NEGROS ACADEMY

Tampocon II, Ayungon Negros Oriental

Division of Negros Oriental
Region VII

LEARNING MODULE
IN
MATHEMATICS 9
3rd Quarter

Name of student:___________________________________________
Section: ____________________________
Teacher: Mrs. Lourdes D. De Jesus, LPT
 QUADRILATERALS
This module shall focus on quadrilaterals that are parallelograms, properties of a parallelogram,
theorems on the different kinds of parallelogram, and problems involving parallelograms

I. Competencies
The learner…
 determines the conditions that make a quadrilateral a parallelogram.
 uses properties to find measures of angles, sides and other quantities involving
parallelograms.
 proves theorems on the different kinds of parallelogram (rectangle, rhombus, square).
 solves problems involving parallelograms.

II. Pre – Test

Find out how much you already know about this module. Encircle the letter of your
answer.

1. Which of the following statements is true?

a. Every square is a rectangle.

b. Every rectangle is a square.
c. Every rhombus is a rectangle.
d. Every parallelogram is a rhombus.

2. How do you describe any two opposite angles in a parallelogram?

a. They are always congruent.

b. They are supplementary.
c. They are complementary.
d. They are both right angles.

3. Which of the following quadrilaterals has diagonals that do not bisect each other?

a. Square
b. Rhombus
c. Rectangle
d. Trapezoid

4. What is the measure of angle M in rhombus HOME?

a. 75°
b. 90° H O
c. 105° 105 o
d. 180°

E M

5. Consecutive angles of a parallelogram are

a. complementary c. adjacent

b. supplementary d. congruent
 6. In rectangle KAYE, YO = 18 cm. Find the length of diagonal AE.

a. 6 cm K E
b. 9 cm
c. 18 cm O 18cm
d. 36 cm
A
Y
7. In quadrilateral RSTW, diagonals RT and SW are perpendicular bisectors of each other.

Quadrilateral RSTW must be a:

I. Rectangle
II. Rhombus
III. Square

a. I c. II and III
b. II d. I, II, and III

8. All of the following are properties of a parallelogram EXCEPT:

a. diagonals bisect each other. c. opposite angles are congruent

b. opposite sides are congruent. d. opposite sides are not parallel

9. What can you say about any two consecutive angles in a parallelogram?

a. They are always congruent.

b. They are always supplementary.

c. They are sometimes complementary.
d. They are both right angles.

10. Which of the following statements could be 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.

II. DISCUSSION

Quadrilateral is a four – sided polygon. The sum of the measures of the interior angles of a
closed quadrilaterals is 360 o. It is denoted by the symbol .

The Quadrilateral Family Tree

QUADRILATERAL

Parallelogram Trapezium Trapezoid

Rectangle Rhombus Kite Isosceles Right

trapezoid trapezoid

Square
Definition
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

A trapezoid is a quadrilateral with exactly one pair of opposite sides are parallel.

A trapezium is a quadrilateral that has no parallel sides.

A rectangle is a parallelogram with four right angles.

A rhombus is a parallelogram with four equal sides.

A square is a quadrilateral with four equal sides.

An isosceles trapezoid is a trapezoid in which the nonparallel sides are equal.

A right trapezoid is a trapezoid with two right angles.

A kite is a trapezium in which two pairs of adjacent sides are equal.

Task
On a construction paper draw each
quadrilaterals then cut each of them and paste it on a
short bond paper with its definition. You should
draw each figure based on its definition.

Properties of a Parallelogram

Parallelogram ABCD

A D Line AB is parallel to line DC,

therefore AB ≅ BC ,
meaning AB is approximately equal to DC.
Diagonal BD
Line AD is parallel to line BC,
B C Therefore AD ≅ BC
Diagonal AC meaning AD is approximately equal to BC.

Each diagonal separates a parallelogram into two equal triangles.

 For diagonal AC the two triangles are ABC and ADC, the two triangles are equal.
 For diagonal BD the two triangles are BAD and BCD, the two triangles are equal.
 The opposite angles of a parallelogram are equal.

A B
angle A is opposite to angle C,
therefore m A is equal to m C or
m A≅m C
m A read as measure of
D C angle A
m B≅m D
ANGLES measure of angle B is equal to measure
of angle D

Consecutive angles of parallelogram are supplementary .

A B
xo 180 - xo read as ANGLE

A and B are supplementary,

B and C are supplementary ,
180 - xo xo C and D are supplementary , and
D C D and A are supplementary .

The diagonals of a parallelogram bisect each other.

A B
7 1 3 6 AC and BD bisect each other at E.

E 1 ≅2 angle 1 is equal to angle 2

3≅ 4 angle 3 is equal to angle 4
5 4 2 8 5≅ 6 angle 5 is equal to angle 6
D C 7≅ 8 angle 7 is equal to angle 9
Parallelogram ABCD with diagonals AC and BD.

EXAMPLE # 1.
HOPE is a parallelogram with diagonals HP and OE. Complete the following
statements and justify your answer.

a. HO = H O

b. HE =

c. HOP =

d. H= E P

e. E =

f. O = is the supplementary of
 ANSWERS :

a. EP HO is parallel to EP, The opposite sides of parallelogram are equal.

b. EP HE is parallel to OP, The opposite sides of parallelogram are equal.

c. EPO Each diagonals separates a parallelogram into two equal triangles.

d. P Opposite angles of parallelogram are equal.

e. O Opposite angles of parallelogram are equal.

f. P Consecutive angles of a parallelogram are supplementary.

EXAMPLE # 2.
CURE is a parallelogram with diagonals CR and EU . Determine the measures of the
numbered angles .

