Weekly Home Learning Plan/Tasks
MATHEMATICS 9
Quarter 3- Week 2
Name: ________________________________________________________
Grade and Section: _____________________________________________
Date
Learning Tasks
Read and understand the concepts that is being discussed below. (Basahin at unawain ang mga konsepto na
ipinaliwanag sa ibaba.) You may watch the video tutorial of Marlyn G. Nuno on youtube for further discussion.
(Maaaring manood ng video tutorial ni Marlyn G. Nuno sa youtube para sa mas detalyadong pagpapaliwanag.)
LESSON 6: MIDLINE THEOREM
In this lesson, we shall focus on other forms of quadrilaterals, but before we start the discussion, we will talk about
the proof of the triangle midline theorem. It will be essential in discussing the median
of a trapezoid and its proof. And then, we will look at the proofs of different special
quadrilaterals and their properties.
Midline Theorem – The segment that joins the midpoints of two sides of a triangle
is parallel to the third side and half as long.
̅̅̅̅̅ and E is the midpoint of ̅̅̅̅
Given: ∆𝐻𝑁𝑆, O is the midpoint of 𝐻𝑁
𝑁𝑆
1
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
Prove: 𝑂𝐸 //𝐻𝑆 and 𝑂𝐸 = 𝐻𝑆
2
MONDAY
1. Proves the
Midline
Theorem
Statements
̅̅̅̅̅ and E is the
1. ∆𝐻𝑁𝑆, O is the midpoint of 𝐻𝑁
midpoint of ̅̅̅̅
𝑁𝑆
⃗⃗⃗⃗⃗ , there is a point T such that 𝑂𝐸
̅̅̅̅ ≅ 𝐸𝑇
̅̅̅̅
2. In 𝑂𝐸
̅̅̅̅ ≅ ̅̅̅̅
3. 𝑁𝐸
𝐸𝑆
4. ∠2 ≅ ∠3
5. ∆𝑁𝐸𝑂 ≅ ∆𝑆𝐸𝑇
6. ∠1 ≅ ∠4
̅̅̅̅
̅̅̅̅̅//𝑆𝑇
7. 𝐻𝑁
̅̅̅̅ ≅ 𝑂𝑁
̅̅̅̅
8. 𝑂𝐻
̅̅̅̅
̅
̅̅̅
9. 𝑂𝑁 ≅ 𝑇𝑆
̅̅̅̅
̅
̅̅̅
10. 𝑂𝐻 ≅ 𝑇𝑆
11. Quadrilateral HOTS is a parallelogram
12.
13.
14.
15.
16.
17.
18.
̅̅̅̅ //𝐻𝑆
̅̅̅̅
𝑂𝐸
̅̅̅̅ + 𝐸𝑇
̅̅̅̅
̅̅̅̅ = 𝑂𝑇
𝑂𝐸
̅̅̅̅ + 𝑂𝐸
̅̅̅̅ = 𝑂𝑇
̅̅̅̅
𝑂𝐸
̅̅̅̅ = 𝑂𝑇
̅̅̅̅
2𝑂𝐸
̅̅̅̅
̅̅̅̅
𝐻𝑆 ≅ 𝑂𝑇
̅̅̅̅
2𝑂𝐸 = ̅̅̅̅
𝐻𝑆
̅̅̅̅ = 1 𝐻𝑆
̅̅̅̅
𝑂𝐸
2
Reasons
Given
Line Postulate
Definition of a midpoint
Vertical Angles Theorem
SAS Postulate
CPCTC
Converse of Alternate Interior Angles Theorem
Definition of a midpoint
CPCTC
Transitive Property
If opposite sides of a quadrilateral are congruent and
parallel, then it is a parallelogram.
Definition of a parallelogram
Segment Addition Postulate
Substitution Property
Addition Property
In a parallelogram, any two opposite sides are congruent.
Substitution Property
Divide both sides by two
EXAMPLE:
̅̅̅̅̅ and 𝐺𝐶
̅̅̅̅ , respectively.
In Δ𝑀𝐶𝐺, A and I are the midpoints of 𝑀𝐺
GIVEN
AI = 10.5, find
MC
ANSWER
MC = 21
CG = 32, find GI
GI = 16
AG = 7 and CI = 8,
find MG + GC
MG + GC = 14 +
16 = 30
AI = 3x – 2 and
MC = 9x – 13
AI = 1/2MC
3x-2=1/2(9x-13)
9x-13=2(3x-2)
M
DISCUSSION
̅̅̅̅̅ ∥ 𝐴𝐼
̅̅̅ , 𝐴𝐼
̅̅̅ intersects the two
𝑀𝐶
sides of the triangle at its midpoint
therefore the length of MC is twice
the length of AI.
I is the midpoint of CG therefore GI
is half the length of CG
Since A and I are midpoints of MG
and CG respectively. Therefore,
MG and CG is twice the length of
AG and CI.
Formula of Midline Theorem
By substitution
By cross – multiplication
C
I
A
G
a. What is the
value of x?
b. What is the
length of AI and
MC?
