the different kinds of parallelogram (rectangle)

What’s In
❖ Flashback
To recall module 1 of this quarter, you have learned how to prove
theorems regarding parallelograms through various examples
using its properties to find the measures of angles, sides and
other quantities involving parallelograms.

And, here are some of the properties of the parallelogram which

were derived from the various theorems relating to it as being
mentioned in lesson 1 of this quarter as shown in the table below.
Properties of Parallelogram
1. The opposite sides are parallel and congruent to each other.
2. The opposite angles are congruent.
3. The diagonals are not the same in length but, it bisects each other.
4. The diagonal creates 2 pairs of alternate interior angles.
5. Any pair of consecutive angles are supplementary but, not with opposite
angles.
6. Each diagonal separates the parallelogram into two congruent triangles.

Study the examples on the table below on how the measures of the angles, sides
and other quantities involving parallelograms were derived. Also, read the
explanation beside each figure to fully understand the process.

Missing Values Diagram and Parts Explanation

a ✓ 𝒂 = 𝟏𝟔 and 𝒃 = 𝟔 since opposite
sides are parallel and congruent to
1. a = _______ 𝑥° each other.
b = _______ 6 ✓ 𝒙 = 𝟏𝟎𝟖° since opposite angle are
b
congruent
x = _______ 𝑦° 108° ✓ 𝒚 = 𝟕𝟐° since the angle is
supplementary to the consecutive
y = _______ 16 angle 108°

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 Missing Values Diagram and Parts Explanation
✓ 𝒛 = 𝟔𝟒° and 𝒚 = 𝟏𝟐 by solving since
2. w = _______ 36 = 3𝑦° by reason that a diagonal
in a parallelogram creates 2 pairs
36° 4𝑥°
x = _______ 64° of alternate interior angles.
✓ To find the value of x, it is given
y = _______ that 𝑧 = 64° and the sum of the
𝑧° angles in a triangle is 180°, and if
z = _______ 𝑤° 3𝑦°
you add 36° + 64° + 4𝑥° = 180°, then
algebraically, 4𝑥 = 180 − 100 and to
simplify the equation, it would give
us the value of 𝒙 = 𝟐𝟎.
✓ While 𝒘 = 𝟖𝟎° since 4𝑥° = 4(20) of
which opposite angles are
congruent by nature.
✓ Likewise, the diagonal separates
the parallelogram into 2 congruent
triangles since their angles are
congruent.

✓ knowing that the diagonals are not

3. m = _______ of the same length but bisects to
9 3𝑚 each other, then 3𝑚 = 𝑚 + 8 and
n = _______ 2𝑛 − 1 = 9. To solve it algebraically
𝑚+8 2𝑛 − 1 in a separate manner,
3𝑚 − 𝑚 = 8 2𝑛 = 9 + 1
2𝑚 = 8 2𝑛 = 10
𝒎=𝟒 𝒏=𝟓

What’s New
❖ Level Up!

So, how was it answering the activity above? Were you able to
get the same answer as provided on the table above? Were the
solutions able to guide you in recalling the past lesson on
parallelogram? Or do you have any way of solving the missing
values in the given problem above?

If you were able to get it right, I guess you have perfectly

understood Lesson 1 of this module especially, if you were
able to arrive at the same answer without looking at the given
explanation on the table.

Based on the activity above, answer the following questions relating to other forms of
parallelogram.
1. What other geometric figure represents the family of parallelograms?
2. Since it’s a family of parallelogram, what significant features does it have as
compared to a regular parallelogram?
3. How are we going to measure the angles, sides and other quantities relating to
a rectangle?
What Is It
8
❖ How should I do it?

To answer those questions in What’s New, look at these properties of a RECTANGLE

as a member of the family of parallelogram.

Properties of Rectangle
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary. a a b
b
4. Diagonals are congruent and bisect each other, d
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not the opposite angles. c c
5. Each diagonal creates two pairs of alternate d
interior angles (𝑎 𝑎𝑛𝑑 𝑏) b b
a a
6. Diagonals create two pairs of isosceles triangles.
7. Diagonals create two pairs of vertical angles in the
middle (𝑐 𝑎𝑛𝑑 𝑑).
8. It has four right angles.

So, with the given properties of a RECTANGLE, what common features does it have as
compared to a regular parallelogram? Would you like to give your observations on the
figure itself and properties stated above? Are there features which are not found in a
regular parallelogram?

To clearly visualize the said properties of a RECTANGLE, study a few samples to help
you measure the angles, sides and other quantities pertaining to it.
Refer to the figure below and find the measure of each quantity.
1. If AE = 5, BC = 6, and DC = 8, find AC, BD, AD and AB.
A B
✓ AC=10 since AE is half of AC and BD=10
AC = 10
because AC=BD since the 2 diagonals
E BD = 10 are congruent.
AD = 6 ✓ BC=AD then, AD=6 and DC=AB then,
AB = 8 AB=8 since opposite sides are congruent
D C and are parallel.

2. If 𝐵𝐷 = 3𝑥 − 7 and 𝐶𝐴 = 𝑥 + 5, find BD, ED, CA and AE.

A B
BD = 11
✓ Since 𝐵𝐷 = 𝐴𝐶, then by solving
E ED = 5.5
algebraically, you will get 𝒙 = 𝟔.
CA = 11
✓ To solve BD and AC, you just substitute
AE = 5.5 the value of 𝑥 = 6 to 𝐵𝐷 = 3𝑥 − 7 and
D C 𝐴𝐶 = 𝑥 + 5 that would give you 11.
✓ To solve for ED and AE which are just
Since BD = AC, then
half of each diagonal since point E is the
3𝑥 − 7 = 𝑥 + 5 midpoint of each diagonals BD and AC,
3𝑥 − 𝑥 = 5 + 7 you just get the half of 11 since
2𝑥 = 12 diagonals bisect each other.
𝒙=𝟔

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 Rectangle BOYS has diagonals BY and OS, which intersect at X.
3. If m∡𝑋𝑂𝐵 = 70°, then find the measures of the following angles:
a. YSO
✓ m∡𝒀𝑺𝑶 = 𝟕𝟎° since diagonal creates a pair of
b. BSO
alternate interior angles.
c. SBO
✓ m∡𝑩𝑺𝑶 = 𝟐𝟎° since angles BSO and YSO are
d. XBO
complementary angles, equal to 90°
e. OXB
f. YXO ✓ m∡𝑺𝑩𝑶 = 𝟗𝟎° since it is a corner angle which is 90°
✓ m∡𝑿𝑩𝑶 = 𝟕𝟎° since it is pair with ∡𝑋𝑂𝐵 that are
base angles of an isosceles triangle.
✓ m∡𝑶𝑿𝑩 = 𝟒𝟎° since it supplements the other base
B O angles 𝑋𝐵𝑂 𝑎𝑛𝑑 𝑋𝑂𝐵 to make it 180°.
At the same time, angles 𝑂𝑋𝐵 𝑎𝑛𝑑 𝑌𝑋𝑆 are
X vertical angles.
✓ m∡𝒀𝑿𝑶 = 𝟏𝟒𝟎° since it is supplementary to
∡𝑂𝑋𝐵 = 40° that is already half the circle.
At the same time, angles 𝑌𝑋𝑂 𝑎𝑛𝑑 𝑆𝑋𝐵 are
S Y another set of vertical angles

Refer to rectangle FACE below to find the measures of the numbered and lettered
angles. ✓
4. If 𝑚∡𝐴𝐹𝐶 = 60°, find 𝑚∡1, 𝑚∡2, 𝑚∡3, 𝑚∡𝐴 𝑎𝑛𝑑 𝑚∡𝐸𝐹𝐶.
✓ 𝒎∡𝟏 = 𝟔𝟎° since the two angles are
F E alternate interior angles
3 ✓ 𝒎∡𝟐 = 𝟑𝟎° since angles 1 and 2 are
𝟔𝟎° complementary angles, equal to 90°
✓ 𝒎∡𝟑 = 𝟗𝟎° since it is a corner angle
1 that measures 90° that is congruent
2 with 𝑚∡𝐴 which is also 90°.
A C ✓ 𝒎∡𝑬𝑭𝑪 = 𝟑𝟎° since it is congruent
with 𝑚∡2 which is considered
alternate interior angles.

Refer to rectangle BEARS below to answer the missing angle.

