9
NOT
Mathematics
Quarter 3, Wk.5 - Module 5
Solving Problems Involving Parallelograms,
Trapezoids and Kites
Department of Education ● Republic of the Philippines
Math- Grade 9
Alternative Delivery Mode
Quarter 3, Wk.5 - Module 5: Solving Problems Involving Parallelograms,
Trapezoids and Kites
First Edition, 2020
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Chairperson:
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9
Mathematics
Quarter 3, Wk.5 - Module 5
Solving Problems Involving
Parallelograms, Trapezoids and Kites
This instructional material was collaboratively developed and reviewed
by educators from public and private schools, colleges, and or/universities.
We encourage teachers and other education stakeholders to email their
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Department of Education ● Republic of the Philippines
Table of Contents
What This Module is About ........................................................................................................... i
What I Need to Know ..................................................................................................................... ii
How to Learn from this Module ................................................................................................... ii
Icons of this Module ...................................................................................................................... iii
What I Know.................................................................................................................................. iii
Lesson 1:
Solving Problems Involving Parallelograms ......................................................... 1
What I Need to Know ..................................................................................................... 1
What’s New ................................................................................................................... 6
What Is It ............................................................................................................................ 8
What’s More .................................................................................................................... 14
What I Have Learned ..................................................................................................... 14
What I Can Do ................................................................................................................. 15
Lesson 2:
Solving Problems Involving Trapezoids ................................................................... 15
What’s In ............................................................................................................................ 15
What I Need to Know ..................................................................................................... 16
What Is It ......................................................................................................................... 16
What’s More .................................................................................................................. 19
What I Have Learned ………………………………………………………….20
What I Can Do .............................................................................................................. 20
Lesson 3:
Solving Problems Involving Kites .................................................................................... 20
What’s In ............................................................................................................................ 20
What I Need to Know ..................................................................................................... 21
What Is It ......................................................................................................................... 21
What’s More .................................................................................................................. 22
What I Have Learned ………………………………………………………….23
What I Can Do .............................................................................................................. 24
Summary ........................................................................................................................25
Assessment: (Post-Test) ...............................................................................................26
Key to Answers ..............................................................................................................27
References .....................................................................................................................29
What This Module is About
Instill in mind the question “How useful are the quadrilaterals in dealing with real-life
situations?”. In this module, you will be given a chance to formulate and learn how to solve
problems involving parallelogram, trapezoids and kites, demonstrate understanding of the
lesson by doing some practical tasks.
This will also tell you about the properties of parallelograms, trapezoids and kites.
Furthermore, the module contains problems involving the median of a trapezoid, the base
angles and diagonals of an isosceles trapezoid, as well as problems involving the diagonals,
angles and sides of parallelograms and problems involving the diagonals and area of kites.
What I Need to Know
There are many ways to apply your understanding in parallelograms, trapezoids and
kites with the application of mathematics principles and concepts. This module is designed
for you to
• Solve problems involving the sides, angles and areas of parallelogram and
the special kinds of parallelogram.
• Solve problems involving the sides, median, angle and areas of trapezoid.
• Solve problems involving the diagonals and area of kite.
i
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
•
Take your time reading the lessons carefully.
•
Follow the directions and/or instructions in the activities and exercises diligently.
•
Answer all the given tests and exercises.
Icons of this Module
What I Need to
Know
This part contains learning objectives that
are set for you to learn as you go along the
module.
What I know
This is an assessment as to your level of
knowledge to the subject matter at hand,
meant specifically to gauge prior related
knowledge
This part connects previous lesson with that
of the current one.
What’s In
What’s New
An introduction of the new lesson through
various activities, before it will be presented
to you
What is It
These are discussions of the activities as a
way to deepen your discovery and understanding of the concept.
What’s More
These are follow-up activities that are intended for you to practice further in order to
master the competencies.
What I Have
Learned
Activities designed to process what you
have learned from the lesson
What I can do
These are tasks that are designed to showcase your skills and knowledge gained, and
applied into real-life concerns and situations.
ii
What I Know
Pre-Assessment
Directions: Find out how much you already know about this module. Please answer all items.
Take note of the items that you were not able to answer correctly and find the right answer
as you go through this module.
A. Multiple Choice
Choose the letter that you think best answers the question.
1. Its area is half of the product of the lengths of its diagonals.
a. rectangle
c. trapezoid
b. square
d. kite
For items 2-3, use the figure on the right.
Given 􀀀EASY is a parallelogram.
2. If 􀀀EASY is a rectangle, and EX = 9, then AY = ____.
a. 9
c. 27
b. 18
d. 36
3. If 􀀀EASY is a rhombus m∠SAY = 52, then m∠ASX = ____.
a. 26°
c. 90°
b. 38°
d. 104°
For items 4-5, use the kite 􀀀PLAY.
4. 􀀀PLAY is a kite, what is the m∠PSY?
a. 45°
c. 180°
b. 90°
d. 360°
5. Find PA if the area of the kite is 60 sq. units when LY =20 and PA = x +4.
a. 2
c. 10
b. 6
d. 16
6. What is x in 􀀀FLIA if FA = 32, LI = 16 when MY = x +6?
a.48
c. 18
b. 24
d. 12
7. Given rectangle ABCD with diagonals AC and BD intersect at E. AE = 15x – 6 and
CE = 12x + 9. What is BD?
a. 5
c. 69
b. 10
d. 138
8. Given parallelogram 􀀀PRAY with m∠P = (14x - 8)° and m∠Y = (6x - 12)°, what is the
m∠A?
a. 10°
c. 90°
b. 48°
d. 132°
9. A 􀀀FOUR is an isosceles trapezoid, FO and UR are the bases. What is the m∠O if
m∠U = (5x + 2)° and m∠R = (6x – 3)°?
a. 5°
c. 54°
b. 27°
d. 153°
10. The perimeter of a parallelogram is 34 cm. If a diagonal is 1 cm less than its length
and 8 cm more than its width, what are the dimensions of this parallelogram?
a. 4 cm × 13 cm
c. 6 cm × 11 cm
b. 5 cm × 12 cm
d. 7 cm × 10 cm
iii
B. Check Me Out
Indicate with a check (✓) mark in the table below the property that corresponds to the
given quadrilaterals.
