MATH 9
QUARTER 3 Week 3
NAME: ____________________________________________GR. & SEC: _____________________
Competency:
The learner:
1. proves the Midline Theorem. (M9GE-lll-d-1)
2. proves theorems on trapezoids and kites. (M9GE-lll-d-2)
To the Learners:
Before starting the module, I want you to set aside other tasks that will disturb you
while enjoying the lessons. Read the simple instructions below to successfully
enjoy the objectives of this kit. Have fun!
1. Follow carefully all the contents and instructions indicated in every page of this
module.
2. Write on your notebook the concepts about the lessons. Writing enhances learning;
that is important to develop and keep in mind.
3. Perform all the provided activities in the module.
4. Let your facilitator/guardian assess your answers using the answer key card.
5. Analyze conceptually the posttest and apply what you have learned.
6. Enjoy studying!
Expectations
This module was designed to help you prove the midline theorem and the theorems
on trapezoids and kites.
After going through this module, you are expected to:
1. Prove the Midline Theorem and solve problem using it.
2. Prove the theorems related to trapezoid and kites.
3. Solve problems involving trapezoids and kites by applying the theorems.
Pre-test
Direction: Choose the letter of the correct answer.
1. In the figure at the right, DE is the midline of ∆𝐴𝐵𝐶 and AC = 12 cm. What is the
length of DE?
B
a. 6 cm
b. 8 cm
D
E
c. 12 cm
d. 24 cm
2. What is true about an isosceles trapezoid?
a. It has exactly one pair of parallel sides.
b. It has congruent legs.
c. It has congruent base angles.
d. All of the above.
MATH 9 QUARTER 3 WEEK 3
A
C
3. In the figure at the right, PQRS is an isosceles trapezoid.
If 𝑚∠𝑃 = 105°. What is the measure of ∠𝑅?
P
105˚
a. 75°
b. 85°
c. 105°
S
d. 180°
4. In the figure at the right, PQ = 16cm and SR = 24 cm.
What is the measurement of ̅̅̅̅
𝑇𝑈?
a. 8 cm
b. 18 cm
c. 20 cm
d. 40 cm
P
T
S
Q
R
Q
U
R
5. What is the length of the diagonal of a kite whose area is 176 sq.cm and the other
diagonal is 16 cm long?
a. 22 cm
b. 24 cm
c. 26 cm
d. 28 cm
Looking Back to your Lesson
In a parallelogram,
1. any two opposite sides are congruent;
2. any two opposite angles are congruent;
3. two consecutive angles are supplementary;
4. the diagonals bisect each other;
5. the diagonals of a parallelogram divides the parallelogram into two congruent
triangles.
The different characteristics of special types of parallelogram
In a rectangle,
1. all four angles are right angles;
2.the diagonals are congruent.
In a rhombus,
1. each diagonal is the perpendicular bisector of the other diagonal;
2. each diagonal bisects a pair of opposite angles.
In a square,
1. all four sides are congruent;
2. all angles are right angles;
3. opposite angles are congruent and supplementary;
5. consecutive angles are supplementary and congruent;
6. the diagonals are congruent;
8. each diagonal bisects a pair of opposite angles.
7. each diagonal is the perpendicular bisector of the other diagonal;
Introduction of the Topic
You have already studied the different parallelograms. Now you need to explore
the other kinds of quadrilaterals. Specifically, trapezoids and kites; their properties;
and how they will be of use to real life situations. But before that, you should study
first the proof of the midline theorem which you can also apply to solve problems.
MATH 9 QUARTER 3 WEEK 3
The Midline Theorem
The segment that joins the midpoints of two sides of a triangle is parallel to the third
side and half as long.
B
1
3 E
D
F
2
Given: In ∆𝐴𝐵𝐶 , D is the midpoint of ̅̅̅̅
𝐴𝐵 and E
̅̅̅̅
is the midpoint of 𝐵𝐶 .
Prove: ̅̅̅̅
𝐷𝐸 || ̅̅̅̅
𝐴𝐶 and 𝐷𝐸 =
C 4
A
1
𝐴𝐶
2
Proof:
Statements
Reasons
1. In ∆𝐴𝐵𝐶 , D is the midpoint of ̅̅̅̅
𝐴𝐵 and
̅̅̅̅ .