5. If 𝑚∡𝑅𝐵𝐴 = 𝑥 + 9 𝑎𝑛𝑑 𝑚∡𝐸𝐴𝐵 = 2𝑥 − 44, find 𝑚∡𝐸𝐴𝐵.
✓ Since the two given angles are alternate interior
angles, then the 2 angles are congruent 𝒎∡𝑹𝑩𝑨 ≅
B 𝒎∡𝑬𝑨𝑩
R
✓ By solution, we get;
S 𝑚∡𝑅𝐵𝐴 = 𝑚∡𝐸𝐴𝐵
𝑥 + 9 = 2𝑥 − 44
9 + 44 = 2𝑥 − 𝑥
𝟓𝟑 = 𝒙
E A ✓ To get the exact measurement of the angle, substitute
the value of x to the equation 𝑚∡𝐸𝐴𝐵 = 2𝑥 − 44
when 𝑥 = 53 will give you
𝒎∡𝑬𝑨𝑩 = 𝟐(𝟓𝟑) − 𝟒𝟒 = 𝟔𝟐°

Are the examples clear enough to illustrate the properties of a rectangle and how its
missing angles, sides and other quantities have been solved? If there are no questions,
you can now answer the next activity.

10
 What’s More

Activity 2: NOW IT’S YOUR TURN!

Instruction: Refer to the given figure below and use the properties to measure the
angles, sides and other quantities pertaining to rectangle. Show your complete
solution on your activity notebook.

1. LMNO is a rectangle, if LM=16, MN=12, and 𝑚∡1 = 60°, find the following:
a. ON = ______ f. m∡2 = ________ L M
2 3
b. OL = ______ g. OX = ______ 1 4
X
c. LN = ______ h. m∡3 = ________
d. LX = ______ i. m∡4 = ________
e. m∡LON= ______ j. m∡LXM = ________ O N

Refer to the diagram on the right side, to answer items 2 – 4.

2. If 𝑈𝑍 = 𝑥 + 21 𝑎𝑛𝑑 𝑍𝑆 = 3𝑥 − 15, R S
find US.
3. If 𝑚∡SUT = 3x + 6 and m∡RUS = 5x − 4, Z
𝑓𝑖𝑛𝑑 𝑚∡SUT.
4. If 𝑚∡RSU = x + 41 an d m∡TUS = 3x + 9,
find m∡RSU. U T

What I Need to Remember

You can use the properties of a rectangle to measure the angles, sides and other
quantities relating to it.

Properties of Rectangle
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary.
4. Diagonals are congruent and bisect each other, not the opposite angles.
5. Each diagonal creates two pairs of alternate interior angles (𝑎 𝑎𝑛𝑑 𝑏)
6. Diagonals create two pairs of isosceles triangles.
7. Diagonals create two pairs of vertical angles in the middle (𝑐 𝑎𝑛𝑑 𝑑).
8. It has four right angles.

11
 Lesson
Special Parallelograms:
3 Square and Its Properties

What I Need to Know

At the end of this lesson, you are expected to:
o prove theorems on the different kinds of parallelogram (square)

What’s In
❖ Flashback
Have a short recall of rectangle and its properties by answering
this short activity.

1. Rectangle GALS has diagonals GL and AS.

If 𝐺𝐿 = 3𝑥 + 6 𝑎𝑛𝑑 𝐴𝑆 = 5𝑥 − 18, G A
then what is the value of x?
Also, find the measure of
diagonal GL?

S L
2. Quadrilateral RSTU is a rectangle
with point Z as intersection. R
If 𝑅𝑍 = 3𝑥 + 8 𝑎𝑛𝑑
S
𝑍𝑆 = 6𝑥 − 28, Z
find ZS.

U T

What’s New
❖ Level Up!

So, how was it answering the activity above? Were you able to
get the answer as part of recalling your past lesson on
rectangle as a parallelogram? Or do you have any way of
solving the missing values in the given problem above?

Based on the activity above, answer the following questions

relating to other forms of parallelogram other than rectangle.

12
 1. What other geometric figure represents the family of parallelograms other than
rectangle?
2. Since it’s a family of parallelogram, what significant features does it have as
compared to a regular parallelogram?
3. How are we going to measure the angles, sides and other quantities relating to
a square?

What Is It

❖ How should I do it?

To answer those questions in What’s New, look at these properties of a SQUARE as a

member of the family of parallelogram.

Properties of Square
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent. a a
a a
3. Consecutive angles are supplementary.
7
4. Diagonals are congruent and bisect each other, b
including the opposite angles into 2 pairs of 45°. b b
5. Diagonals create 4 congruent isosceles triangles. b
6. Diagonals create 4 congruent vertical angles since
the diagonals are perpendicular to each other. a a
7. It has four right angles. a a
8. It has four congruent sides.
9. Diagonal of a square is equal to the length of its
side multiplied by the square root of 2.

So, with the given properties of a SQUARE, what common features does it have as
compared to a rectangle? Would you like to give your observations on the figure itself
and properties stated above? Are there features which are not found in a rectangle?