Quadrilaterals
6. Two pairs of opposite angles are congruent
7. Diagonals are congruent
8. Diagonals are perpendicular to each other
9. Exactly one pair of sides is parallel
10. No sides are parallel
iv
Kite
5. Opposite angles are supplementary
Trapezium
4. All angles are right angles
Trapezoid
3. Two pairs of opposite sides are parallel
Square
2. Two pairs of opposite sides are congruent
Rhombus
1. All sides are congruent
Rectangle
Parallelogram
Property
Solving Problems Involving
Parallelograms
Lesson
1
What I Need to Know
In this lesson, you will learn the conditions that guarantee that a quadrilateral is a
parallelogram. After which, you will be able to determine the properties of a parallelogram &
properties of diagonals on special quadrilaterals and use these to find measures of angles,
sides, and other quantities involving parallelograms.
What’s In
Let’s start this module by answering the following activities to help you refresh your
mind the properties of quadrilaterals.
Activity 1: Which is Which?
Identify whether the following quadrilaterals are parallelograms or not. Put a check mark (✓)
under the appropriate column and answer the questions that follow.
Quadrilateral
Parallelogram
Not Parallelogram
1. trapezoid
2. rectangle
3. rhombus
4. square
Process Questions:
1. Which of the quadrilaterals are parallelograms? Why?
2. Which of the quadrilaterals are not parallelograms? Why?
You’ve just determined kind of quadrilaterals that are parallelograms. Before
discussing the properties of parallelogram let check your readiness in Activity 2.
What’s New
Activity 2: Check Your Guess.
Write TRUE if your guess on the statement is true, and write FALSE if your guess is false in
the second column.
6
Statement
My Guess (True or False)
1. A quadrilateral is a parallelogram if both pairs of
opposite sides are parallel.
2. A quadrilateral is a parallelogram if both pairs of
opposite sides are congruent.
3. A quadrilateral is a parallelogram if both pairs of
opposite angles are congruent.
4. A quadrilateral is a parallelogram if any two
consecutive angles are complementary.
5. A quadrilateral is a parallelogram if exactly one pair of
adjacent sides is perpendicular.
6. A quadrilateral is a parallelogram if one pair of
opposite sides are both congruent and parallel.
This time, you are ready to learn more about quadrilaterals that are parallelograms
from a deeper perspective. Answering Activity 3 will help you.
Activity 3: Draw Me a Parallelogram
Do the following.
1. On a graphing paper, draw a parallelogram similar to the one below. Name your
parallelogram ABCD.
̅̅̅̅ . What do you notice?
2. Draw diagonal 𝐴𝐶
3. Using a protractor, find the measures of the opposite angles of parallelogram ABCD.
Are the angles congruent?
4. Using a protractor, find the measures of each pair of non-opposite angles. Add their
measures. Are the angles supplementary?
5. Using a ruler, find the lengths of each pair of opposite sides. Are their lengths equal?
6. Draw diagonal ̅̅̅̅
𝐵𝐷 . What do you notice?
Were you careful in doing the above activity? You actually described inductively
the following properties of a parallelogram and the conditions that guarantee that a
quadrilateral is a parallelogram.
7
What Is It
Quadrilaterals that are Parallelograms
Conditions that Guarantee that a Quadrilateral is a Parallelogram
1. A quadrilateral is a parallelogram if both pairs of opposite sides are congruent.
2. A quadrilateral is a parallelogram if both pairs of opposite angles are congruent.
3. A quadrilateral is a parallelogram if pairs of consecutive angles are supplementary.
4. A quadrilateral is a parallelogram if the diagonals bisect each other.
5. A quadrilateral is a parallelogram if each diagonal divides a parallelogram into two
consecutive triangles.
6. A quadrilateral is a parallelogram if one pair of opposite sides are congruent and
parallel.
Properties of a Parallelogram
1. In a parallelogram, any two opposite sides are congruent.
2. In a parallelogram, any two opposite angles are congruent.
3. In a parallelogram, any two consecutive angles are supplementary.
4. The diagonals of a parallelogram bisect each other.
5. A diagonal of a parallelogram forms two congruent triangles.
Now that you already know the properties of a parallelogram, you can use these in
solving the following problems.
Example 1:
Quadrilateral JACK is a parallelogram. If m∠K = 110, what is m∠A?
K
C
Solution:
Step 1:
Step 2:
Example 2:
A
J
Opposite angles of a parallelogram are congruent. Hence, ∠A ≅∠K.
Then, m∠A = m∠K
(Congruent angles have equal measures)
Replace m∠K with 110. Then,
m∠A = 110.
Therefore, the measure of ∠A is 110°.
NCOV is a parallelogram. If m∠V = x + 15 and m∠C = 40, what is x?
N
C
O
V
Opposite angles of a parallelogram are congruent. Hence, ∠V ≅ ∠C.
Then, m∠V = m∠C (Congruent angles have equal measures)
Step 2:
Substitute the values of ∠V and ∠C. Then, solve for x.
Solution:
Step 1:
m∠V
x + 15
x
x
=
=
=
=
8
m∠C
40
40 - 15
25
Therefore, x is 25.
Example 3:
The figure below is a parallelogram. If m∠O = 2x + 10 and m∠E = x + 30.
What is the m∠O?
E
V
Solution:
Step 1:
O
L
Step 2:
Opposite angles of a parallelogram are congruent. Hence, ∠O ≅ ∠E.