E is the midpoint of 𝐵𝐶
2. In a ray opposite ̅̅̅̅
𝐷𝐸 , there is a point F
̅̅̅̅ ≅ ̅̅̅̅
such that 𝐷𝐸
𝐸𝐹
1. Given
3. ̅̅̅̅
𝐸𝐵 ≅ ̅̅̅̅
𝐸𝐶
3. Definition of Midpoint
4. ∠2 ≅ ∠3
4. Vertical angles are congruent.
5. ∆𝐷𝐵𝐸 ≅ ∆𝐹𝐶𝐸
5. SAS Congruence Postulate (2,3 & 4)
6. ∠1 ≅ ∠4
̅̅̅̅ || 𝐶𝐹
̅̅̅̅
7. 𝐴𝐵
6. CPCTC
̅̅̅̅
8. ̅̅̅̅
𝐷𝐴 ≅ 𝐷𝐵
̅̅̅̅
̅̅̅̅ ≅ 𝐹𝐶
9. 𝐷𝐵
8. Definition of Midpoint
̅̅̅̅ ≅ 𝐹𝐶
̅̅̅̅
10. 𝐷𝐴
10. Transitive Property (8 & 9)
11. Quadrilateral ADFC is a Parallelogram
12. ̅̅̅̅
𝐷𝐸 || ̅̅̅̅
𝐴𝐶
11. Definition of Parallelogram
12.Ray 𝐷𝐹 contains ̅̅̅̅
𝐷𝐸
13. 𝐷𝐸 + 𝐸𝐹 = 𝐷𝐹
13. Segment Addition Postulate
14. 𝐷𝐸 + 𝐷𝐸 = 𝐷𝐹
14. Substitution (2)
15. 2𝐷𝐸 = 𝐷𝐹
̅̅̅̅ ≅ 𝐷𝐹
̅̅̅̅
16. 𝐴𝐶
15. Addition (14)
17. 2𝐷𝐸 = 𝐴𝐶
17. Substitution (15 & 16)
18. 𝐷𝐸 =
2. In a ray, there is exactly one point at a
given distance from the endpoint of the ray.
(Point Plotting Theorem)
7. If two lines are cut by a transversal such
that a pair of alternate interior angles are
congruent, then the lines are parallel.
9. CPCTC- Corresponding Parts of Congruent
Triangles are Congruent
16. Property of Parallelogram
18. Multiplication Property of Equality (17)
1
𝐴𝐶
2
Example
̅̅̅̅ 𝑎𝑛𝑑 𝐵𝐶
̅̅̅̅ respectively.
In ∆𝐴𝐵𝐶, D and E are the midpoints of 𝐴𝐵
B
D
A
E
C
MATH 9 QUARTER 3 WEEK 3
1. If DE = 13, then AC = 26.
4. If BC = 38, then BE = 19.
2. If BC = 38, then BE = 19.
5. If AC = 57, then DE = 28.5.
3. If AD = 11.5, then AB = 23.
6. If DE = x – 2 and AC = x + 10,
then DE = 12
Properties of Trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
The parallel sides are the called the bases. A trapezoid has two pairs of base angles.
The base angles of a trapezoid are two consecutive angles whose common side is a base.
The nonparallel sides are called the legs of the trapezoid. If the legs of a trapezoid are
congruent, then the trapezoid is an isosceles trapezoid.
Theorems on Trapezoid
The Midsegment Theorem. The median of a trapezoid is parallel to each base
and its length is one half the sum of the lengths of the bases.
Look at the proof of the theorem and see how the statements flow to arrive at the
conclusion.
A
B
Given: Trapezoid ABCD with median PQ
̅̅̅̅ ∥ 𝐴𝐵
̅̅̅̅ || 𝐷𝐶
̅̅̅̅ and 𝑃𝑄 = 1 (𝐷𝐶 + 𝐴𝐵)
Prove: 𝑃𝑄
2
P
Q
D
C
E
Proof:
Statements
Reasons
̅̅̅̅
1. Trapezoid 𝐴𝐵𝐶𝐷 with median 𝑃𝑄
1. Given
2. Let ̅̅̅̅
𝐴𝑄 and ̅̅̅̅
𝐷𝐶 be extended and
meet at E.
3. ̅̅̅̅
𝐴𝐵 || ̅̅̅̅
𝐷𝐶
2. The Line Postulate at E
̅̅̅̅ ≅ 𝐶𝑄
̅̅̅̅
4. 𝐵𝑄
4. Definition of median of a trapezoid
5. ∠𝐴𝐵𝑄 ≅ ∠𝐸𝐶𝑄
5. If two parallel lines are cut by a
transversal, then the alternate interior
angles are congruent.
6. ∠𝐵𝑄𝐴 ≅ ∠𝐶𝑄𝐸
6. Vertical angles are congruent.
7. ∆𝐵𝑄𝐴 ≅ ∆𝐶𝑄𝐸
̅̅̅̅ ≅ 𝐸𝐶
̅̅̅̅ , 𝐴𝑄
̅̅̅̅ ≅ ̅̅̅̅
8. 𝐴𝐵
𝐸𝑄
7. ASA Postulate (4,5 & 6)
̅̅̅̅ || 𝐷𝐸
̅̅̅̅ and 𝑃𝑄 =
9. 𝑃𝑄
3. Definition of a trapezoid
8. CPCTC- Corresponding Parts of
Congruent Triangles are Congruent
1
𝐷𝐸
2
9. The Midline Theorem
10. ̅̅̅̅
𝑃𝑄 || ̅̅̅̅
𝐴𝐵
10. A line parallel to one of two parallel line
is parallel to the other line.
11. 𝐷𝐸 = 𝐷𝐶 + 𝐸𝐶
11. Definition of Betweenness
12. 𝐴𝐵 = 𝐸𝐶
12. Definition of congruent segments
13. 𝐷𝐸 = 𝐷𝐶 + 𝐴𝐵
13. Substitution (11 &12)
14. 𝑃𝑄 =
1
(𝐷𝐶
2
+ 𝐴𝐵)
14. Substitution (9 & 13)
Consider the following examples, see how the theorem is used to solve the given
problems.
Q
P
12
Example 1
1. In the figure, ̅̅̅̅̅
𝑀𝑁 is the median of trapezoid PQRS.
M
N
Find MN.
Solution:
R
S
28
MATH 9 QUARTER 3 WEEK 3
1
By Midsegment Theorem
1
2
Substitution
𝑀𝑁 = 2 (𝑃𝑄 + 𝑆𝑅)
𝑀𝑁 = (12 + 28)
𝑴𝑵 = 𝟐𝟎 units
Example 2
̅̅̅̅ is the median of trapezoid LMNO.
In the figure, 𝑃𝑄
Find x, LM, and ON.
Solution:
1
P
By Midsegment Theorem
𝑃𝑄 = 2 (𝐿𝑀 + 𝑂𝑁)
1
25 = [(3𝑥 + 2) + (4𝑥 − 1)]
2
Substitution
4x-1
𝑂𝑁 = 3𝑥 + 2
𝑂𝑁 = 3(7) + 2
𝑂𝑁 = 21 + 2
𝑶𝑵 = 𝟐𝟑 𝒖𝒏𝒊𝒕𝒔
Theorems on Isosceles Trapezoid
Isosceles trapezoids have certain properties that you need to learn to have a
better understanding of its functions. Go through and study the proof of each theorem
on isosceles trapezoid.
Theorem: The base angles of an isosceles trapezoid are congruent.
Given: ABCD is an isosceles trapezoid
A
B
Prove: ∠𝐷 ≅ ∠𝐶
D
Proof:
Statements
E
F
C
Reasons
1. ABCD is an isosceles trapezoid
̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅
2. 𝐴𝐷
1. Given
3. Draw 𝐴𝐸 ⊥ 𝐷𝐶 and 𝐵𝐹 ⊥ 𝐷𝐶
3. There is exactly one perpendicular
from a point to a line.
̅̅̅̅ ≅ 𝐵𝐹
̅̅̅̅
4. 𝐴𝐸
4. Parallel lines are everywhere
equidistant.
5. ∠𝐴𝐸𝐷 𝑎𝑛𝑑 ∠𝐵𝐹𝐶 are right angles.
5. Definition of perpendicularity
6. ∆𝐴𝐸𝐷 𝑎𝑛𝑑 ∆𝐵𝐹𝐶 are right triangles
6. Definition of right triangles
7. ∆𝐴𝐸𝐷 ≅ ∆𝐵𝐹𝐶
7. Hypotenuse-Leg Theorem
8. ∠𝐷 ≅ ∠𝐶
8. CPCTC
2. Definition of Isosceles Trapezoid
Theorem: Opposite angles of an isosceles trapezoid are supplementary.
