QUARTER 3 NOTES

Quadrilateral is a polygon with four vertices, four angles and four sides.
Trapezoid is a quadrilateral with one pair of parallel sides.
Kite is a quadrilateral with two pairs of distinct adjacent sides that are congruent but not parallel.
Parallelogram is a quadrilateral with two pairs of parallel and congruent sides.
 Rectangle is a parallelogram with two pairs of opposite sides parallel and congruent and all
angles are right.

 Rhombus is a parallelogram with four congruent sides and the angles are oblique.

. Square is a parallelogram with four congruent sides and four congruent angles.

Determine whether the following statement is true otherwise give your reason why it is false.
1. A square is a parallelogram. True
2. A kite is a trapezoid. False. Kite is a trapezium.
3. All parallelograms are rhombus. False. Can be square or rectangle.
4. An isosceles trapezoid has parallel sides that are congruent. True
5. A rhombus is a parallelogram. True
6. Every rhombus is a parallelogram. True
7. A trapezium is a parallelogram. False. Trapezium has no parallel side.
8. A quadrilateral is a trapezoid. False. A trapezoid is a quadrilateral.
9. A trapezoid can only have one angle. False. More than one.
10. Some kites are parallelogram. False. Kite has no parallel side.

Write TRUE if the statement is correct otherwise change the underline word to make it correct.
1. A quadrilateral is a polygon has no parallel sides. Parallelogram
2. A parallelogram has 2 pairs of parallel sides. True
3. A rectangle is a kind of square. False. Square is a kind of rectangle.
4. All rectangles are parallelogram. True
5. Trapezoid has 1 pair of parallel sides.True

Let the students study the parallelogram figure.

A B

C D

ANALYSIS:
1. How many sides does this figure has? The figure has 4 sides.
2. Look at side AB and CD. Are they parallel with each other? Yes.
3. How about side AC and BD? Yes.
 4. What can you say about them? Two pairs of sides are parallel.
5. Therefore,how many parallel sides are there? There are two pairs of parallel sides.
6. Measure side AB and CD. Do they have the same measurement? AB and CD has the same
measurement.
7. How about AC and BD? BD and AC has the same measurement.

This figure is called parallelogram. What can you say about parallelogram? A parallelogram is
a quadrilateral with two pairs of parallel side and each opposite sides has the same
measurement.
A quadrilateral is a parallelogram if:
 both pairs of opposite sides are congruent
 both pairs of opposite angles are congruent
 one pair of opposite sides are both parallel and congruent
 an angle of a quadrilateral is supplementary to both consecutive angles
 diagonals bisect each other.

Set A:

Use the figure at the right to answer the following: T I

1. Which angles of ⧠TIME are congruent?
Ans: ∠ T and ∠ M
E M
∠ I and ∠ E
2. Which angles of ⧠ TIME are supplementary?
Ans: ∠ T and ∠ I
∠ I and ∠ M
∠ M and ∠ E
∠ E and ∠ T

3. Which sides of ⧠TIME are parallel?

Ans: TI||EM
TE||IM
4. Which sides of ⧠ TIME are congruent?
Ans: TI ≃EM
TE≃IM

Given ⧠QUAD , complete the statement.

1. If QD=16 , then UA =¿ ¿. U A
2. If DE=5 , then UD=¿ ¿ .
3. If QU =7.5 , then AD=¿ ¿. E
Q
4. If m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3UAD=98 , then m 𝐸𝑀𝐵𝐸𝐷 D .3 ADQ= ¿.
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 ¿
5. IfUA=2 AD , and UA=15, then AD=¿ ¿.

In ⧠QUED , m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 D is 30 greater than m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 E .

Find the measures of each the angles.
Answer: m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 E & m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3Q is 75o
m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 D & m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3U is 105 o

In ⧠ ABCD , m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 B is twice m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 A . Find the

measure of all the angles.
Answer: m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 B & m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 D is 120o
m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3 A & m 𝐸𝑀𝐵𝐸𝐷 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 .3C is 60 o

Theorems on the different kinds of a Parallelogram:

 Rectangle
1. If a parallelogram has one right angle, then it has four right angles and the parallelogram
is a rectangle.
2. The diagonals of a rectangles are congruent.
3. The diagonals of a rectangle bisect each other.
 Square
1. A square is a rectangle all of whose sides are congruent.
Diagonals of a square bisect each other.