Then, m∠O = m∠E (Congruent angles have equal measures)
Substitute 2x + 10 for m∠O and x + 30 for m∠E. Then, solve for x.
Step 3:
m∠O = m∠E
2x + 10 = x + 30
2x - x = 30 - 10
x = 20
Substitute 20 for x in m∠O = 2x + 10 to solve for m∠O.
m∠O = 2x + 10
m∠O = 2(20) + 30
m∠O = 40 + 10
m∠O = 50
Therefore, the measure of ∠O is 50°.
Example 4:
Quadrilateral ABCD is a parallelogram. If m∠A = 60, what is m∠B?
D
C
A
Solution:
Step 1:
Step 2:
B
Non-opposite angels of a parallelogram are supplementary ∠A and ∠B
are supplementary m∠A + m∠B = 180.
Substitute 60 for m∠A and solve for m∠B
m∠A + m∠B
60 + m∠B
m∠B
m∠B
=
=
=
=
180
180
180 - 60
120
Therefore, the measure of angle B is 120°.
Example 5:
Quadrilateral LATE is a parallelogram. If m∠L = x – 60 and m∠E = 2x,
What is m∠E?
L
A
E
Solution:
Step 1:
Step 2:
T
Non opposite angles of a parallelogram are supplementary ∠E
and ∠A are supplementary m∠L + m∠E = 180
Substitute x – 60 for m∠L, and 2x for m∠E
m∠L + m∠E =
x – 60 + 2x =
9
180
180
x + 2x
3x
3𝑥
3
x
=
=
=
=
180 + 60
240
240
3
80
Substitute 80 for x in m∠E = x – 60. Then solve for m∠E.
Step 3:
m∠E = x - 60
m∠E = 80 - 60
m∠E = 20
Therefore, the measure of angle E is 20°.
Example 6:
In the parallelogram below, m∠M = 2x and m∠T = 4x – 80. What is m∠A?
H
T
Solution:
M
A
Opposite angles of a parallelogram are congruent. Hence, ∠M ≅ ∠T
Then, m∠M = m∠T (Congruent angles have equal measures)
Substitute 2x for m∠M and 4x – 80 for m∠T. Then solve for x.
m∠M = m∠T
2x = 4x - 80
2x – 4x = -80
-2x = -80 (divide both sides by -2)
x = 40
Step 1:
Step 2:
Step 3:
Substitute 40 for m∠M = 2x
m∠M = 2x
m∠M = 2(40)
m∠M = 80
Non-opposite angles of a parallelogram are supplementary ∠M and
∠A are supplementary m∠M + m∠A = 180
Substitute 80 for m∠M and solve for m∠A.
Step 4:
Step 5:
m∠M + m∠A =
80 + m∠A =
m∠A =
180
180 - 80
100
Therefore, the measure of angle A is 100°.
Example 7:
Quadrilateral FELY is a parallelogram. If FY = 14 cm long, how long is
̅̅̅̅?
F
E
side LE
Solution:
Step 1:
Step 2:
Y
L
Opposite sides of a parallelogram are congruent. Hence, ̅̅̅̅
FY ≅ ̅̅̅̅
LE .
̅̅̅̅
̅̅̅̅
Then, FY = LE
(Congruent segments have equal lengths)
Substitute 14 for FY in the equation FY = LE
LE = 14
Therefore, LE is 14 cm long.
10
Example 8:
Quadrilateral GEOM is a parallelogram. If MO = 2x + 3 and GE = 4x – 15.
What is MO?
M
O
E
G
Solution:
̅̅̅̅̅ ≅ ̅̅̅̅̅
Opposite sides of a parallelogram are congruent. Hence, MO
GE.
̅̅̅̅̅
̅̅̅̅̅
Then, MO = GE (Congruent segments have equal lengths)
Substitute 2x + 3 for MO and 4x – 15 for GE. Solve for x.
Step 1:
Step 2:
MO
2x + 3
2x – 4x
-2x
x
Step 3:
=
=
=
=
=
GE
4x – 15
-15 – 3
-18 (divide both sides by -2)
9
Substitute 9 for x in MO = 2x + 3
MO
MO
MO
MO
=
=
=
=
2x + 3
2(9) + 3
18 + 3
21
̅̅̅̅̅ is 21 units.
Therefore, the length of MO
Example 9:
FERN is a parallelogram. If FR = 12 cm, what is the length of ̅̅̅̅
FS?
E
F
S
Solution:
R
N
The diagonals of a parallelogram bisect each other.
̅̅̅̅ bisects FR
̅̅̅̅. We can say that FS is half of FR.
Thus, NE
1
● FR
2
1
● 12
2
FS =
FS =
FS =
6
Therefore, the length of FS is 6 cm.
̅̅̅̅ and RY
̅̅̅̅ intersecting
Example 10: Quadrilateral PRAY is a parallelogram with diagonals PA
at G. If YG = 3x – 7 and GR = x + 21, find YG.
Solution:
Step 1:
Draw the figure.
A
Y
G
Step 2:
P
R
The diagonals of a parallelogram bisect each other. Thus YG = GR.
11
Step 3:
Substitute 3x – 7 for YG and x + 21 for GR in the equation YG = GR.
YG = GR
3x – 7 = x + 21
3x - x = 21 + 7
2x = 28 (divide both sides by 2)
x = 14
Step 4:
Substitute 14 for x in the equation YG = 3x – 7.
YG = 3x - 7
YG = 3(14) - 7
YG = 42 - 7
YG = 35
Therefore, the length of YG is 35 units.
The Properties of the Diagonals of Special Quadrilaterals
A diagonal of a quadrilateral is a segment which connects any two non-consecutive
vertices. In the following quadrilateral, AC and BD are the diagonals.
C
D
A
B
The following are the properties of the diagonals of special quadrilaterals.
1. The diagonals of a rectangle are congruent.
2. The diagonals of a square are congruent.
3. The diagonals of a square are perpendicular
4. Each diagonal of a square bisects a pair of opposite angles.
5. The diagonals of a rhombus are perpendicular.
6. Each diagonal of a rhombus bisects a pair of opposite angles
E
V
Example 11: The figure at the right is a rectangle.
If the diagonal LV = 2x the diagonal
OE = 12 cm, find x.
Solution:
L
O
Step 1.
The diagonals of a rectangle are congruent. Hence, ̅̅̅̅
LV ≅ ̅̅̅̅
OE. Then,
̅̅̅̅ since congruent segments have equal lengths.
̅̅̅̅ = OE
LV
̅̅̅̅. Then solve for x.
̅̅̅̅ and 12 for OE
Step 2.
Substitute 2x for LV
̅̅̅̅
̅̅̅̅ = OE
LV
2x = 12
D
x = 6
C
Therefore, the value of x is 6 cm.
Example 12: Quadrilateral ABCD at the right is a square. Find m∠CAB.
Solution:
Step 1:
Step 2:
A
Quadrilateral ABCD is a square and a square is a rectangle.
Therefore, m∠DAB = 90.
But each diagonal of a square bisects a pair of opposite angles.
Hence, m∠CAB =
1
m∠DAB
2
12
B
Step 3:
1
Substitute 90 for m∠DAB. m ∠CAB = 2 (90) = 45. Therefore, the
measure of ∠CAB is 45°.
Example 13: 􀀀EDNA is a square. If m∠END is 3(x +5), what is x?
Solution:
Step 1:
m∠DCB = 90 since 􀀀ABCD is a square
Step 2:
Each diagonal of a square bisects a pair of opposite angles. Hence,
m∠ACB=45.
3(x + 5) = 45
3x + 15 = 45
3x = 45 - 15
3x = 30
x = 10
C
D
Therefore, the value of x is 10 units.
Example 14: The figure at the right is a rhombus.
If m ∠CAB = 30, what is the m∠CAD?
Solution:
Step 1.
A
B
Each diagonal of a rhombus bisects pair of opposite angles.
m∠CAD = m∠CAB
Step 2.
Substitute 30 for m ∠CAB in the above equation m∠CAD = 30.
Therefore, the measure of ∠CAD is 30°.
H
Example 15: Quadrilateral BETH is a rhombus. If m∠TBE = 35, what is m∠HEB?
T
M
Solution:
Step 1.
Step 2.
Step 3.
The diagonals of a rhombus are perpendicular.
Hence, ∠BME is a right angle and its measure is 90°.
B
So, m∠BME = 90
The sum of the measures of the angles of a triangle is 180°
m∠TBE + m∠BME+ m∠HEB = 180
Substitute 35 for m∠TBE and 90 for m∠BME in the above equation.
m∠TBE + m∠BME+ m∠HEB = 180
35 +
90 + m∠HEB = 180
125 + m∠HEB = 180
m∠HEB = 180 - 125
D
m∠HEB = 55
E
C
M
Therefore, the measure of the ∠HEB is 55°.
Example 16: 􀀀 ABCD is a rhombus. If AM = 16 cm, what is the length of CM?
A
Solution:
Step 1: The diagonals of a rhombus bisect each other. Hence, CM = AM
Step 2. Substitute 16 cm for AM in the above equation. Therefore, CM = 16 cm.
After applying the different properties of a parallelogram and properties of the
diagonals of special quadrilaterals, you are now ready to answer more problems involving
parallelograms. You should always remember what you have learned in the past. It pays
best to instill in mind what had been taught. Now, prepare for the next activities.
13
B
What’s More
Activity 4: Find my Measure.
Answer the following problems below. Show your complete solutions. Write your answers in
your activity notebook.
A. Quadrilateral ABCD is a parallelogram. The diagonals are AC and BD that intersects
at E.
1. If AB = x + 6 and CD = 14, what is x?
2. If BE = 2x and DE = 6, what is x?
3. If AD = 4x + 15 and BC = 2x + 21, then what is the length of AD?
4. If AE = 15 and CE = x + 4, what is x?
B. 􀀀 ABCD is a rectangle with diagonals AC and BD.
5. AC = 2x + 15, BD = 3x + 10. Find AC.
6. BD = 6x + 5, AC = 5x + 14. Find BD.
C. 􀀀 FERM is a rhombus. ̅̅̅̅̅
𝑀𝐸 and ̅̅̅̅
𝐹𝑅 intersects at I.
7. If m∠IFE = x +20, m∠IEF = x + 26, find x.
8. If m∠IMR = 4x + 20, m∠IRM = 2x + 10, find x.
D. 􀀀 BETH is a square and M is the intersection of the diagonals.
9. If BM = x + 12, EM = 2x – 20, what is the length of ̅̅̅̅
𝐻𝐸 ?
̅̅̅̅?
10. If HM = 44 – x, TM = 4 + 3x, what is the length of 𝐵𝑇
Based on your findings, what is the most special among the kinds of
parallelogram? Why? Yes, you’re right! The Square is the most special parallelogram
because all the properties of the parallelograms and the theorems on rectangles and
rhombuses are true to all squares.
What I Have Learned
Activity 5: Show Me What You’ve Got!
Solve each problem completely and accurately on your activity notebook. Show your solution
and write the properties you applied to justify each step in the solution process. You may
illustrate each given, to serve as your guide. Be sure to box your answer.
Quadrilateral MECQ is a parallelogram.
a. If m∠M= x + 15 and m∠C = 2x + 5, what is m∠M?
̅̅̅̅ ?
b. If ME = 3y + 3 and QC = y + 13, how long is 𝑄𝐶
c. 􀀀MECQ is a rectangle and its perimeter is 56 cm. One side is 5 cm less than twice
the other side. What are its dimensions and how long is its area?
d. What is the perimeter and the area of the largest square that can be formed from
rectangle MECQ in item c?
14
What I Can Do
Activity 6: Create Me a Parallelogram.
Do as directed.
Create a problem and/or draw a parallelogram picture frame with unknown
measurement of angles/sides Show and solve one of the properties of a parallelogram.
4
Understanding and
applying the concept
learned
Construction and
presentation of
problem and solution
3
Demonstrate well
understanding and
applying the of
concept
Shows a wellconstructed problem,
with correct and
appropriate solution
and/or discussion.
2
1
Demonstrate the
understanding and
applying of concept
Demonstrate limited
understanding and
applying of concept
Shows a good
problem, with correct
and appropriate
solution and/or
discussion but
somehow needs
improvement.
Shows a problem
with appropriate
solution and/or
discussion but
somehow needs
improvement
Solving Problems Involving
Trapezoid
Lesson
2
What’s In
Another kind of quadrilateral that is equally important as parallelogram
is the trapezoid. A trapezoid is a quadrilateral with exactly one pair of parallel sides. The
parallel sides of a trapezoid are called the bases and the non-parallel sides are called the
legs. The angles formed by a base and a leg are called base angles.
Let’s start this lesson by answering the following activities to help you refresh the
properties involving trapezoids.
Activity 7: Draw Me a Trapezoid.
If the length of the upper base of a trapezoid is 4 cm, and the lower base is 6 cm, what do
you think is the length of the median?
Do the following:
1. Using a ruler draw a segment 4 cm long. Name the segment ̅̅̅̅
𝐻𝑇.
15
̅̅̅̅̅.
̅̅̅̅. Name the segment 𝑀𝐴
Draw another segment 6 cm long parallel to segment 𝐻𝑇
Connect points H and M.
Connect points T and A.
̅̅̅̅̅ and ̅̅̅̅
Using a ruler, carefully determine the midpoints of 𝐻𝑀
𝑇𝐴. Name the midpoint of
̅̅̅̅ as G and E, respectively.
̅̅̅̅̅ and 𝑇𝐴
𝐻𝑀
6. Connect points G and E. Carefully measure the length of ̅̅̅̅
𝐺𝐸 .
2.
3.
4.
5.
What did you discover? Did you discover that:
̅̅̅̅ is parallel to 𝐻𝑇
̅̅̅̅̅.
̅̅̅̅ and 𝑀𝐴
a. 𝐺𝐸
1
̅̅̅̅+ ̅̅̅̅̅
b. ̅̅̅̅̅
GE= (HT
MA)
2
Activity 8: Draw Me an Isosceles Trapezoid.
Do the following.
1. Using a ruler, draw isosceles trapezoid ABCD with base angles, ∠A and ∠B, on a
graphing paper.
2. Using a protractor, find the measures of ∠A and ∠B. What do you notice?
3. Find also the measure of ∠D and ∠C. What do you notice?
4. Draw the diagonals AC and BD. Using a ruler, find their lengths. Are the lengths
equal?
D
C
A
B
What did you discover? Perhaps you discovered the following
a. The base angles of an isosceles trapezoid are congruent.
b. The diagonals of an isosceles trapezoid are congruent.
What I Need to Know
In this lesson, you will be able to apply the properties involving trapezoid in finding
the length of the sides, the area, and the measure of the angles of the trapezoid.
What Is It
Trapezium and Isosceles Trapezoid
A trapezoid, in plane geometry, is a quadrilateral (four-sided) figure with two parallel
sides, or bases, of unequal length. The perpendicular distance between the bases is known
as the altitude. The sides that are not parallel are called legs, and a line from the midpoint of
16
one leg to the midpoint of the other is called the median. The multiplication of the altitude
and the median yields the area of the trapezoid. When the legs of a trapezoid are of equal
length, the figure is called an isosceles trapezoid.
In an isosceles trapezoid,
• the legs are congruent
• the base angles are congruent
• the diagonals are congruent
The median of a trapezoid is parallel to the bases.
The length of the median is one-half of the sum of the lengths of the two bases.
•
To find the median of a trapezoid:
upper base + lower base
2
The sum of all the angles of a trapezoid is 360°.
Theorems related to isosceles trapezoids as follows:
• The base angles of an isosceles trapezoid are congruent.
• Opposite angles of an isosceles trapezoid are supplementary.
• The diagonals of an isosceles trapezoid are congruent.
Example 1:
ABCD is a trapezoid with median ̅̅̅̅
𝐸𝐹 . If DC = 8 cm and AB = 14 cm, find EF.
D
8
C
E
F
14
A
B
Solution:
1
(AB + DC)
2
Step 1:
Write the formula. EF =
Step 2:
Substitute the values of DC and AB into the formula. Solve for EF.
EF
=
EF
=
1
(14 + 8)
2
1
(22)
2
EF = 11
Therefore, the length of EF is 11 cm.
Example 2:
ABCD is a trapezoid with median EF. If DC = x + 5, EF = 2x + 1 and
AB = 4x – 10, find EF.
D
x+5
C
E
2x + 1
4x – 10
A
Solution:
Step 1:
B
Write the formula.
EF =
Step 2:
F
1
(AB + DC)
2
Substitute the values of EF, AB and DC into the formula. Solve for x.
17
2x + 1
=
2x + 1
=
2(2x +1)
4x + 2
4x – 5x
-x
x
=
=
=
=
=
1
(4x – 10 + x + 5)
2
1
( 5x –5)
2
5x – 5
5x – 5
-5 – 2
-7
7
Step 3:
Substitute 7 for x. Solve for EF
EF = 2x + 1
EF = 2(7) + 1
EF = 14 + 1
EF = 15
Therefore, the length of EF is 15 cm.