Given: Isosceles Trapezoid MASK
Prove: ∠𝐴 𝑎𝑛𝑑 ∠𝐾 are supplementary
∠𝑀 𝑎𝑛𝑑 ∠𝑆 are supplementary
Proof:
Statements
1. Isosceles Trapezoid MASK
MATH 9 QUARTER 3 WEEK 3
Reasons
1. Given
M
Q
O
50 = 7𝑥 + 1
7𝑥 = 50 − 1
7𝑥 = 49
𝑥=7
Find the measure of 𝐿𝑀 and 𝑂𝑁
𝐿𝑀 = 4𝑥 − 1
𝐿𝑀 = 4(7) − 1
𝐿𝑀 = 28 − 1
𝑳𝑴 = 𝟐𝟕 𝒖𝒏𝒊𝒕𝒔
3x+2
L
N
2. ∠𝑀 ≅ ∠𝐾 , ∠𝐴 ≅ ∠𝑆
2. An isosceles trapezoid has congruent base
angles.
̅̅̅̅ || 𝑀𝐾
̅̅̅̅̅
3. 𝐴𝑆
3. Definition of trapezoid
4. ∠𝑀 𝑎𝑛𝑑 ∠𝐴 are supplementary
∠𝑆 𝑎𝑛𝑑 ∠𝐾 are supplementary
4. If two parallel lines are cut by a transversal,
the interior angles on the same side of
transversal are supplementary.
5. ∠𝐾 𝑎𝑛𝑑 ∠𝐴 are supplementary
∠𝑆 𝑎𝑛𝑑 ∠𝑀 are supplementary
5. Substitution (4 & 2)
Look at the example. See how the theorem is used to solve the given problem.
Example
Find the measure of each angle in the given isosceles trapezoid.
Solution:
𝒎∠𝑵 = 𝟖𝟐°
Base angles of an isosceles trapezoid are congruent.
𝑚∠𝐶 = 180° − 82°
Opposite angles of an isosceles trapezoid are supplementary.
𝒎∠𝑪 = 𝟗𝟖
Since base angles of an isosceles trapezoid are congruent, then 𝒎∠𝑶 = 𝟗𝟖.
Theorem: The diagonals of an isosceles trapezoid are congruent.
A
B
Given: ABCD is an isosceles trapezoid
̅̅̅̅ ≅ 𝐵𝐷
̅̅̅̅
Prove: 𝐴𝐶
C
D
Proof:
Statements
Reasons
1. Isosceles Trapezoid ABCD where
̅̅̅̅
𝐴𝐷 ≅ ̅̅̅̅
𝐵𝐶 and ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐶𝐷
1. Given
2. ∠𝐷 ≅ ∠𝐶
2. The base angles of an isosceles
trapezoid are congruent.
3. ̅̅̅̅
𝐷𝐶 ≅ ̅̅̅̅
𝐷𝐶
3. Reflexive Property
4. ∆𝐴𝐷𝐶 ≅ ∆𝐵𝐶𝐷
4. SAS Postulate – If 2 sides and the
included angle of one triangle are
congruent to the corresponding part of
another triangle, the triangles are
congruent.
̅̅̅̅ ≅ 𝐵𝐷
̅̅̅̅
5. 𝐴𝐶
5. CPCTC
Look at the example. See how the theorems are used to solve for the measure of
the median, diagonals and angles of the given trapezoid.
Example
Given isosceles trapezoid GRAB whose median is ED, diagonals GA and RB,
𝑚∠𝐵𝐺𝑅 = 111°, 𝐺𝐴 = 63 𝑐𝑚 and 𝐺𝐵 = 18 𝑐𝑚. Find EB, RB, ED, 𝑚∠𝐺𝐵𝐴 and 𝑚∠𝑅𝐴𝐵.
45 cm.
G
R
E
B
D
67 cm
Solve for EB (Def. of median of a trapezoid)
𝐸𝐵 =
𝐸𝐵 =
1
𝐺𝐵
2
1
(18)
2
𝑬𝑩 = 𝟗 𝒄𝒎
MATH 9 QUARTER 3 WEEK 3
A
Solve for RB (Diagonals are congruent)
̅̅̅̅
̅̅̅̅ ≅ 𝐺𝐴
𝑅𝐵
𝑅𝐵 = 𝐺𝐴
𝑹𝑩 = 𝟔𝟑 𝒄𝒎
Solve for ED (Midsegment Theorem)
1
𝐸𝐷 = (𝐺𝑅 + 𝐵𝐴)
2
1
𝐸𝐷 = (45 + 67)
2
1
𝐸𝐷 = (112)
2
𝑬𝑫 = 𝟓𝟔 𝒄𝒎
Solve for 𝑚∠𝐺𝐵𝐴 (Same side interior angles
are supplementary)
𝑚∠𝐺𝐵𝐴 = 180 – 𝑚∠𝐵𝐺𝑅
𝑚∠𝐺𝐵𝐴 = 180 − 111
𝒎∠𝑮𝑩𝑨 = 𝟔𝟗
Solve for 𝑚∠𝑅𝐴𝐵 (Base angles are congruent)
𝑚∠𝑅𝐴𝐵 = 𝑚∠𝐺𝐵𝐴
𝒎∠𝑹𝑨𝑩 = 𝟔𝟗
Properties of Kite
In this section, you will study about kite and its theorems.
A kite is a quadrilateral with two distinct pairs of equal adjacent sides.
Refer to the KITE JKLM, where ̅̅̅
𝐽𝐾 ≅ ̅̅̅̅
𝐾𝐿 and ̅̅̅̅
𝐽𝑀 ≅ ̅̅̅̅
𝑀𝐿.
1.One of the diagonals divides the kite into two congruent triangles,
while the other diagonal divides the kite into two isosceles triangles.
• Diagonal ̅̅̅̅̅
𝐾𝑀 divides the KITE JKLM into two congruent triangles ∆𝐾𝐽𝑀 ≅ ∆𝐾𝐿𝑀.
̅̅̅
• Diagonal 𝐽𝐿 divides the KITE JKLM into two isosceles triangles ∆ 𝐽𝐾𝑙 𝑎𝑛𝑑 ∆𝐽𝑀𝐿.
2. One of the diagonals is a bisector of the other diagonal.
̅̅̅.
• Diagonal ̅̅̅̅̅
𝐾𝑀 bisect ̅̅̅
𝐽𝐿, therefore ̅̅̅
𝐽𝐶 ≅ ̅𝐶𝐿
3. Only one pair of opposite angles is congruent.
• ∠ 𝑀𝐽𝐾 ≅ ∠𝑀𝐿𝐾
4. One of the diagonals bisects the opposite angles of the kite.
• ∠ 𝐽𝐾𝑀 ≅ ∠𝑀𝐾𝐿 and ∠ 𝐽𝑀𝐾 ≅ ∠𝐿𝑀𝐾
Theorems on Kite
Theorem: The diagonals of a kite are perpendicular to each other.
C
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
Given: ABCD is a kite with 𝐵𝐶 ≅ 𝐵𝐴 and 𝐷𝐶 ≅ 𝐷𝐴
Prove: ̅̅̅̅
𝐵𝐷 is the perpendicular bisector of ̅̅̅̅
𝐴𝐶
B
D
A
Proof:
Statements
̅̅̅̅ ≅ 𝐵𝐴
̅̅̅̅ and 𝐷𝐶
̅̅̅̅ ≅ 𝐷𝐴
̅̅̅̅ 1. Given
1. ABCD is a kite with 𝐵𝐶
Reasons
2. B and D lie on the perpendicular
bisector of AC
3. ̅̅̅̅
𝐵𝐷 is the perpendicular bisector of ̅̅̅̅
𝐴𝐶
2. Converse of the Perpendicular
Bisector Theorem
4. 𝐴𝐶 ⊥ 𝐵𝐷
4.Definition of perpendicular bisector.
3. Through any two points, there is
exactly one line.
Look at the example; see how the theorem is used to find the measure of the angles in
a kite.
Example: Find 𝑚∠1, 𝑚∠2, 𝑎𝑛𝑑 𝑚∠𝑅𝐴𝐷 in the kite DEAR. Given 𝑚∠𝐸𝐴𝐷 = 37.
D
E
2
1
R
MATH 9 QUARTER 3 WEEK 3
37˚
A
Solution:
𝒎∠𝟏 = 90
𝑚∠1 + 𝑚∠2 + 𝑚∠𝐸𝐴𝐷 = 180
Diagonals of a kite are perpendicular
The sum of the measures of the interior
angles of a triangle is 180.