9x-13=6x-4
9x-6x=-4+13
3x=9
x=3
AI = 3x -2
AI = 3(3) – 2
AI = 7
MC = 9x – 13
MC = 9(3) – 13
MC = 14
Combining like terms by applying
APE
Simplifying
Substitution
Simplify
Substitution
Simplify
Learning Task 1: Directions: Answer the questions that follows based on the given figure. Choose the correct answer
inside the box. Write the letter of the correct answer before each number.
A. 16
̅̅̅̅
B. 𝐴𝐶
C. 27
̅̅̅̅
D. 𝑂𝑁
̅̅̅̅
E. 𝐵𝐶
F. 34
̅̅̅̅̅
G. 𝑀𝑂
̅̅̅̅̅
H. 𝑀𝑁
I. 22
J. ̅̅̅̅
𝐴𝐵
______ 1. Which side is parallel to ̅̅̅̅
𝐴𝐵?
______ 2. In Δ𝑂𝑀𝑁, which side the midline?
̅̅̅̅ ?
______ 3. Which side is parallel to 𝐴𝐶
̅̅̅̅ ?
______ 4. Which side is parallel to 𝐵𝐶
______ 5. In Δ𝑁𝑂𝑀, which side the midline?
______ 6. In Δ𝑀𝑁𝑂, which side the midline?
______ 7. In Δ𝑁𝑂𝑀, if the ̅̅̅̅
𝐴𝐶 = 13.5 how long is ̅̅̅̅̅
𝑀𝑁 ?
̅̅̅̅ = 17 how long is 𝑂𝐶
̅̅̅̅ ?
______ 8. In Δ𝑂𝑀𝑁, if the 𝐴𝐵
̅̅̅̅ = 11 how long is 𝑀𝑂
̅̅̅̅̅?
______ 9. In Δ𝑀𝑁𝑂, if the 𝐵𝐶
̅̅̅̅̅ = 32 how long is 𝐵𝐶
̅̅̅̅ ?
______ 10. In Δ𝑀𝑁𝑂, if the 𝑀𝑂
Check the box:
Completely done.
Incompletely done. Reason: ______________________________________________________
______________________________________
Signature over Printed Name
Read and understand your Learners’ Material on page 310. (Basahin at unawain ang inyong libro sa pahina 310.)
LESSON 7: TRAPEZOID
Definition of Trapezoid – a quadrilateral with one pair of parallel sides.
Midsegment Theorem of Trapezoid – The median of a trapezoid is parallel to each base and its length is one half
the sum of the lengths of the bases.
TUESDAY
2. Proves
theorems on
trapezoid.
̅ at P.
Given: Trapezoid MINS with median ̅̅̅̅
𝑇𝑅 that intersects the diagonal 𝐼𝑆
1
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅
̅
̅
̅
̅̅̅̅
̅
̅
̅
̅
Prove: 𝑀𝑆//𝑇𝑅//𝐼𝑁 and 𝑇𝑅 = (𝑀𝑆 + 𝐼𝑁)
2
Statements
̅̅̅̅ that intersects
1. Trapezoid MINS with median 𝑇𝑅
̅ at P.
the diagonal 𝐼𝑆
̅̅̅̅ + 𝑃𝑅
̅̅̅̅ = 𝑇𝑅
̅̅̅̅
2. 𝑇𝑃
1
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅ = 1 𝑀𝑆
̅̅̅̅ and ̅̅̅̅
̅̅̅
3. 𝑇𝑃 //𝑀𝑆 and 𝑇𝑃
𝑅𝑃 //𝐼𝑁
𝑅𝑃 = ̅𝐼𝑁
4.
5.
6.
7.
̅̅̅̅//𝐼𝑁
̅̅̅̅
𝑀𝑆
̅̅̅̅//𝑇𝑃
̅̅̅̅ + 𝑅𝑃
̅̅̅̅ //𝐼𝑁
̅̅̅̅
𝑀𝑆
̅̅̅̅//𝑇𝑅
̅̅̅̅ //𝐼𝑁
̅̅̅̅
𝑀𝑆
̅̅̅̅ + 1 𝐼𝑁
̅̅̅̅ = 1 𝑀𝑆
̅̅̅̅
𝑇𝑅
8.
̅̅̅̅ + 𝐼𝑁
̅̅̅̅ = 1 (𝑀𝑆
̅̅̅̅ )
𝑇𝑅
2
2
2
2
2
Reasons
Given
Segment Addition Postulate
Midline Theorem
Definition of Trapezoid
Transitive Property
Addition Property
Substitution Property
Factoring
Definition of Isosceles Trapezoid – a trapezoid with opposite sides that are congruent.
Properties of Isosceles Trapezoid

The base angles of an isosceles trapezoid are congruent.
̅̅̅̅
Given: Isosceles Trapezoid AMOR with ̅̅̅̅̅
𝑀𝑂 //𝐴𝑅
Prove: ∠𝐴 ≅ ∠𝑅 and ∠𝐴𝑀𝑂 ≅ ∠𝑂
Statements
̅̅̅̅̅ //𝐴𝑅
̅̅̅̅.
1. Isosceles Trapezoid AMOR with 𝑀𝑂
̅̅̅̅̅ ≅ 𝑂𝑅
̅̅̅̅
2. 𝐴𝑀
̅̅̅̅ where E lies on 𝐴𝑅
̅̅̅̅
̅̅̅̅̅ //𝑂𝑅
3. From M, draw 𝑀𝐸
4. Quadrilateral MORE is a parallelogram
5. ̅̅̅̅̅
𝑀𝐸 ≅ ̅̅̅̅
𝑂𝑅
̅̅̅̅̅ ≅ ̅̅̅̅̅
6. 𝑀𝐴
𝑀𝐸
7. ∆𝐴𝑀𝐸 is an isosceles triangle
8. ∠𝐴 ≅ ∠1
9. ∠1 ≅ ∠𝑅
10. ∠𝐴 ≅ ∠𝑅
11. ∠𝐴 and ∠𝐴𝑀𝑂 are supplementary angles.
∠𝑂 and ∠𝑅 are supplementary angles.
12. ∠𝐴𝑀𝑂 ≅ ∠𝑂


Reasons
Given
Definition of Isosceles Trapezoid
Two points determine a line
Definition of Parallelogram
In a parallelogram, any two opposite sides are congruent.
Transitive Property
Definition of Isosceles Triangle
Base angles of an isosceles triangles are congruent.
Corresponding Angles Theorem
Transitive Property
Same Side Interior Angle Theorem
Supplements of congruent angles are also congruent
Opposite angles of an isosceles trapezoid are supplementary.
Diagonals of an isosceles trapezoid are congruent.
Learning Task 2A: Directions: Complete the two-column proof by choosing the correct word/s inside the box. Write
the letter of the correct answer on the space provided. (Kumpletuhin ang two-column proof sa pamamagitan ng
pagpili ng tamang salita o mga salita sa loob ng kahon. Isulat ang letra ng tamang sagot sa nakalaang sulatan.)
A. ∠𝑆𝑇𝑅
B. Transitive Property
C. Definition of Isosceles Trapezoid
̅̅̅̅
̅̅̅̅ ∥ 𝐴𝑆
D. Isosceles trapezoid ARTS with 𝑅𝑇
E. ∠𝐴𝑅𝑆 and ∠𝑆 are supplementary. ∠𝐴 and ∠𝑇 are supplementary.
Opposite angles of an isosceles trapezoid are supplementary.
Given: Isosceles
Trapezoid ARTS with
̅̅̅̅
̅̅̅̅
𝑅𝑇 //𝐴𝑆
Prove: ∠𝐴𝑅𝑆 and ∠𝑆
are supplementary.
∠𝐴 and ∠𝑇 are
supplementary.
Statements
1. (1)
̅̅̅̅ ≅ ̅𝑆𝑇
̅̅̅
2. 𝐴𝑅
3. (3) and ∠𝐴𝑅𝑇 ≅ ∠𝑆
4.
5.
6.
∠𝐴 + ∠𝐴𝑅𝑇 = 1800 and
∠𝑆 + ∠𝑇 = 1800
∠𝐴 + ∠𝑇 = 1800 and ∠𝑆 +
∠𝐴𝑅𝑇 = 1800
(5)
Reasons
Given
(2)
The base angles of an isosceles trapezoid are
congruent.
Same Side Interior Angle Theorem
(4)
Definition of Supplementary Angles
Answer:
1. ____________
2. ____________
3. ____________
4. ____________
5. ____________
Learning Task 2B: Directions: Complete the two-column proof by choosing the correct word/s inside the box. Write
the letter of the correct answer on the space provided. (Kumpletuhin ang two-column proof sa pamamagitan ng
pagpili ng tamang salita o mga salita sa loob ng kahon. Isulat ang letra ng tamang sagot sa nakalaang sulatan.)
A. Given
̅̅̅̅̅
B. ̅̅̅̅
𝑂𝑅 ≅ 𝑀𝐴
C. Δ𝑂𝑅𝐴 ≅ Δ𝑀𝐴𝑅
D. Reflexive Property
E. The base angles of an isosceles trapezoid are congruent.

Diagonals of an isosceles trapezoid are congruent.
Given: Isosceles
Trapezoid ROMA with
̅̅̅̅
̅̅̅̅̅ and 𝐴𝑂
diagonals 𝑅𝑀
̅̅̅̅̅ ≅ ̅̅̅̅
Prove: 𝑅𝑀
𝐴𝑂
Statements
1. Isosceles Trapezoid ROMA with
̅̅̅̅.
̅̅̅̅̅ and 𝐴𝑂
diagonals 𝑅𝑀
2. (2)
3. ∠𝑂𝑅𝐴 ≅ ∠𝑀𝐴𝑅
̅̅̅̅ ≅ 𝐴𝑅
̅̅̅̅
4. 𝑅𝐴
5. (5)
̅̅̅̅
̅̅̅̅̅ ≅ 𝐴𝑂
6. 𝑅𝑀
Reasons
(1)
Definition of Isosceles Trapezoid
(3)
(4)
SAS Postulate
CPCTC
Answer:
6. ____________
7. ____________
8. ____________
9. ____________
10. ____________
Check the box:
Completely done.