To clearly visualize the said properties of a SQUARE, study a few samples to help you
measure the angles, sides and other quantities pertaining to it.
1. MATH is a square with intersection at S.
a. If 𝑀𝐴 = 8, 𝑡ℎ𝑒𝑛 𝐴𝑇 =_______ d. 𝑖𝑓 𝐻𝑆 = 2, 𝑡ℎ𝑒𝑛 𝐻𝐴 = ______ and 𝑀𝑇 = _____
b. 𝑚∡HST = _______ e. 𝑚∡HMT =_______
c. 𝑚∡MAT = _______
✓ 𝑨𝑻 = 𝟖 since square has 4 congruent sides and are also
M A parallel
7 ✓ 𝒎∡𝐇𝐒𝐓 = 𝟗𝟎° since diagonals are perpendicular to each
other
S ✓ 𝒎∡𝐌𝐀𝐓 = 𝟗𝟎° since square has 4 right angles
1
✓ 𝑯𝑨 = 𝟒 since diagonals bisect each other and 𝐻𝑆 = 𝐻𝐴
2
✓ 𝑴𝑻 = 𝟒 since diagonals are congruent
✓ 𝒎∡𝐇𝐌𝐓 = 𝟒𝟓° since diagonals bisect opposite angles
H T

13
 2. Use square ABCD and the given information to find each value.
a. If 𝑚∡AEB = (3𝑥)° , a. ∡𝐴𝐸𝐵 is a right angle since it’s a product of
find the value of x. 2 diagonals that are perpendicular to each
b. If 𝑚∡BAC = (9𝑥)° , other, then by solution; we divide 90 by 3
find the value of x. 90 3𝑥
= then, 𝟑𝟎 = 𝒙
c. If 𝐴𝐵 = 2𝑥 + 4 𝑎𝑛𝑑 𝐶𝐷 = 3𝑥 − 5, 3 3
b. ∡𝐵𝐴𝐶 is an acute angle which is a product
find the value of x, side BC and
of a bisection of a diagonal, then
diagonal BD.
𝑚∡𝐵𝐴𝐶 = 45°. So, by solving, we divide
d. The perimeter of the square 45 by 9
is 32cm. Find the length of 45 9𝑥
the diagonal BD. = then, 𝟓 = 𝒙
9 9
A B c. Side AB and CD are congruent, then
2𝑥 + 4 = 3𝑥 − 5
4 + 5 = 3𝑥 − 2𝑥
E 𝟗=𝒙
Side 𝐴𝐵 = 𝐶𝐷 = 𝐴𝐷 = 𝐵𝐶, then
𝑩𝑪 = 2𝑥 + 4 = 2(9) + 4 = 𝟐𝟐

To find diagonal BD, we solve it by using

D C Pythagorean theorem 𝑎2 + 𝑏 2 = 𝑐 2
wherein diagonal BD represents side c and
d. Perimeter = 4 (sides) sides BC and CD are sides 𝑎 𝑎𝑛𝑑 𝑏,
32 = 4𝑠 respectively. Substitute the values of BC
𝟖=𝒔 and CD, then solve for BD;
If one side of the square is 8cm, 𝑎2 + 𝑏 2 = 𝑐 2
(𝐵𝐶) + (𝐶𝐷)2 = (𝐵𝐷)2
2
then, its diagonal should be
222 + 222 = 𝑐 2
𝑩𝑫 = 𝟖√𝟐.
√484 ∙ 2 = √𝑐 2
𝟐𝟐√𝟐 = 𝒄 = 𝑩𝑫

What’s More

Activity 3: NOW IT’S YOUR TURN!

Instruction: Refer to the given figure below and use the properties to measure the
angles, sides and other quantities pertaining to square. Show your complete solution
on your activity notebook.

1. EFGH is a square, if EF=10, find the following:

E F
a. FG = ______ f. m∡EIF = ________
7
b. m∡EFG = ______ g. m∡1 = ______ I
c. EG = ______ h. m∡3 = ________
d. EI = ______ i. HF = ________ 1 4
H 2 3 G
e. IF = ______ j. m∡FHG = ________

14
Refer to the diagram on the right side, to answer items 2 – 5.
FISH is a square with intersection at T when 𝐼𝑆 = 6. F I
2. find IH.
3. If 𝑚∡HTS = (5𝑥)° , find x. T

4. If 𝐹𝐼 = 3𝑥 + 6 𝑎𝑛𝑑 𝐼𝑆 = 4𝑥 − 2, find:
a. 𝑥 b. side FI
5. The perimeter of the square is 48cm. H S
Find the length of the diagonal IH.