Substitution
Simplify
Addition Property of Equality
90 + 𝑚∠2 + 37 = 180
𝑚∠2 + 127 = 180
𝒎∠𝟐 = 53
Since ∆𝐷𝐸𝐴 ≅ ∆𝐷𝑅𝐴 by SSS Postulate, then ∠𝐸𝐴𝐷 ≅ ∠𝑅𝐴𝐷 by CPCTC.
Thus, 𝒎∠𝑹𝑨𝑫 = 37.
Theorem: The area of a kite is half the product of the lengths of its diagonals.
P
Given: Kite ROPE
O
1
Prove: Area of kite ROPE = 2 (𝑂𝐸)(𝑃𝑅)
W
E
R
Proof:
Statements
Reasons
1. Kite ROPE
1. Given
2. 𝑂𝐸 ⊥ 𝑃𝑅
2. The diagonals of a kite are
perpendicular to each other.
3. Area of kite ROPE = Area of ∆𝑂𝑃𝐸 + Area of ∆𝑂𝑅𝐸
3. Area Addition Postulate
4. Area Formula for Triangles
1
4. Area of ∆𝑂𝑃𝐸 = 2 (𝑂𝐸)(𝑃𝑊)
1
2
Area of ∆𝑂𝑅𝐸 = (𝑂𝐸)(𝑊𝑅)
1
5. Substitution Property
1
5. Area of Kite ROPE = 2 (𝑂𝐸)(𝑃𝑊) +2 (𝑂𝐸)(𝑊𝑅)
6. Area of Kite ROPE = 2 (𝑂𝐸)(𝑃𝑊 + 𝑊𝑅)
6. Distributive Property (5)
7. PW + WR = PR
7. Segment Addition Postulate
1
8. Area of Kite ROPE =
8. Substitution Property (6 & 7)
1
(𝑂𝐸)(𝑃𝑅)
2
Look at the example; see how the theorem is used to solve the given problems.
Example
1. Given: SA = 14 cm and TN = 6 cm. What is the area of the kite?
N
S
T
A
Solution:
𝐴𝑟𝑒𝑎 𝑜𝑓 𝐾𝑖𝑡𝑒 𝑁𝐴𝑆𝑇 = 1/2 ( 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑠)
1
= 2 (𝑆𝐴)(𝑇𝑁)
1
= 2 (14 𝑐𝑚)(6 𝑐𝑚)
1
= 2 (84 𝑐𝑚2 )
𝑨𝒓𝒆𝒂 𝒐𝒇 𝑲𝒊𝒕𝒆 𝑵𝑨𝑺𝑻 = 𝟒𝟏 𝒄𝒎𝟐
MATH 9 QUARTER 3 WEEK 3
Activities
Your goal in this section is to apply the properties and theorems on triangle,
trapezoids, and kites in doing the activities that follow.
Activity 1: Midline Theorem
O and P are the midpoints of ̅̅̅̅
𝐿𝑀and ̅̅̅̅̅
𝑀𝑁, respectively. Complete each statement.
1.
2.
3.
4.
5.
If
If
If
If
If
M
OP = 18, then LN = _________.
LN = 34, then OP = _________.
O
P
MO = 12.5, then ML = __________.
L
N
MN = 29, then PN = __________.
OP = 3x – 2 and LN = 9x – 13, then x = _____, OP = _______ and LN = ______.
Activity 2: Median of a trapezoid
Using trapezoid LIKE with median ̅̅̅̅
𝑁𝑂, find the indicated base/median.
1. If LI = 23 and EK = 36, find NO.
L
2. If LI= 15.5 and EK = 22.5, find NO.
N
3. If NO = 19 and LI = 13, find EK.
4. If NO = 21.5 and EK = 28, find LI.
5. If LI = 3x + 8, EK = 6x + 4 and NO = 24, find LI and EK.
I
O
E
K
Activity 3: Isosceles Trapezoid
A. Given isosceles trapezoid GRAB whose median is ED and diagonals GA and RB.
Determine the relationship between each of the following.
37 cm
̅̅̅̅ and 𝐵𝐴
̅̅̅̅
1. 𝐺𝑅
G
R
̅̅̅̅ and 𝑅𝐴
̅̅̅̅
2. 𝐺𝐵
E
D
3. ̅̅̅̅
𝐺𝐴 and ̅̅̅̅
𝑅𝐵
4. ̅̅̅̅
𝐸𝐷 and ̅̅̅̅
𝐵𝐴
B
A
52 cm
5. ∠𝐵 and ∠𝐴
B. Given isosceles trapezoid GRAB whose median is ED and diagonals GA and RB,
𝑚∠𝐺𝐵𝐴 = 74° , 𝑅𝐵 = 56 𝑐𝑚 and 𝐺𝐵 = 21 𝑐𝑚 .Find GE, GA, ED, 𝑚∠𝐵𝐺𝑅 and 𝑚∠𝑅𝐴𝐵.
Activity 4: Kite
In kite MATH, 𝑚∠𝑇𝐴𝑆 = 52° and𝑚∠𝑇𝐻𝐴 = 33°, MA = 10, MH = 17 and MT = 16
A
Find the measure of the following.
10
1. AT
2. TH
S 52˚
3. MS
4. AS (Hint: Pythagorean Theorem) M
5. 𝑚∠𝑀𝑆𝐴
6. 𝑚∠𝑀𝐴𝐻
7. 𝑚∠𝑀𝐻𝐴
8. 𝑚∠𝐴𝑇𝐻
9. 𝑚∠𝑀𝐴𝑇
10. 𝑚∠𝑀 + 𝑚∠𝐴 + 𝑚∠𝑇 + 𝑚∠𝐻
17
T
33˚
H
Remember
The Midline Theorem. The segment that joins the midpoints of two sides of a triangle
is parallel to the third side and half as long.
Theorem on Trapezoid
The Midsegment Theorem. The median of a trapezoid is parallel to each base and
its length is one half the sum of the lengths of the bases.
Theorems on Isosceles Trapezoid
● The base angles of an isosceles trapezoid are congruent.
MATH 9 QUARTER 3 WEEK 3
● Opposite angles of an isosceles trapezoid are supplementary.
● The diagonals of an isosceles trapezoid are congruent.
Theorems on Kite
● The diagonals of a kite are perpendicular to each other.
● The area of a kite is half the product of the lengths of its diagonals.
Check your Understanding
Draw out conclusions by applying the different theorems on trapezoid and kites.
1. In an isosceles trapezoid MARE, where (MA) ̅ ∥ (ER) ̅, what is the relationship
between ∠M and ∠A? ∠E and ∠R? Explain.
2. WXYZ is an isosceles trapezoid, how do you compare (WY) ̅ and (XZ) ̅? Why?
3. A and B are the midpoints of the legs (TQ) ̅ and (SR ) ̅ respectively of trapezoid
QRST, how would you find the measure of (AB) ̅? Why?
4. A and B are the midpoints of the legs TQ and SR of trapezoid QRST, what is the
relationship?
between AB and QR? AB and TS? Explain.
5. In trapezoid ABCD, if AC = BD, what can you say about trapezoid ABCD? Why is
that so?
6. In kite RSTV. RS = RV and TS = TV. What can you conclude about ∆RTV and ∆RTS
? Why?
Post-test
Direction: Choose the letter of the correct answer.
1. In the figure at the right, DE is the midline of ∆𝐴𝐵𝐶 and AC = 12 cm. What is the
length of DE?
B
a. 6 cm
b. 8 cm
D
E
c. 12 cm
d. 24 cm
A
C
2. What is true about an isosceles trapezoid?
a. It has exactly one pair of parallel sides.
b. It has congruent legs.
c. It has congruent base angles.
d. All of the above.
3. In the figure at the right, PQRS is an isosceles trapezoid.
If 𝑚∠𝑃 = 105°. What is the measure of ∠𝑅?
a.
b.
c.
d.
75°
85°
105°
180°
P
S
105˚
Q
R
4. In the figure at the right PQ = 16cm and SR = 24 cm.
P
̅̅̅̅?
Q
What is the measurement of 𝑇𝑈
a. 8 cm
T
U
b. 18 cm
c. 20 cm
S
R
d. 40 cm
5. What is the length of the diagonal of a kite whose area is 176 sq.cm and the other
diagonal is 16 cm long?
a. 22 cm
b. 24 cm
c. 26 cm
d. 28 cm
MATH 9 QUARTER 3 WEEK 3