C U
60o 30o

70o B The sum of the measures of the interior

1 angles of parallelogram is 360 o.
5 2
3 4
E R

Solution:
Solving for 1,

360−2(70) 360−140 220

1¿ = = =110
2 2 2

Solving for 2& 3

Since diagonal separates the parallelogram into two equal triangle, then the alternate
interior angles are equal.
C U
Therefore , 60 30 o

2 = 60 0
110o
o
3 = 30 70 B 70o
o

1= 110 o
5 2=60 o
0
3 = 30 4
Solving for 4& 5 E R

Total angle of a triangle is 180 o.

Then 4 = 180 – ( 30 + 110 ) 5 = 180 – ( 60 + 70 )

= 180 –140 = 180 –130
= 40 o = 50 o
EXAMPLE #3

HERO is a parallelogram.
Answer the following: H E
1. Given: HE = 2x
OR = x + 5
Z
Find : HE _____

2. Given : E = 5y – 26
O = 2y + 40 O R

Find : O = _____

3. Given : HZ = 4a – 5
RZ = 3a + 5

Find: HZ = _____

Solutions
1. Since HERO is a parallelogram, HE = OR.
Equating the two equations to find the value of x.

2x = x + 5
2x – x = 5 combining like terms

x= 5
Substitute the value of x to HE.
HE = 2x
= 2(5)
HE = 10

2. In a parallelogram opposite angles are equal,

therefore E= O.

5y – 26 = 2y + 40
5y – 2y = 40 + 26 combining like terms

3y = 66

3 y 66
=
3 3

y = 22
Substitute the value of y to O.
O = 2y + 40
= 2( 22) +40
= 44 + 40
 O = 84
3. Diagonals of a parallelogram bisect each other. Diagonal OE bisect diagonal HR and HR
bisect OE, their intersection at point Z.

Therefore , HZ = RZ.
Equating the two equations since they are equal:

4a – 5 = 3a + 5

4a -3a = 5 + 5 combining like terms

a = 10

Substitute the value of a to HZ = 4a – 5.

HZ = 4a – 5

HZ = 4(10 )

HZ = 40

Properties of Special Parallelogram

Rectangle , rhombuses (or rhombi) , and squares are called special parallelograms because
their interior angles and/ or side lengths are all equal to each other.

The diagonals of a rectangle are equal and bisect opposite angles.

H E HELP is a rectangle with diagonals HL and PE.

HL ≅ PE

P L

If a parallelogram has a right angle, then it has four right angles and the parallelogram
is a rectangle.

L O
LOVE is a rectangle with L is a right angle.

Therefore,
L = 90 o, O = 90o , V = 90o,

and E = 90 o
E V
 The diagonals of square and rhombus are perpendicular to each other .

M I C O

Right angle

E N D L

SQUARE RHOMBUS

MINE is a square with diagonals EI and MN. COLD is a rhombus with diagonals CL and OD.
 The diagonals of a square are • The diagonals of a rhombus bisect each other.
perpendicular to each other. • Four sides of a rhombus are equal.
 Four sides of a square are equal. • Opposite angles of a rhombus are equal and
 Four angles of a square are equal. the diagonals bisect the opposite angles.
 The diagonals of a square bisect opposite angles.

EXAMPLE # 4.
Given square HARD. If m HGA = 20x + 30, then what is x?

H A
Solution:
Since diagonals of a square are perpendicular to each
other then HGA = 90o.
G
Then substitute the value of G to the given equation:
HGA = 20x + 30
90 = 20x + 30
D R 90 - 30 = 20x
60 = 20x
60 20 x
=
20 20
30 = x , therefore x = 30

EXAMPLE # 5.
Given rhombus PRAY. If RA = 4 and R = 30, then what is the value of the
following a. Y = ____, b. PY = _____ ?

P R Solution:
All sides of a rhombus are equal, then:
PY = RA, RA = 4
PY = 4
 Opposite angles of a rhombus are equal, then
Y= R, R = 30
Y A Y = 30

To check your knowledge

On the blank before each number , write A if the statement is always true, S if the statement
is sometimes true, or N if the statement is never true.

_____1. A square is a rhombus .

_____2. A rhombus is a square .

_____3. A rectangle is a square.
_____4. A rhombus is a rectangle.
_____5. A square is a parallelogram.
_____6. A rhombus is a parallelogram.
_____7. A quadrilateral is a parallelogram.
_____8. The diagonals of a square are perpendicular.
_____9. The diagonals of a rectangle bisect each other.
_____10. The opposite sides of a parallelogram are equal.

Perform

A. Determine if the given properties hold true for the special parallelograms. Place a check mark to
indicate your answer.

Property Rectangl Rhombus Square

e

1. Diagonals are equal.

2 Diagonals bisect each other

3 Diagonals are perpendicular.

4 Opposite sides are equal.

5 Opposite angles are equal.

6 A diagonal bisect opposite angles.

7. Consecutive angles are supplementary

8 Both pairs of opposite sides are equal.

9. The sum of the measures of interior angles are equal.

10. Diagonals form four equal triangles.

B. Determine the unknown measures.

1. Given rectangle KNOW . WO = 20, SN = 12.

Find: W O
WS = ____ KS = _____

WN = ____ KN = _____ S

SO = ____

KO= ____
K N

2. Given rhombus MAKE. MA = 15x -10 and AK = 10x + 15. ( Show your solutions)

Find : M A
x = _____

Perimeter of rhombus MAKE: _____

E K

3. Given parallelogram ABCD. Find the measures of the other angles of parallelogram ABCD
if D = 120o .
( Remember , opposite angles of parallelogram are equal , and the sum of interior angles Is 360 o)

A B

120o
D C

Challenge 2
Find the measures of the numbered angles.
( Show your solutions)

102o 1
20 o
8 5 4
3

60 0
6
24o
10 55 o

140 o
9