What I Need to Remember

You can use the properties of a square to measure the angles, sides and other
quantities relating to it.

Properties of Square
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary.
4. Diagonals are congruent and bisect each other, including the opposite
angles into 2 pairs of 45°.
5. Diagonals create 4 congruent isosceles triangles.
6. Diagonals create 4 congruent vertical angles since the diagonals are
perpendicular to each other.
7. It has four right angles.
8. It has four congruent sides.
9. Diagonal of a square is equal to the length of its side multiplied by the
square root of 2.

15
 Lesson
Special Parallelograms:
4 Rhombus and Its Properties

What I Need to Know

At the end of this lesson, you are expected to:
o prove theorems on the different kinds of parallelogram (rhombus)

What’s In
❖ Flashback
Have a short recall of square and its properties by answering this
short activity.

1. MORE is a square with intersection at S when 𝑀𝑂 = 7,

find the value of: M O
a. OR
b. OE
S
c. 𝑚∡MEO
d. 𝑚∡MER
2. The perimeter of the square
MORE is 60cm. Find the
length of its diagonal EO. E R

What’s New
❖ Level Up!

So, how was it answering the activity above? Were you able to
get the answer as part of recalling your past lesson on square
as a parallelogram? Or do you have any way of solving the
missing values in the given problem above?

Based on the activity above, answer the following questions

relating to other forms of parallelogram other than rectangle
and square.

1. What other geometric figure represents the family of parallelograms other than
rectangle and square?
2. Since it’s a family of parallelogram, what significant features does it have as
compared to a regular parallelogram?
3. How are you going to measure the angles, sides and other quantities relating to
a rhombus?

16
 What Is It

❖ How should I do it?

To answer those questions in What’s New, try to look at these properties of a

RHOMBUS as a member of the family of parallelogram.

Properties of Rhombus
1. Opposite sides are congruent and parallel.
b a
2. Opposite angles are congruent.
b a
3. Consecutive angles are supplementary.
4. Diagonals are NOT congruent but, bisect each c
other. c c
5. Diagonals bisect opposite angles (𝑎 𝑎𝑛𝑑 𝑏). c
6. Diagonals create 4 congruent vertical angles since a b
the diagonals are perpendicular to each other (𝑐). a b
7. It has four congruent sides.

So, with the given properties of a RHOMBUS, what common features does it have as
compared to a rectangle and square? Would you like to give your observations on the
figure itself and properties stated above? Are there features which are not found in a
rectangle and square?

To clearly visualize the said properties of a RHOMBUS, let us study a few samples to
help us measure the angles, sides and other quantities pertaining to it.

1. Given RSTV as a rhombus with 1

center W and 𝑚∡RST = 64°. Find a. 𝒎∡𝐑𝐒𝐖 = 𝟑𝟐° since ∡RSW = ∡RST
2
the measure of the b. 𝒎∡𝐒𝐑𝐕 = 𝟏𝟏𝟔° since consecutive angles
following. S T are supplementary, wherein
a. 𝑚∡RSW ∡RST + ∡SRV = 180°
b. 𝑚∡SRV c. 𝒎∡𝐑𝐕𝐓 = 𝟔𝟒° since opposite angles are
W congruent, ∡RST = ∡RVT
c. 𝑚∡RVT
d. 𝒎∡𝐒𝐖𝐓 = 𝟗𝟎° since diagonals that are
d. 𝑚∡SWT perpendicular create 4 congruent vertical
R V angles

2. In rhombus DLMP with center O,

1
𝐷𝑀 = 24, 𝑚∡LDO = 43°, a. 𝑶𝑴 = 𝟏𝟐 since 𝑂𝑀 = 𝐷𝑀
2
and DL = 13. Find the b. 𝒎∡𝐃𝐎𝐋 = 𝟗𝟎° since diagonals that are
measure of the following. perpendicular create 4 congruent vertical
a. OM D L
angles
b. m∡DOL c. 𝒎∡𝐃𝐋𝐎 = 𝟒𝟕° since the sum of
c. m∡DLO ∡LDO + DLO + ∡DOL = 180°
O
d. 𝐦∡𝐃𝐌𝐋 = 𝟒𝟑° since it is congruent
d. m∡DML
with ∡LDO.
e. DP e. 𝑫𝑷 = 𝟏𝟑 since all sides are congruent
P M

17
 3. Quadrilateral DKLM is a rhombus
a. Segment 𝐷𝐴 = 𝐴𝐿, then their
with center A. Find the measures are the same and by solving,
measure of the following if: 4𝑥 = 5𝑥 − 3
3 = 5𝑥 − 4𝑥
a. 𝐷𝐴 = 4𝑥 𝑎𝑛𝑑 𝟑=𝒙
𝐴𝐿 = 5𝑥 − 3, K
Then, substitute the value of 𝑥 = 3 to
D
𝑫𝑨 = 4𝑥 = 4(3) = 𝟏𝟐 and since
𝑓𝑖𝑛𝑑 𝐷𝐿 𝐷𝐴 = 𝐴𝐿, both segments are 12,
b. 𝐷𝐾 = 6𝑦 + 4 𝑎𝑛𝑑 therefore, 𝑫𝑳 = 𝟐𝟒
A b. Sides 𝐷𝐾 = 𝐾𝐿 = 𝐿𝑀 = 𝐷𝑀, then
𝐾𝐿 = 5𝑦 + 8, 6𝑦 + 4 = 5𝑦 + 8
𝑓𝑖𝑛𝑑 𝐿𝑀 6𝑦 − 5𝑦 = 8 − 4
M L
𝒚=𝟒
Then, substitute the value of 𝑦 = 4 to
𝐷𝐾 = 6𝑦 + 4 since 𝐷𝐾 = 𝐿𝑀, then
4. The diagonals of rhombus MINE 𝑳𝑴 = 𝟔(𝟒) + 𝟒 = 𝟐𝟖
with center S are 10 and 24cm. Find
the length of each side of
the rhombus.
M I The two diagonals are not of the same length, but they bisect
each other such that segment MN is shorter than segment EI,
then 𝑀𝑆 𝑎𝑛𝑑 𝑁𝑆 have measures of 5cm each while
S 𝐸𝑆 𝑎𝑛𝑑 𝐼𝑆 have measures of 12cm each.

And to solve for the 4 congruent sides of the rhombus, we can

use one small triangle and its parts to solve for the 3rd missing
E N
side using Pythagorean theorem, 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 .
x By solving, you will get
M I
52 + 122 = 𝑥 2
25 + 144 = 𝑥 2
5 12 169 = 𝑥 2
√169 = √𝑥 2
S 𝟏𝟑 = 𝒙
Therefore, each side of the rhombus measures 13 cm.

What’s More

Activity 4: NOW IT’S YOUR TURN!

Instruction: Refer to the given figure below and use the properties to measure the
angles, sides and other quantities pertaining to rhombus. Show your complete solution
on your activity notebook.
W X
1. WXYZ is a rhombus. If 𝑊𝑋 = 5, 𝑊𝑌 = 6, 𝑍𝑋 = 8 2 3
and 𝑚∡WXY = 60°, find the following: 1
4
a. XY = _____ d. m∡2 = _______ g. WO = _____ O
b. m∡ZWX = _____ e. m ∡3 = _______ h. OX = _____
c. m∡1 = _____ f. m∡4 = _______ i. YZ = _____
Z Y

18
 Refer to the diagram on the right side, to answer items 2 – 4.
2. FEAR is a rhombus with intersection at S when
F E
𝐹𝐸 = 5𝑥 − 6 𝑎𝑛𝑑 𝑅𝐴 = 7𝑥 − 12.
Find AE.
3. If 𝐹𝑆 = 6𝑥 − 9 𝑎𝑛𝑑 𝐴𝑆 = 3𝑥, find FA. S

4. The diagonals of rhombus FEAR

are 16 and 30 cm. find the length R A
of each side of the rhombus.

What I Need to Remember

You can use the properties of a rhombus to measure the angles, sides and other
quantities relating to it.

Properties of Rhombus
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary.
4. Diagonals are NOT congruent but, bisect each other.
5. Diagonals bisect opposite angles (𝑎 𝑎𝑛𝑑 𝑏).
6. Diagonals create 4 congruent vertical angles since the diagonals are
perpendicular to each other (𝑐).
7. It has four congruent sides.

What I Can Do
❖ Extra Challenge

Use rectangle STUV and the given information to find each measure below.