Example 3:
In isosceles trapezoid ABCD with A and B as base angles.
If mA = x + 20 and mB = 2x. Find mA.
D
C
A
Solution:
Step 1:
Base angles of an isosceles trapezoid are congruent. Hence,
A  B. Then, mA = mB (the measures two angles congruent
are equal)
Step 2:
Substitute x + 20 for mA and 2x for mB.
x + 20 = 2x
x – 2x = -20
x = 20
Substitute 20 for x in mA = x + 20. Then solve for mA.
mA = x + 20
= 20 + 20
mA = 40
Therefore, the measure of A is 40˚.
Step 3:
Example 4:
B
In isosceles trapezoid ABCD, AC = 4x + 4 and BD = 2x + 10. What is the
length of AC?
D
C
A
Solution:
Step 1:
Step 2.
B
The diagonals of an isosceles trapezoid are congruent. Hence,
AC  BD. Then, AC = BD (Congruent segments have equal lengths)
Substitute 4x + 4 for AC and 2x + 10 for BD.
4x + 4 = 2x + 10
18
4x – 2x
2x
x
Step 3:
Example 5:
=
=
=
10 – 4
6
3
Substitute 3 to AC = 4x + 4 = 4(3) + 4 = 12 + 4 = 16.
Therefore, the length of AC is 16.
Find the longer base of a trapezoid with shorter base 5, height 4 and area 24
square units.
Solution:
Step 1:
Draw and label the figure. Represent the longer leg by x.
5
A = 24
4
x
Step 2:
Substitute the data in the formula.
1
h( b1 + b2)
2
1
24 = (4)( x + 5)
2
A=
24 = 2( x + 5)
2(x + 5 ) = 24
2x + 10 = 24
2x = 24 – 10
2x = 14
x=7
Therefore, the longer base is 7 units.
You’ve just applied the different properties/theorems concerning trapezoid. Now,
you are now ready to answer the next activities.
What’s More
Activity 9: You Can Do it!
Answer the following problems below. Show your complete solution in your activity
notebook.
1. KLMN is a trapezoid with median ̅̅̅̅
𝑂𝑃.
a. If OP = 22, NM = x + 4, and KL = x + 8, what is NM?
b. If OP = 24, NM = x – 3 and KL = x + 7, what is KL?
2. ABCD is an isosceles trapezoid.
a.
b.
c.
d.
If mA = 2x + 10 and mB = 3x – 20 what is mA?
If mA = 2x + 15 and mB = 4x – 11 what is mB?
If mD = x + 15 and mC = 2x –85 what is mD?
If mC = 3y + 12 and mD = 2y + 50, what is mC?
19
What I Have Learned
Activity 10: Let’s Solve More!
Solve each problem completely and accurately on your activity notebook. Show your solution
and write the properties you applied to justify each step in the solution process. You may
illustrate each given, to serve as your guide. Be sure to box your answer.
Quadrilateral NCOV is an isosceles trapezoid with NV || CO. PH is its median.
1. If CO = 3x – 2, NV = 2x + 10 and PH = 14, how long is each base?
2. If m∠N = 2x + 5 and m∠C = 3x – 10, what is m∠V?
3. One base is twice the other and PH is 6 cm long. If its perimeter is 27 cm, how long
is each leg?
4. PH is 8.5 in long and one leg measures 9 in. What is its perimeter if one of the
bases is 3 in more than the other?
What I Can Do
Activity 11: Attic Making.
Do as directed.
Create a problem and/or draw an attic of the house has unknown measurements.
Show and solve that the midsegment of a triangle is half of the base.
4
Understanding and
applying the concept
learned
Construction and
presentation of
problem and solution
Lesson
3
3
Demonstrate well
understanding and
applying the of
concept
Shows a wellconstructed problem,
with correct and
appropriate solution
and/or discussion.
2
1
Demonstrate the
understanding and
applying of concept
Demonstrate limited
understanding and
applying of concept
Shows a good
problem, with correct
and appropriate
solution and/or
discussion but
somehow needs
improvement.
Shows a problem
with appropriate
solution and/or
discussion but
somehow needs
improvement
Solving Problems Involving
Kites
What’s In
Have you ever experienced making a kite? Have you tried joining a kite festival in
your community?
20
A kite is defined as quadrilateral with two pairs of adjacent and congruent sides. Note
that a rhombus (where all adjacent sides are equal) is a special kind of kite.
.
Let’s start this lesson by answering the following activities to help you refresh the
properties involving kite.
Activity 12: Cute Kite.
Do the following:
T
̅̅̅̅ ≅ 𝑈𝑇
̅̅̅̅ ≅ 𝑇𝐸
̅̅̅̅ and 𝐶𝐸
̅̅̅̅
1. Draw kite CUTE where 𝑈𝐶
like what is shown at the right. Consider
X
̅̅̅̅ and 𝑈𝐸
̅̅̅̅ that meet at X.
U
diagonals 𝐶𝑇
2. Measure each of the angles about point X.
a. What is the measure of each angle?
b. What kind of angle is formed by the
C
intersecting diagonals?
c. What is true about the lines that form right angle?
3. Describe this property of this kite with respect to the segments formed by
intersecting diagonals.
What I Need to Know
In this lesson, you will be able to apply the properties involving kite in finding the area
of the kite.
What Is It
There are two theorems related to kites as follows:
• In a kite, the perpendicular bisector of at least one diagonal is the other diagonal.
• The area of a kite is half the product of the lengths of its diagonals.
Example 1:
The perimeter of a kite is 64 cm. The length of one of its sides is 14 cm more
than half the length of another. Find the length of each side of the kite.
Solutions:
Step 1:
Let x be the length of one side of a kite. The length of the other side is
1
x + 14.
2
Step 2:
Since consecutive sides of a kite are congruent. Then the perimeter is
Perimeter = 2(x) + 2 (
Step 3:
1
x + 14)
2
Solve for x.
Perimeter
64
21
=
2(x) + 2 (
=
2x + 2 (
1
x + 14)
2
1
x + 14)
2
E
64
64 – 28
36
x
Step 4:
Substitute 12 in
=
=
=
=
2x + x + 28
2x + x
3x (divide both sides by 3)
12
1
x + 14.
2
1
(12) + 14 = 6 + 14 = 20
2
Therefore, the length of the sides of the kite are 12 cm and 20 cm.
Example 2:
The diagonals of a kite have lengths of 13 cm and 9 cm. Find the area of the
kite.
Solution:
The area of a kite is half the product of the lengths of its diagonals. Therefore,
the area is
1
(13)(9) = 58.5 square centimeters.
2
The area of a kite is 180 cm2 and the length of the diagonal is 36 cm. How
long is the other diagonal?
Solution:
Step 1:
Let d1 = 36 cm and x be the d2.
Step 2:
Substitute the given to the area formula of kite.
Example 3:
A =
180 =
1
(d1 ● d2)
2
1
(36)(x)
2
180 = 18x
x = 10
Therefore, the length of the other diagonal is 10 cm.
What’s More
Activity 15: Play a Kite
Consider the figure that follows and answer the given questions.
A. Quadrilateral PLAY is a kite.
1. Given: PA = 12 cm; LY = 6 cm
Questions:
• What is the area of kite PLAY?
• How did you solve for its area?
• What theorem justifies your answer?
2. Given: Area of kite PLAY = 135 cm2; LY = 9 cm
Questions:
• How long is PA?
• How did you solve for PA?
• What theorem justifies your answer above?
M
1
L
3 N
B. Consider kite KLMN on the right.
1.
2.
3.
4.
If LM = 6, what is MN?
If LN = 7 cm and KM = 13 cm, what is the area?
If m∠1 = 31, what is the m∠LMN?
If m∠2 = 22, what is the m∠3?
22
2
K
What I Have Learned
Activity 16: Like ko si Kite.
Solve each problem completely and accurately on your activity notebook. Show your solution
and write the properties you applied to justify each step in the solution process. You may
illustrate each given, to serve as your guide. Be sure to box your answer.
Quadrilateral LIKE is a kite with LI ≅ IK and LE ≅ KE.
1. LE is twice LI. If its perimeter is 21 cm, how long is ̅̅̅̅
𝐿𝐸 ?
2. What is its area if one of the diagonals is 4 more than the other and IE + LK = 16
in?
̅̅̅ and 𝐿𝐾
̅̅̅̅ ?
3. IE = (x – 1) ft and LK = (x + 2) ft. If its area is 44 ft2, how long are 𝐼𝐸
Activity 15: Find the Right Word
To identify the appropriate word/s that best describe the properties of quadrilaterals.
Use the words from the Answer Bank below to complete the story about.
Mary and her lessons on Geometry. One word may be used several times. Mary's
Geometry teacher announced that the class would have a quiz on quadrilaterals the next
day. That night Mary began to study.
She noted that there are six classifications of quadrilaterals and they are named
according to their properties. When all angles of a quadrilateral are right angles, it is a
___________. In any ___________, ___________, or ___________, diagonals are
perpendicular to each other. ___________ are line segments that joins two nonconsecutive
vertices of a quadrilateral. She remembered what her teacher said that a square is a
parallelogram with four ___________ and four ___________. From this definition, she
understood that a square is either a ______________ or a ____________. If all sides of a
parallelogram are congruent, then the parallelogram is a ____________. Besides the
general properties of quadrilaterals, the parallelogram has some properties of its own. Its
opposite _________ and _________ are congruent.
When a quadrilateral has exactly one pair of parallel sides, it is a _______________.
A trapezoid is said to be an _________________ when the legs are congruent, the base
angles are congruent and the diagonals are congruent. A quadrilateral with no parallel sides
is called ____________.
There is one more quadrilateral that Mary noted. A _________ is a quadrilateral with
two distinct pairs of adjacent, congruent sides. Like a rhombus, the diagonals of a kite are
____________________ to each other. Finally, she kept in her mind that the sum of the
__________ interior angles of every quadrilateral is ______.
The next day, Mary felt confident when taking the quiz.
ANSWER BANK
rhombus
diagonals
trapezoid
perpendicular
360°
rectangle
congruent sides
angles
four
isosceles
23
Square
kite
congruent angles
sides
trapezium
What I Can Do
Activity 16: Kite Making.
Read and understand the situation below then answer or perform what are asked. Show
your answer on your activity notebook.
Bryan, your classmate, who is also an SK Chairman in your Barangay Maria Cristina,
organized a KITE FLYING FESTIVAL. He informed your school principal to motivate
students to join the said KITE FLYING FESTIVAL.
1. Suppose you are one of the students in your barangay, how will you prepare the
design of the kite?
2. Make a design of the kite assigned to you.
3. Illustrate every part or portion of the kite including their measures.
4. Using the design of the kite made, determine all the mathematics concepts or
principles involved.
Criteria
Poor
(1 pt)
Fair
(2 pts)
Design is functional and
has a pleasant visual
appeal. Design includes
most parts of a kite.
Design lacks some
details.
Design
Design is basic, lacks
originality and
elaboration. Design is
not detailed for
construction.
Planning
Overall planning is
random and
incomplete. Student
is asked to return for
more planning more
than once.
Plan is perfunctory. It
presents a basic design
but is not well thought
out. Contains little
evidence of forward
thinking or problem
solving.
Construction
Work time is not used
well. Construction is
haphazard. Framing
is loose. Covering is
not even and tight.
Not all components of
a kite are present.
Materials not used
resourcefully.
Work time is not always
focused. Construction
is of fair quality. All
components of a kite
are present. Materials
may not be used
resourcefully.