Incompletely done. Reason: ______________________________________________________
______________________________________
Signature over Printed Name
Read and understand the lesson below. (Basahin at unawain ang aralin na ipinaliwanag sa ibaba.)
LESSON 8: KITE
Definition of Kite – a quadrilateral with two pairs of adjacent sides that are congruent, a rhombus is a special kind
of kite.
Properties of Kite

In a kite, the perpendicular bisector of at least one is the other diagonal.

The area of a kite is half the product of the lengths of its diagonals.
̅̅̅̅ intersect at point W.
̅̅̅̅ and 𝑂𝐸
Given: Kite ROPE with diagonals 𝑃𝑅
1
̅̅̅̅
̅̅̅̅
Prove: Area of kite ROPE = (𝑈𝐸 )(𝑃𝑅).
2
WEDNESDAY
Statements
̅̅̅̅ intersect at
1. Kite ROPE with diagonals ̅̅̅̅
𝑃𝑅 and 𝑂𝐸
point W.
̅̅̅̅
̅̅̅̅ ⊥ 𝑂𝐸
2. 𝑃𝑅
3. Area of kite ROPE = Area of ∆𝑂𝑃𝐸 + Area of ∆𝑂𝑅𝐸
1
̅̅̅̅ )(𝑃𝑊
̅̅̅̅̅ ) and Area of ∆𝑂𝑅𝐸
4. Area of ∆𝑂𝑃𝐸 = (𝑂𝐸
2
1
̅̅̅̅ )(𝑊𝑅
̅̅̅̅̅)
= (𝑂𝐸
Reasons
Given
1
Factoring
6.
7.
̅̅̅̅ )(𝑃𝑊
̅̅̅̅̅ + 𝑊𝑅
̅̅̅̅̅)
Area of kite ROPE = = (𝑂𝐸
2
̅̅̅̅̅ + 𝑊𝑅
̅̅̅̅̅ = 𝑃𝑅
̅̅̅̅
𝑃𝑊
1
̅̅̅̅ )(𝑃𝑅
̅̅̅̅ )
Area of Kite ROPE = (𝑈𝐸
8.
̅̅̅̅ ≅ 𝑇𝑋
̅̅̅̅
𝐶𝑋
CPCTC
2
3. Proves
theorems on
kite.
5.
2
Diagonals of a kite are perpendicular to each other.
Area Addition Postulate
Formula for Area of Triangles
Segment Addition Postulate
Substitution
Learning Task 3A: Directions: Complete the two-column proof by choosing the correct word/s inside the box. Write
the letter of the correct answer on the space provided. (Kumpletuhin ang two-column proof sa pamamagitan ng
pagpili ng tamang salita o mga salita sa loob ng kahon. Isulat ang letra ng tamang sagot sa nakalaang sulatan.)
A. Given
̅̅̅̅ ≅ 𝑀𝐴
̅̅̅̅̅
B. 𝑂𝑅
C. Δ𝑊𝑂𝑅 ≅ Δ𝑊𝐷𝑅
D. Definition of a Kite
E. The base angles of an isosceles trapezoid are congruent.

A diagonal of a kite is an angle bisector of a pair of opposite angles.
Given: Kite WORD with
Statements
diagonals ̅̅̅̅
𝑂𝐷 and ̅̅̅̅̅
𝑊𝑅.
1. Kite WORD with diagonals ̅̅̅̅
𝑂𝐷 and ̅̅̅̅̅
𝑊𝑅.
̅̅̅̅̅ is angle
̅̅̅̅̅
̅̅̅̅
̅̅̅̅̅
̅̅̅̅
Prove: 𝑊𝑅
2. 𝑊𝑂 ≅ 𝑊𝐷 and 𝑅𝑂 ≅ 𝑅𝐷
bisector of ∠𝑂𝑊𝐷 and
̅̅̅̅̅ ≅ 𝑊𝑅
̅̅̅̅̅
3. 𝑊𝑅
Reasons
(1)
(2)
Reflexive Property
∠𝑂𝑅𝐷.
4.
5.
6.
(3)
(4)
̅̅̅̅̅ is angle bisector of ∠𝑂𝑊𝐷 and ∠𝑂𝑅𝐷.
𝑊𝑅
SSS Postulate
(5)
Definition of Angle Bisector
Answer:
1. ____________
2. ____________
3. ____________
4. ____________
5. ____________
Learning Task 3B: Directions: Complete the two-column proof by choosing the correct word/s inside the box. Write
the letter of the correct answer on the space provided. (Kumpletuhin ang two-column proof sa pamamagitan ng
pagpili ng tamang salita o mga salita sa loob ng kahon. Isulat ang letra ng tamang sagot sa nakalaang sulatan.)
A. Given
B. UC = UT; CE = TE
C. Definition of a Kite
D. Definition of congruent segments
̅̅̅̅
̅̅̅̅ is the perpendicular bisector of 𝐶𝑇
E. 𝑈𝐸
Given: Kite CUTE with
̅̅̅̅
̅̅̅̅ and 𝐶𝑇
diagonals 𝑈𝐸
intersect at point X.