1. 𝑚∡3 = _____ If 𝑚∡4 = 30°, 𝑓𝑖𝑛𝑑 𝑚∡3

2. 𝑚∡4 = _____ If 𝑚∡6 = 57°, 𝑓𝑖𝑛𝑑 𝑚∡4

3. 𝑚∡2 = _____ If 𝑚∡5 = 26°, 𝑓𝑖𝑛𝑑 𝑚∡2

4. KT = ____ If 𝑆𝐾 = 15, 𝑓𝑖𝑛𝑑 𝐾𝑇

5. TU = ____ If 𝑆𝑈 = 15 𝑎𝑛𝑑 𝑆𝑇 = 12, 𝑓𝑖𝑛𝑑 𝑇𝑈

6. SV = ____ If 𝐾𝑉 = 5 𝑎𝑛𝑑 𝑆𝑇 = 8, 𝑓𝑖𝑛𝑑 𝑆𝑉

19
 Assessment (Post Test)

Instructions: Choose the letter of the correct answer. Write your chosen answer on
a separate sheet of paper.
1. How do you call a parallelogram that has 2 pairs of congruent and parallel sides?
a. square c. rectangle
b. trapezoid d. rhombus

2. Which of the following statement is FALSE?

a. The diagonals of a rectangle are congruent.
b. The diagonals of an isosceles trapezoid are congruent.
c. The diagonals of a square are perpendicular and bisect each other.
d. The diagonals of a rhombus are congruent and perpendicular to each other.
P A
3. What would be the value of 𝑏 if the rectangle PALE
1
has diagonals 𝐴𝐸 = 8𝑏 − 6 and 𝑃𝐿 = 4𝑏 + 10 ?
3
a. 12 c. 6
b. 10 d. 4 2
E L
4. Which of the following should be the measure of angle 3 on rectangle PALE when
the measure of angle 1 is 40°?
a. 70° b. 80° c. 100° d. 120°
P E
5. Square PEAR has the measures 𝑃𝐸 = 6𝑥 − 2 and
𝐴𝑅 = 4𝑥 + 6, find the exact measure of side PE.
a. 12 c. 20
b. 22 d. 24
R A
6. The perimeter of the square PEAR is 24 cm. Find the length of diagonal ER.
a. 6√2 b. 9√2 c. 12√2 d. 14√2
M A
7. What is the measure of angle ANE in rhombus
MANE with center R if 𝑚∡AMN = 34°? R
a. 34° c. 68°
b. 46° d. 86°
E N
8. What is the measure of diagonal MN of rhombus MANE if 𝑀𝑅 = 11?
a. 18 b. 22 c. 26 d. 28

9. Perpendicular lines create what kind of angles?

a. vertical angles b. acute angles c. obtuse angles d. right angles
F A
10. Which of the following is the exact measure of the diagonal
2𝑥 + 7
TA in parallelogram FAST?
a. 8 c. 35 3𝑥 − 1
b. 31 d. 46
T S