24
Good
(3 pts)
Design incorporates artistic
elements and is original and
well elaborated. Engineering
design is well detailed for
construction including four
parts of a kite.
Plan is well thought out.
Problems have been
addressed prior to
construction. Measurements
are included. Materials are
listed and gathered before
construction. Student works
cooperatively with adult
leader and plans time well.
Work time is focused.
Construction is of excellent
quality. All components of the
kite are present. Care is
taken to attach pieces
carefully. Materials are used
resourcefully. Student
eagerly helps others when
needed. Student works
cooperatively with adult
leader.
Summary
SQUARE
In any square:
• opposite sides are parallel
• all sides are equal in length
• all angles are right angles
• diagonals bisect each other
at right angles
• diagonals bisect the angles
at each vertex
PARALLELOGRAM
In any parallelogram:
• opposite sides are equal in
length
• opposite angles are equal in
size
• diagonals bisect each other
TRAPEZOID
In any trapezoid:
• one pair of opposite sides is
parallel
RHOMBUS
In any rhombus:
• opposite sides are parallel
• opposite angles are equal in
size
• diagonals bisect each other
at right angles
• diagonals bisect the angles
at each vertex
ISOSCELES TRAPEZOID
In an isosceles trapezoid:
• the legs are congruent
• the
base
angles
are
congruent
• the diagonals are congruent.
RECTANGLE
In any rectangle:
• opposite sides are equal in
length
• diagonals are equal in length
• diagonals bisect each other
KITE
In any kite:
• diagonals are perpendicular to
each other
• two pairs of distinct adjacent
sides are equal in length
Conditions that Guarantee that a Quadrilateral is a Parallelogram
• A quadrilateral is a parallelogram if both pairs of opposite sides are congruent.
• A quadrilateral is a parallelogram if both pairs of opposite angles are congruent.
• A quadrilateral is a parallelogram if pairs of consecutive angles are supplementary.
• A quadrilateral is a parallelogram if the diagonals bisect each other.
• A quadrilateral is a parallelogram if each diagonal divides a parallelogram into two
consecutive triangles.
• A quadrilateral is a parallelogram if one pair of opposite sides are congruent and
parallel.
Properties of a Parallelogram
• In a parallelogram, any two opposite sides are congruent.
• In a parallelogram, any two opposite angles are congruent.
• In a parallelogram, any two consecutive angles are supplementary.
• The diagonals of a parallelogram bisect each other.
• A diagonal of a parallelogram forms two congruent triangles.
The Properties of the Diagonals of Special Quadrilaterals
•
•
•
•
•
•
The diagonals of a rectangle are congruent.
The diagonals of a square are congruent.
The diagonals of a square are perpendicular
Each diagonal of a square bisects a pair of opposite angles.
The diagonals of a rhombus are perpendicular.
Each diagonal of a rhombus bisects a pair of opposite angles
25
Properties of Trapezoid
In an isosceles trapezoid,
• the legs are congruent
• the base angles are congruent
• the diagonals are congruent
The median of a trapezoid is parallel to the bases.
The length of the median is one-half of the sum of the lengths of the two bases.
•
To find the median of a trapezoid:
upper base + lower base
2
The sum of all the angles of a trapezoid is 360°.
Theorems related to isosceles trapezoids as follows:
• The base angles of an isosceles trapezoid are congruent.
• Opposite angles of an isosceles trapezoid are supplementary.
• The diagonals of an isosceles trapezoid are congruent.
Properties of Kite
• In a kite, the perpendicular bisector of at least one diagonal is the other diagonal.
• The area of a kite is half the product of the lengths of its diagonals.
Assessment: (Post-Test)
Direction: Choose the letter of the correct answer.
1. In the figure at the right, DC= 20 cm and AB = 36 cm. What is FE?
a. 16 cm
c. 28cm
b. 56 cm
d. 46 cm
2. The figure below is a parallelogram. If AD = 2x – 10 and BC = x
+ 30, then BC = ___.
a. 50
c. 70
b. 60
d. 80
3. Quadrilateral BEST is a parallelogram. If m∠B = (x + 40)° and m∠E = (2x + 20)°,
what is the m∠B?
a. 50°
c. 70°
b. 60°
d. 80°
4. The figure in the right is rhombus. If m∠I = (4x)° and m∠E = (2x +
60)°, what is the m∠I?
a. 100°
c. 120°
b. 110°
d. 130°
5. The diagonals of a rectangle have lengths 5x – 11 and 2x + 25. Find
the lengths of the diagonals.
a. 12
c. 49
b. 24
d. 60
6. The figure below is a parallelogram. The diagonals AC and
BD intersect at E. if AE = 2x and EC = 12, what is x?
a. 5
c. 7
b. 6
d. 8
7. Quadrilateral CDEF is a parallelogram. If m∠C = y° and m∠E = (2y – 40)°, then m∠D
is
a. 80°
b. 110°
26
c. 140°
d. 170°
8. Refer to rectangle FIND. If m∠4 = 38, what is m∠3?
a. 42°
c. 62°
b. 52°
d. 72°
9. In an isosceles trapezoid, the altitude drawn from an endpoint of the shorter base to
the longer base divides the longer base in segments of 5 cm and 10 cm long. Find
the lengths of the bases of the trapezoid.
a. 5 cm and 15 cm
c. 15 cm and 25 cm
b. 10 cm and 20 cm
d. 20 cm and 30 cm
10. One side of a kite is 5 cm less than 7 times the length of another. If the perimeter is
86 cm, find the length of each side of the kite.
a. 4 cm, 4 cm, 39 cm, 39 cm
c. 6 cm, 6 cm, 37 cm, 37 cm
b. 5 cm, 5 cm, 38 cm, 38 cm
d. 7 cm, 7 cm, 36 cm, 36 cm
27