̅̅̅̅ is the
Prove: 𝑈𝐸
perpendicular bisector of
̅̅̅̅.
𝐶𝑇
Statements
̅̅̅̅
̅̅̅̅ and 𝐶𝑇
1. Kite CUTE with diagonals 𝑈𝐸
intersect at point X.
̅̅̅̅ ≅ 𝑇𝐸
̅̅̅̅
2. ̅̅̅̅
𝑈𝐶 ≅ ̅̅̅̅
𝑈𝑇 ; 𝐶𝐸
3.
(8)
4.
(10)
Reasons
1.
(6)
2. (7)
3. (9)
4. If a line contains two points each of
which is equidistant from the endpoints f
a segment, then the line is the
perpendicular bisector of the segment.
Answer:
6. ____________
7. ____________
8. ____________
9. ____________
10. ____________
Check the box:
Completely done.
Incompletely done. Reason: ______________________________________________________
______________________________________
Signature over Printed Name
Read and understand the lesson below. (Basahin at unawain ang aralin na ipinaliwanag sa ibaba.)
THURSDAY
LESSON 9: SOLVE PROBLEMS INVOLVING PARALLELOGRAM, TRAPEZOID AND KITE
4. Solves
problems
involving
parallelogram,
trapezoids
and kites.
In this lesson, we shall focus on solving problems involving the relationship of sides and angles in
parallelograms, trapezoids, and kites using their properties and different theorems. We need to remember all the
definitions, properties, and theorems that we have already discussed regarding parallelograms, trapezoids, and
kites in the previous lessons.
Steps in Geometric Problem Solving:
1. Read the problem carefully.
2. Recognize the relationship of the given figure.
3.
4.
5.
Pay attention to the labels.
Use appropriate definition, property, postulate, or theorem.
Answer the question.
EXAMPLES:
1. Given: Quadrilateral WISH is a parallelogram
a. If m∠𝑊 = (x + 15)0 and m∠𝑆 = (2x + 5)0, what is m∠𝑊?
m∠𝑊 = m∠𝑆
In a parallelogram, any two opposite angles are congruent.
(x + 15)0 = (2x + 5)0
Substitution
(x – x + 15 – 5)0 = (2x – x + 5 – 5)0
Addition Property of Equality
x = 100
Subtraction and Addition Property
m∠𝑊 = ((10) + 15)0
Substitution
m∠𝑊 = 250
Addition Property
̅̅̅̅ = y + 13, how long is 𝐻𝑆
̅̅̅̅?
̅̅̅̅ = 3y + 3 and 𝐻𝑆
If 𝑊𝐼
̅̅̅̅
̅̅̅̅ ≅ 𝐻𝑆
In a parallelogram, any two opposite sides are congruent.
𝑊𝐼
3y + 3 = y + 13
Substitution
3y – y + 3 – 3 = y – y + 13 – 3
Addition Property of Equality
2y = 10
Subtraction and Addition Property
y=5
Dividing both sides by 2
̅̅̅̅
Substitution
𝐻𝑆 = (5) + 13
̅̅̅̅ = 18
Addition Property
𝐻𝑆
b.
c.
Quadrilateral WISH is a rectangle, and its perimeter is 56 cm. One side is 5 cm less than twice the
other side. What are the dimensions and how large is its area?
Perimeter of Rectangle = 2L + 2W
Formula for Perimeter of Rectangle
56 = 2L + 2(2L – 5) cm
Substitution
56 = 2L + 4L – 10 cm
Distributive Property
56 + 10 = 6L – 10 + 10 cm
Addition Property of Equality
6L = 66 cm
Addition Property
L = 11 cm
Dividing both sides by 6
56 = 2(11) + 2W cm
Substitution
56 = 22 + 2W cm
Multiplication Property
56 – 22 = 22 – 22 + 2W cm
Addition Property of Equality
2W = 34 cm
Subtraction Property
W = 17 cm
Dividing both sides by 2
Area of Rectangle = LW
Formula for Area of Rectangle
Area of Rectangle = 11 cm * 17 cm
Substitution
Area of Rectangle = 187 cm2
Multiplication Property
d.
What is the perimeter and the area of the largest square that can be formed from Rectangle WISH
from the previous question?
L = 11 cm
Determine the smaller number from the length and width of the
rectangle
Area of Square = s2
Formula for Area of Square
Area of Square = (11 cm)2
Substitution
Area of Square = 121 cm2
Multiplication Property
2.
̅̅̅̅//𝑷𝑻
̅̅̅̅ and 𝑬𝑹
̅̅̅̅ is its median.
Given: Isosceles trapezoid POST with 𝑶𝑺
a. If ̅̅̅̅
𝑂𝑆 = 3x – 2, ̅̅̅̅
𝑃𝑇 = 2x + 10 and ̅̅̅̅
𝐸𝑅 = 14, how long is each base?
1
Formula for length of median
̅̅̅̅ + 𝑃𝑇
̅̅̅̅ = (𝑂𝑆
̅̅̅̅)
𝐸𝑅
2
1
Substitution
14 = ((3x – 2) + (2x + 10))
2
28 = 5x + 8
Combining like terms and simplifying
28 − 8 = 5x + 8 − 8
Addition Property of Equality
5x = 20
Subtraction Property
𝑥=4
Dividing both sides by 5
̅̅̅̅ = 3(4) – 2
Substitution
𝑂𝑆
̅̅̅̅
Multiplication and Subtraction Property
𝑂𝑆 = 10
̅̅̅̅ = 2(4) + 10
Substitution
𝑃𝑇
̅̅̅̅ = 18
Multiplication and Addition Property
𝑃𝑇
b. If m∠𝑃 = (2x + 5)0 and m∠𝑂
m∠𝑃 and m∠𝑂 are supplementary
(2x + 5)0 + (3x – 10)0 = 1800
(5x – 5)0 = 1800
= (3x – 10)0, what is m∠𝑇?
Same Side Interior Angles are Supplementary
Substitution
Addition and Subtraction Property
(5x – 5 + 5)0 = (180 + 5)0
5x = 1850
x = 370
m∠𝑃 and m∠𝑇 are congruent
m∠𝑇 = (2(37) + 5)0
m∠𝑇 = 790
Addition Property of Equality
Addition Property
Divide both sides by 5
In an isosceles trapezoid, base angles are congruent
Substitution
Learning Task 4: Directions: Illustrate the following and solve for what is required. Show your complete solution in
a separate sheet of paper. Write the letter of the correct answer before each number.
_____ 1. One side of a rectangle is 3 m more than the other. If the perimeter of the rectangle is 30 m, what are its
dimensions?
A. L = 4 m and W = 7 m
C. L = 6 m and W = 9 m
B. L = 5 m and W = 8 m
D. L = 7 m and W = 10 m
_____ 2. A rhombus with a perimeter of 60 in has a side with a length of (8x) in. Find x.
A. 1.675
B. 1.875
C. 2.275
D. 7.5
_____ 3. One base of a trapezoid is 4 cm less than twice the other. If the median measures 13 cm, what is the
length of the longer base?
A. 10
B. 12
C. 16
D. 20
̅̅̅̅//𝑃𝑇
̅̅̅̅. If m∠O = (10x + 20)O and m∠P= 8x – 2)O , what is x?
______4. Isosceles trapezoid POST with 𝑂𝑆
A. 9
B. 10
C. 11
D. 12
_____ 5. In parallelogram ABCD, if BC is 3x+25 and DA is 5x-15. What is the length of BC and DA?
A. 85
B. 86
C. 87
D. 88
Check the box:
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Signature over Printed Name
Read and understand the lesson below. (Basahin at unawain ang aralin na ipinaliwanag sa ibaba.)
LESSON 9: SOLVE PROBLEMS INVOLVING PARALLELOGRAM, TRAPEZOID AND KITE
̅̅̅̅//𝑷𝑻
̅̅̅̅ and 𝑬𝑹
̅̅̅̅ is its median.
Given: Isosceles trapezoid POST with 𝑶𝑺
̅̅̅̅ is 6 cm long. If its perimeter is 27 cm, how long is each leg?
a. One base is twice the other and 𝐸𝑅
1
Formula for length of median
̅̅̅̅ + 𝑃𝑇
̅̅̅̅ = (𝑂𝑆
̅̅̅̅)
𝐸𝑅
2
1
Substitution
6 = ((x) + (2x))
2
12 = 3x
Combining like terms and simplifying
𝑥=4
Dividing both sides by 3
Perimeter of Isosceles Trapezoid = 2L Formula of Perimeter of Isosceles Trapezoid
+ B1 + B2
27 = 2L + 4 + 2(4)
Substitution
27 = 2L + 12
Multiplication and Addition Property
27 – 12 = 2L + 12 – 12
Addition Property of Equality
2L = 15
Multiplication and Addition Property
L = 7.5 cm
Dividing both sides by 2
3.
FRIDAY
5. Solves
problems
involving
parallelogram,
trapezoid and
kites.
̅̅̅̅ is 8.5 inches long and one leg measures 9 inches. What is its perimeter if one of the bases is 3
𝐸𝑅
inches more than the other?
1
Formula for length of median
̅̅̅̅ + 𝑃𝑇
̅̅̅̅ = (𝑂𝑆
̅̅̅̅)
𝐸𝑅
2
1
Substitution
8.5 = ((x) + (x + 3))
2
17 = 2x + 3
Combining like terms and simplifying
17 – 3 = 2x + 3 – 3
Addition Property of Equality
2x = 14
𝑥=7
Dividing both sides by 2
Perimeter of Isosceles Trapezoid = 2L Formula of Perimeter of Isosceles Trapezoid
+ B1 + B2
Perimeter of Isosceles Trapezoid =
Substitution
2(9) + 7 + (7 + 3)
Perimeter of Isosceles Trapezoid = 18 Multiplication and Addition Property
+ 7 + 10
Perimeter of Isosceles Trapezoid = 35 Addition Property
inches
b.
4.
Given: Quadrilateral LIKE is a kite with ̅̅̅
𝑳𝑰 ≅ ̅̅̅̅
𝑰𝑲 and ̅̅̅̅
𝑳𝑬 ≅ ̅̅̅̅
𝑲𝑬.
̅̅̅̅ is twice 𝐿𝐼
̅ . If its perimeter is 21 cm, how long is 𝐿𝐸
̅̅̅̅ ?
a. 𝐿𝐸
Perimeter of Kite = 2S1 + 2S2
Formula of Perimeter of Kite
̅ + 2(2𝐿𝐼
̅)
Substitution
21 = 2𝐿𝐼
̅
Multiplication Property
21 = 2S1 + 4𝐿𝐼
̅
Combining like terms
21 = 6𝐿𝐼
̅ = 3.5 cm
Dividing both sides by 6
𝐿𝐼
̅̅̅̅ = 7 cm
̅̅̅̅ is twice 𝐿𝐼
̅
𝐿𝐸
𝐿𝐸
What is the area if one of the diagonals is 4 more than the other and ̅̅̅
𝐼𝐸 + ̅̅̅̅
𝐿𝐾 = 16 inches?
̅̅̅ + 𝐿𝐾
̅̅̅̅ = 16
Given
𝐼𝐸
x + (x + 4) = 16
Substitution
2x + 4 = 16
Combining like terms
2x + 4 – 4 = 16 – 4
Addition Property of Equality
2x = 12
Subtraction Property
x=6
Dividing both sides by 2
1
Formula of Area of Kite
Area of Kite = D1D2
b.
2
1
̅̅̅ )(𝐿𝐾
̅̅̅̅)
Area of Kite = (𝐼𝐸
Substitution
Area of Kite = (6)(10)
Substitution
2
1
2
Area of Kite = 30 inches
c.
2
Multiplication Property
̅̅̅ = (x – 1) ft and 𝐿𝐾
̅̅̅̅ = (x + 2) ft. If its area is 44 ft2, how long are 𝐼𝐸
̅̅̅ and 𝐿𝐾
̅̅̅̅ ?
𝐼𝐸
1
Formula of Area of Kite
Area of Kite = D1D2
2
1
̅̅̅ )(𝐿𝐾
̅̅̅̅)
Area of Kite = (𝐼𝐸
1
Substitution
2
Substitution
44= (𝑥 − 1)(𝑥 + 2)
2
88 = x2 + x – 2
x2 + x – 2 – 88 = 0
x2 + x – 90 = 0
(x – 9)(x + 10) = 0
x = 9 or -10
̅̅̅
𝐼𝐸 = ((9) – 1) ft and ̅̅̅̅
𝐿𝐾 = ((9)+ 2) ft
̅̅̅
𝐼𝐸 = 8 ft and ̅̅̅̅
𝐿𝐾 = 11 ft
Simplifying
Transposition
Subtraction Property
Factoring
Zero Product Rule but only consider 9 since there is no negative
measure.
Substitution
Addition and Subtraction Property
Learning Task 5: Directions: Illustrate the following and solve for what is required. Show your complete solution in
a separate sheet of paper. Write the letter of the correct answer on the space provided.
_____ 1. One base of a trapezoid is 6 cm less than twice the other. If the median measures 15 cm, what is the
length of the longer base?
A. 10
B. 12
C. 16
D. 20
̅̅̅̅//𝑀𝑃
̅̅̅̅̅. If m∠A = (10x + 20)O and m∠C= 8x – 2)O , what is the 𝑚∠𝐴?
_____ 2. Isosceles trapezoid CAMP with 𝐶𝐴
A. 90
B. 100
C. 110
D. 120
_____ 3. In parallelogram HIJK, if IJ is x+24 and HK is 5x-15. What is the length of IJ and HK?
A. 20
B. 25
C. 30
D. 35
_____ 4. STUV is a trapezoid and QR is a median, if ST= 10 and VU=12, what is the length of the median?
A. 10
B. 12
C. 11
D. 13
_____ 5. STUV is a trapezoid and QR is a median, if ST=10 and VU=8, how about QR?
A. 6
B. 8
C. 7
D. 9
Check the box:
Completely done.
Incompletely done. Reason: ______________________________________________________
______________________________________
Signature over Printed Name
WRITTEN WORK #2
SATURDAY
5. To
answer
correctly
the
Written
Work #2
A.
Q
Write the letter of the correct answer before each number.
_____ 1. What are the bases of this trapezoid?
I.
QR
III. UV
II.
QS
IV. ST
A. I and II
C. I and IV
U
T
R
V
S
and
Performa
nce task
#2
B. III and IV
D. I, II, and IV
_____ 2. If QR= 9 and ST=15, what is the length of the median?
A. 10
B. 11
C. 12
D. 13
B
A
_____ 3. If ST=8 and QR=6, how about UV?
A. 6
B. 7
C. 8
D. 9
F
E
_____ 4. Trapezoid ABCD at the right is an isosceles trapezoid. If AB = 36 and
CD = 50, what is the length of median EF?
D
C
A. 39
B. 43
C. 44
D.
45
_____ 5. If BF = FC and the length of BF = 9x + 6 and F = 10x - 4, what is the length of BD?
A. 10
B. 96
C. 146
D. 192
______6. ABCD is a trapezoid, if the length of AB is equal to 36 and CD is equal to 50, what is the length of
median?
A. 39
B. 44
C. 43
D. 49
_____ 7. ABCD is a trapezoid, EF is a median. If the length of BF is equal to 9x+6 and FC is equal to 10x-4, what is
the length of BC?
A. 200
B. 192
C. 195
D. 189
̅̅̅̅̅ and 𝑍𝑋
̅̅̅̅ intersect at V. If 𝑊𝑋 = 5, 𝑉𝑍 = 6 𝑎𝑛𝑑 𝑉𝑌 = 3,
Quadrilateral 𝑊𝑋𝑌𝑍 is a kite with diagonals 𝑊𝑌
𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔:
̅̅̅̅̅
_____ 8. What is the length of 𝑊𝑉
A. 7
B. 4
C. 6
D. 3
_____ 9. What is the length of diagonal ̅̅̅̅
𝑋𝑍 and ̅̅̅̅̅
𝑊𝑌
A. 6 and 10
B. 4 and 7
C. 8 and 6
D. 8 and 7
_____ 10. What is the area of the Kite WXYZ?
A. 20 sq. units
B. 30 sq. units
C. 40 sq. units
D. 50 sq. units
_____ 11. QRST is a trapezoid and UV is a median, if QR= 9 and ST=15, what is the length of the median?
A. 10
B. 12
C. 11
D . 13
_____ 12. QRST is a trapezoid and UV is a median, if ST=8 and QR=6, how about UV?
A. 6
B. 8
C. 7
D. 9
_____ 13. ABCD is a trapezoid, if the length of AB is equal to 36 and CD is equal to 50, what is the length of
median?
A. 39
B. 44
C. 43
D. 49
_____ 14. ABCD is a trapezoid, EF is a median. If the length of BF is equal to 9x+6 and FC is equal to 10x-4, what
is the length of BC?
A. 200
B. 192
C. 195
D. 189
_____ 15. GOLD is a square with the diagonals are GL and OD If GL = 10x – 11 and OD= 8x-1. Find the diagonal
GL?
A. 49 units
B. 39 units
C. 36 units
D. 47 units
B. Determine whether each statement is TRUE or FALSE. Write the word TRUE if the statement is correct or else
write the word FALSE.
___________ 1. A kite is a polygon.
___________ 2. Every quadrilateral is a kite.
___________ 3. The area of a kite is equal to the product of the length of its diagonals.
___________ 4. One of the diagonals of a kite is the perpendicular bisector of the other diagonal.
___________ 5. A kite has two distinct pairs of congruent consecutive sides.
B.
̅̅̅̅ and 𝐵𝐷
̅̅̅̅. If AB = 5, EB = 4, and EC = 6, find the
Quadrilateral ABCD below is a kite with diagonals 𝐴𝐶
following:
A. 3 units
B. 2√13 units
̅̅̅̅
______ 1. The length of 𝐴𝐸
̅̅̅̅
______ 2. The length of diagonal 𝐵𝐷
̅̅̅̅
______ 3. The length of diagonal 𝐴𝐶
______ 4. The area of kite ABCD
______ 5. The length of ̅̅̅̅
𝐵𝐶
C. 8 units
D. 9 units
E. 36 sq. units
Performance Task 2
Who Am I?
Identify what kind of quadrilaterals are being described.
1.
Description
I am a Parallelogram
My diagonals are congruent
I have two pairs of congruent sides
I have all right angles
2.
I am a Parallelogram
My diagonals are perpendicular
All my sides are congruent
My diagonals bisect my angles
3.
I am a Parallelogram
I am a Rectangle
I am a Rhombus
I have four congruent sides
My diagonals are perpendicular and congruent
4.
I am a Quadrilateral
I have exactly one pair of parallel sides
The angles that connect my bases are
supplementary
5.
I am a Quadrilateral
I have two pairs of consecutive sides that are
congruent
WHO Am I?
Draw Me!
Check the box:
Completely done.
Incompletely done. Reason: ______________________________________________________
______________________________________
Signature over Printed Name
MODE OF DELIVERY/ RETRIEVAL
Modular:
 Printed modules were distributed ahead of time.
 Teachers may follow up their students in any form of communications.
 Students orand
parents
may ask the assistance
their teacher in any form of communication or they may ask assistance of the LR facilitators.
Comments
suggestions
for this of
week:
______________________________________________________________________________________________
Students should submit their outputs every Friday through the Schools drop off points/ neighboring teachers. Failed to submit the output
______________________________________________________________________________________________
means that the student is considered absent on the specific week.
________________________________
Parent’s Signature
Prepared by:
ALDEN P. ENCIO
Teacher II
Checked by:
Reviewed by:
Noted by:
EVELYN A. LAZO
Master Teacher II
ALMIDA T. CAMITAN
Head Teacher IV
MILDRED M. DE LEON
Principal III