20
 21
What’s More: Activity 2 What’s More: Activity 2 What’s More: Activity 4
1. a. 16 f. 30° 1. a. 5 f. 30°
b. 12 g. 10 4. 𝑚∡RSU = m∡TUS b. 120° g. 3
c. 20 h. 30° 𝑥 + 41 = 3𝑥 + 9 c. 60° h. 4
d. 10 i. 60° 41 − 9 = 3𝑥 − 𝑥 d. 60° i. 5
e. 90° j. 120° 32 = 2𝑥
e. 30°
𝟏𝟔 = 𝒙
2. 𝑈𝑍 = 𝑍𝑆 2. 𝐹𝐸 = 𝑅𝐴
then,
𝑥 + 21 = 3𝑥 − 15 5𝑥 − 6 = 7𝑥 − 12
𝑚∡RSU = 𝑥 + 41
21 + 15 = 3𝑥 − 𝑥 −6 + 12 = 7𝑥 − 5𝑥
36 = 2𝑥
𝒎∡𝐑𝐒𝐔 = 𝟏𝟔 + 𝟒𝟏 = 𝟓𝟕°
6 = 2𝑥
𝟏𝟖 = 𝒙 𝟑=𝒙
then, then, 𝐴𝐸 = 𝐹𝐸 = 5𝑥 − 6
𝑈𝑆 = 𝑈𝑍 + 𝑍𝑆 What’s More: Activity 4 𝐹𝐸 = 5(3) − 6
𝑈𝑆 = (𝑥 + 21) + (3𝑥 − 15)
then, 𝐹𝐴 = 𝐹𝑆 + 𝐴𝑆 𝐹𝐸 = 15 − 6
𝑈𝑆 = (18 + 21) + (3(18) − 15)
𝐹𝐴 = (6𝑥 − 9) + (3𝑥) 𝑭𝑬 = 𝟗
𝑈𝑆 = 39 + 39
𝑼𝑺 = 𝟕𝟖 𝐹𝐴 = 6(3) − 9 + 3(3) where, 𝑅𝐴 = 7𝑥 − 12
𝑭𝑨 = 𝟗 + 𝟗 = 𝟏𝟖 𝑅𝐴 = 7(3) − 12
3. 𝑚∡𝑆𝑈𝑇 + 𝑚∡𝑅𝑌𝑆 = 90° 𝑅𝐴 = 21 − 12
(3𝑥 + 6) + (5𝑥 − 4) = 90 𝑹𝑨 = 𝟗
8𝑥 + 2 = 90 4. 𝑠𝑖𝑑𝑒𝑠 𝑎 = 8, 𝑏 = 15, 𝑐 =?
3. 𝐹𝑆 = 𝐴𝑆
8𝑥 = 88 𝑐 = √82 + 152
𝒙 = 𝟏𝟏
6𝑥 − 9 = 3𝑥
𝑐 = √64 + 225 6𝑥 − 3𝑥 = 9
then, 𝑚∡𝑆𝑈𝑇 = 3𝑥 + 6
𝑚∡𝑆𝑈𝑇 = 3(11) + 6 𝑐 = √289 3𝑥 = 9
𝑚∡SUT = 33 + 6 𝒄 = 𝟏𝟕 𝒙=𝟑
𝒎∡𝐒𝐔𝐓 = 𝟑𝟗° then, one side of the
rhombus is 17cm.
What’s More: Activity 3
What’s More: Activity 1 2. 𝒂 = 𝟒𝟐° 𝒂𝒏𝒅 𝒃 = 𝟏𝟑𝟖° 1. a. 10 f. 90°
2𝑥 = 𝑥 + 9 b. 90° g. 45°
1. 𝒚 = 𝟖𝟓° 𝒂𝒏𝒅 𝒘 = 𝟓𝟎° 2𝑥 − 𝑥 = 9 c. 10√2 h. 45°
3𝑥 45 𝒙=𝟗
= d. 5√2 i. 10√2
3 3 3𝑦 = 𝑦 + 10
𝒙 = 𝟏𝟓 e. 5√2 j. 45°
3𝑦 − 𝑦 = 10 2. 𝑰𝑯 = 𝟔√𝟐
3𝑥 + 𝑦 + 5𝑧 = 180 2𝑦 = 10
3. 5𝑥 = 90 𝑡ℎ𝑒𝑛, 𝒙 = 𝟏𝟖
3(15) + 85 + 5𝑧 = 180 𝒚=𝟓
3. 35 + 14𝑥 + 5 = 180 4. a. 𝐹𝐼 = 𝐼𝑆
45 + 85 + 5𝑧 = 180
14𝑥 + 40 = 180 3𝑥 + 6 = 4𝑥 − 2
130 + 5𝑧 = 180
5𝑧 = 50 14𝑥 = 180 − 40 6 + 2 = 4𝑥 − 3𝑥
14𝑥 140 𝟖=𝒙
𝒛 = 𝟏𝟎 =
14 14 b. 𝐹𝐼 = 3𝑥 + 6
𝒙 = 𝟏𝟎
7𝑎 = 4𝑎 = 21 𝑭𝑰 = 𝟑(𝟖) + 𝟔 = 𝟑𝟎
5𝑦 145
7𝑎 − 4𝑎 = 21 𝑡ℎ𝑒𝑛, = 5. 𝑃 = 4𝑠
3𝑎 21 5 5
= 𝒚 = 𝟐𝟗 48 4𝑠
3 3 =
4. 𝒚 = 𝟒𝟒° , 𝒛 = 𝟑𝟏° 4 4
𝒂=𝟕
𝑎𝑛𝑑 𝑥 = 105° 𝟏𝟐 = 𝒔
then, 𝑰𝑯 = 𝟏𝟐√𝟐
required.
Remember: This portion of the module contains all the answers. Your HONESTY is
Answer Key
References
Books

Lasic-Calamiong, Lanilyn, et.al. “Understanding Mathematics Grade 9”

Vicarish Publication and Trading, Inc., 1946-A, F. Torres St., corner Diamante Ext.,
Pasigline Sta. Ana, Manila, Philippines. Copyright 2014 ISBN:978-971-689-571-1

Websites

Clip Arts
Are taken from phone app named BITMOJI

Congratulations!
You are now ready for the next module. Always remember the following:

1. Make sure every answer sheet has your

▪ Name
▪ Grade and Section
▪ Title of the Activity or Activity No.
2. Follow the date of submission of answer sheets as
agreed with your teacher.
3. Keep the modules with you AND return them at
the end of the school year or whenever
face-to-face interaction is permitted.

22