Quadrilaterals

1. QUADRILATERALS
Lance Gabriel A. Bacerra
Grade 9 – DITA
Feb. 14, 2022

2. Lesson # 1:
Introduction to Quadrilaterals
&
Angle Sum Theorem of Quadrilaterals

3. What is a Quadrilateral?
A quadrilateral is a plane figure that has
four sides or edges, and also has four
corners or vertices. The angles are present
at the four vertices or corners of the
quadrilateral. If ABCD is a quadrilateral
then angles at the vertices are ∠A, ∠B, ∠C
and ∠D. The sides of a quadrilateral are AB,
BC, CD and DA.
If we join the opposite vertices of the
quadrilateral, we get the diagonals. In the
figure AC and BD are the diagonals of
quadrilateral ABCD.

4. Classifications of Quadrilaterals:
 Parallelogram
- It is a quadrilateral with two pairs of parallel sides.
 Rhombus - is a parallelogram with four congruent sides.
 Rectangle - is a parallelogram with four right angles.
 Square - is a parallelogram with four congruent sides and four right angles.
 Trapezium - It is a quadrilateral with no parallel sides.
 Kite - is a trapezium in which two pairs of adjacent sides are congruent.
 Trapezoid - is a quadrilateral with exactly one pair of parallel sides. The
parallel sides are the bases of a trapezoid and the nonparallel
sides are its legs.
 If the legs are congruent, the trapezoid is said to be an isosceles trapezoid.
 If one leg is perpendicular to its bases or forms a right angle, the trapezoid is said to be
a right trapezoid.

5. Square
Isosceles Trapezoid
Kite
Rectangle Rhombus
Parallelogram Trapezoid Trapezium
Quadrilateral
Right Trapezoid

6. You can name a quadrilateral through its consecutive vertices.
 Vertex – a point by which two segments intersect.
 Side – a segment named by two endpoints.
 Angles - union of two rays.
 Diagonal – a segment that joins two opposite vertices.
 Opposite Sides, Angles, Vertices – counterpart sides, angles or vertices.
 Consecutive Sides, Angles, Vertices – succeeding sides, angles or vertices.
Naming Quadrilaterals

7. Solutions and Answers for Assessment 1

8. Lesson # 2A:
Parallelograms
and
their Properties

9. Properties of Parallelogram:
1. Opposite sides are parallel.
2. Opposite sides are congruent.
3. Opposite angles are congruent.
4. Consecutive angles are supplementary.
5. Diagonals bisect each other.
6. A diagonal separates a parallelogram
into two congruent triangles.
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 both 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 congruent triangles.
6. A quadrilateral is a parallelogram if two pair of opposite sides are both parallel and congruent.
P E
R A

10. Lesson # 2B:
Special
Parallelograms

11. SPECIAL PARALLELOGRAMS:
 Rectangle
 Rhombus
 Square
PROPERTIES OF RECTANGLE:
 All properties of parallelogram
 It has four right angles/equiangular (definition of a rectangle)
 Its diagonals are congruent
PROPERTIES OF RHOMBUS:
 All properties of parallelogram
 It has four equal sides/equilateral (definition of a rhombus)
 Its diagonal is perpendicular
 Each diagonal bisects the angle of the rhombus.
PROPERTIES OF SQUARE:
 All properties of a parallelogram
 All properties of a rectangle
 All properties of a rhombus

12. Solutions and Answers for Assessment 2

13. Lesson # 3:
Trapezoid
&
Kites

14. TRAPEZOID - is a quadrilateral with exactly one pair of parallel sides.
 In a trapezoid, the parallel sides are the bases. The nonparallel sides are the legs
 In a trapezoid, the angles on both ends of each base are called base angles.
 In a trapezoid, the line segment joining the midpoints of the legs of the trapezoid is called the median of a
trapezoid.
 In a trapezoid, an altitude is any segment from a point on one base perpendicular to the line containing the
the other base.
Theorem:
The median of a trapezoid is parallel to its base, and its length is half the sum of the length of the bases.
𝑴𝒆𝒅𝒊𝒂𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒕𝒓𝒂𝒑𝒆𝒛𝒐𝒊𝒅 =
𝟏
𝟐
(𝑩𝟏 + 𝑩𝟐)

15. RIGHT TRAPEZOID – a trapezoid whose one leg is
perpendicular to its bases.
ISOSCELES TRAPEZOID - a trapezoid whose legs are
congruent.
Properties of Isosceles Trapezoid:
- Each pair of base angles is congruent
- Diagonals are congruent
Base
leg
Altitude
Base angles
leg
Base
Median
Midpoint Midpoint

16. E T R
3m 3m
C
S
A
Examples:
A. Quadrilateral CARE is a trapezoid:
1. Parallel sides
CE and RA
2. The bases of a trapezoid
CE and RA
3. Median of a trapezoid
TS
4. Legs of a trapezoid
ER and CA
5. Base angles of a trapezoid
∠A and ∠R
∠E and ∠C

17. I
K T
E
Another special quadrilateral that is not a parallelogram is a kite.
Like a rhombus, the diagonals of a kite are perpendicular.
KITE is a quadrilateral with two pairs of congruent adjacent sides
and no congruent opposite sides.
The common vertices of the congruent sides of kite are the ends
of the kite. In the kite at the right, I and E are the ends.
The line containing the ends of a kite is a symmetry line for the
kite. In the given kite, line IE is a symmetry line.

18. Vertex angles are angles formed by two congruent sides. In the given figure above, vertex angles
are ∠𝐾𝐼𝑇 𝑎𝑛𝑑 ∠𝐾𝐸𝑇.
Non-vertex angles are the angles formed by two sides that are not congruent. In the given figure
above, non-vertex angles are ∠𝐸𝐾𝐼 𝑎𝑛𝑑 ∠𝐸𝑇𝐼.
Properties of a Kite
1. Two pairs of adjacent sides of a kite are congruent.
2. Diagonals are perpendicular to each other.
3. Non-vertex angles are congruent.
4. Diagonal between non-vertex angles is bisected by other diagonal.
5. Vertex angles are bisected by diagonal.

19. Solutions and Answers for Assessment 3