Midline Theorem
The segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third
side and half as long.

ACTIVITY : Show Me!

1. The class will be divided into several groups.
2. Each group will be given an illustration of a triangle showing the Midline Theorem.
A

D E

B C
3. Let them measure BA , BD , AD , AC , AE , CE , BC and DE .
ANALYSIS:
1. What can be said about the measure of BD and AD ? How about BA and BD ?
2. What can be said about the measure of AE and CE ? How about AC and CE ?
3. What do you call point D and point E in BA and AE ?
4. What can be said about the measure of BC and DE ?
5. What do you call DE in the given triangle?
6. Based upon your findings in the activity, explain the Midline Theorem.

ABSTRACTION:

“The Midline Theorem states that the segment that joins the midpoints of two sides of a triangle is
parallel to the third side and half as long.”
A
In Δ ABC , x and y are the midpoints of BA
 and AC respecttively, then
X Y BC
XY =
2 and BC=2 XY
B C

TRAPEZOID AND ITS PROPERTIES

A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a
base. The non-parallel sides are called legs. Base angles of a trapezoid are two consecutive angles
whose common side is a base. If the legs of a trapezoid are congruent, the trapezoid is an isosceles
trapezoid.

PROPERTIES/THEOREMS

TRAPEZOIDS IN TIMBUKTU
In the African nation of Mali, an ancient city rests, surrounded by desert. During the Middle Ages,
Timbuktu was center of trade, religion and education. The local builders constructed fantastic mosques
out of mud, clay and wood. Today, the trapezoid mosques and tombs of Timbuktu are famous around
the world.

The United Nations has declared Timbuktu as World Heritage Site. Scholars are especially interested
on how Timbuktu helped spread Islam throughout Northwestern Africa. In addition, the city houses
many rear manuscipts and books from Middle Ages. Unfortunately, Timbuktu is at risk. In 2012, militants
took over the city. They attempted to burn the library. They destroyed 6 of the 13 most historical
buildings in the city. While the militants were later expelled, they’ve vowed to return and finish their work
of destruction.

Ask:
1. What did you feel after listening to the story of Timbuktu? Why?
2. What Mathematical FIGURE made Timbuktu a famous site? Answer: Trapezoid
3. What other real-life objects are trapezoid in shape?

ANALYSIS:
1. How many pair/s of parallel sides does a trapezoid have? How do you define a trapezoid? Answer:
One; Trapezoid is a quadrilateral with one pair of parallel sides.
2. How do you call a trapezoid whose base angles are equal in measure and whose legs are also equal
in length? Answer: Isosceles Trapezoid
3. What are the three properties of an isosceles trapezoid? Answer: (1) Its base angles are congruent;
(2) Its legs are congruent; (3) Its legs are congruent.

OTHER PROPERTIES:
1. Consecutive angles between the bases of an isosceles trapezoid are supplementary.
2. Opposite angles of an isosceles trapezoid are supplementary.

Find the measure of the indicated angle and side. You may also refer to the Workbook for some
exercises.
1. Find m ∠ D .

Answer: 1000
2. If KB=15 , JB=5 , find FM .

Answer: 20
3. Find m ∠F .

Answer: 1310

4. If JN=6 , KM =15 , find KN .

Answer: 9
TRIVIA:
A quadrilateral with no parallel sides is a trapezium. The US and UK have their definitions swapped
over.

Trapezoid Trapezium
US One pair of parallel sides No pair of parallel sides
UK No pair of parallel sides One pair of parallel sides

In Greek, both words mean “little table”.

Recall the definition of a trapezoid.

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

4. Ask one student to draw a trapezoid on the board. Let him label the parallel sides.

The illustration should be similar like this: