Math 9 Quarter 3

1. MATH 9
Quarter 3: Week 1

2. Isosceles Trapezoid Kite Rectangle Square
Rhombus Trapezoid Parallelogram
TYPES OF QUADRILATERALS

3. What is
Quadrilateral?

4. A quadrilateral is a four-
sided polygon with
four straight sides, four
vertices, and four angles.
The sum of all its interior
angles is always 360

7. CONDITIONS THAT MAKE
A QUADRILATERAL
A PARALLELOGRAM
Lesson 1

8. What is Parallelogram?

9. Parallelogram is a
quadrilateral with two pairs
of opposite sides parallel to
each other.

10. This is not a parallelogram because it has
only one pair of parallel sides.
This is a parallelogram because it has
two pairs of parallel sides.
DID YOU GET IT RIGHT?
Which of the two figures below is a
parallelogram?
A B

11. THEOREM
1. A quadrilateral is a parallelogram if its opposite sides
are congruent.
Conditions that make a Quadrilateral a
Parallelogram
M
A T
H
AT MH
MA TH

12. Example 1
CUTE is a quadrilateral.
Determine the length of UT that will make CUTE a parallelogram.
E
U
T
C
8 cm
10 cm
10 cm

13. THEOREM
1. A quadrilateral is a parallelogram if its opposite sides are congruent.
2. A quadrilateral is a parallelogram if the diagonal
of a quadrilateral form two congruent triangles.
Conditions that make a Quadrilateral a
Parallelogram
C
A R
E

14. THEOREM
1. A quadrilateral is a parallelogram if its opposite sides are congruent.
2. A quadrilateral is a parallelogram if the diagonal
of a quadrilateral form two congruent triangles.
Conditions that make a Quadrilateral a
Parallelogram
C
A R
E

15. THEOREMS
1. A quadrilateral is a parallelogram if its opposite sides are congruent.
2. A quadrilateral is a parallelogram if the diagonal of a quadrilateral form
two congruent triangles.
3. A quadrilateral is a parallelogram if its diagonals bisect
each other.
Conditions that make a Quadrilateral a
Parallelogram
E
M
L
I
S
MS LS
IS ES

16. Example 3
CUTE is a quadrilateral with diagonals CT and UE intersecting at
O.
Determine the length of OT that will make CUTE a parallelogram.
E
U
T
C
5
m
7 cm
7 cm O

17. THEOREMS
1. A quadrilateral is a parallelogram if its opposite sides are congruent.
2. A quadrilateral is a parallelogram if the diagonal of a quadrilateral form two congruent
triangles.
3. A quadrilateral is a parallelogram if its diagonals bisect each other.
4. A quadrilateral is a parallelogram if its opposite
angles are congruent.
Conditions that make a Quadrilateral a
Parallelogram
E
H O
P
H P
E O

18. Example 4
CUTE is a quadrilateral.
Determine the length of E that will make CUTE a parallelogram.
E
U
T
C
130
130
50

19. THEOREMS
1. A quadrilateral is a parallelogram if its opposite sides are congruent.
2. A quadrilateral is a parallelogram if the diagonal of a quadrilateral form two
congruent triangles.
3. A quadrilateral is a parallelogram if its diagonals bisect each other.
4. A quadrilateral is a parallelogram if its opposite angles are congruent.
5. A quadrilateral is a parallelogram if any two
consecutive angles are supplementary.
Conditions that make a Quadrilateral a
Parallelogram

20. THEOREMS
5. A quadrilateral is a parallelogram if any two
consecutive angles are supplementary.
Conditions that make a Quadrilateral a
Parallelogram
G
I
E
V G I = 180
I V = 180
V E = 180
E G = 180

21. Example 5
CUTE is a quadrilateral.
Determine the U that will make CUTE a parallelogram.
E
U
T
C
100
100
80

22. Task 1
Opposite sides are congruent
Opposite angles are congruent
Opposite sides are congruent
Consecutive angles are supplementary
D E
F
G

23. Solve the following
equations
5 Minute Drill
1. 2b + 4 = 10
2. a - 2 = 2
3. 3x - 2 = 8
4. b - 6 = 14
5. 2m + 3 = - 4
6. 7x - 4 = 10
7. y + 3 = -12
8. x + 21 = -1
9. 3b - 9 = -6
10. 4y + 8 = -24

24. a) opposite sides are congruent.
b) the diagonals bisect each other.
c) opposite angles are congruent
d) Consecutive angles are supplementary.
PROPERTIES OF PARALLELOGRAM

26. Task 2

27. Using Properties to Find
Measures of Angles, Sides and
other Quantities Involving
Parallelograms
Lesson 2

28. A quadrilateral is a parallelogram if opposite sides are congruent and parallel.
Using Properties to Find Measures of Angles and Sides Involving Parallelograms
Example 1: is a parallelogram.
If │HE│ = 3x and │PL│ = 12 cm, find the value of x.
Step 1: Draw the given.
Step 2: Identify relationships.
HE PL and HP EL
Step 3: Formulate the equation and solve.
H
L
E
P
3x
12 cm

29. A quadrilateral is a parallelogram if opposite sides are congruent and parallel.
Using Properties to Find Measures of Angles and Sides Involving Parallelograms
Example 1: is a parallelogram.
If │HE│ = 3x and │PL│ = 12 cm, find the value of x.
Step 3: Formulate the equation and solve.
Solution:
│HE│ = │PL│
3x = 12
x = 4
H
L
E
P
3x
12

30. A quadrilateral is a parallelogram if opposite sides are congruent and parallel.
Using Properties to Find Measures of Angles and Sides Involving Parallelograms
Example 2: Given is parallelogram HOPE. Find the value of y and the lengths of the given sides.
Formulate the equation and solve
Solution: HE OP
│HE│ = │OP│
5y - 20 = 3y + 10
5y – 3y = 10 + 20
2y = 30
y = 15
H
P
O
E
5y - 20
3y + 10

31. A quadrilateral is a parallelogram if opposite angles are congruent and
consecutive angles are supplementary
Using Properties to Find Measures of Angles and Sides Involving Parallelograms
Example 3: is a parallelogram. If m = 62, find the measures of
the other 3 angles.
Solution:
S
U
A
G
S 62

32. A quadrilateral is a parallelogram if opposite angles are congruent and
consecutive angles are supplementary
Using Properties to Find Measures of Angles and Sides Involving Parallelograms
Example 4: Quadrilateral STAR is a parallelogram.
Find the value of x when mS = 3x + 10 and mR = 2x + 60
Solution:
+ = 180
R A
T
S
2x + 60
3x + 10

33. A quadrilateral is a parallelogram if opposite angles are congruent and
consecutive angles are supplementary
Using Properties to Find Measures of Angles and Sides Involving Parallelograms
Example 5: FOUR is a parallelogram. If m = x, m = 88, mUFO = 32,
mRUF = 2y, find the values of x, y, z and m
Solution:
R U
O
F
88
32
(2y)
x
z

34. A quadrilateral is a parallelogram if the diagonals bisect each other.
Using Properties to Find Measures of Angles and Sides Involving Parallelograms
Example 6: The figure below is a parallelogram. Find the value of x.
Solution:
AM NM

37. Solve the following
equations
5 Minute Drill
1. 5x + 5 = 19 – 2x
2. 2x - 2 = 2 - x
3. 3x - 4 = x + 16
4. 4x - 7 = 5 – 2x
5. -2x - 13 = -3x - 5
6. 3x + 4 = x +18
7. 9 + 6x = 3x + 13
8. x + 12 = 2x - 4
9. 3x - 10 = -6 + x
10. 3x + 8 = -24 + x

38. PROVING THEOREMS ON THE
DIFFERENT KINDS OF
PARALLELOGRAM
(Rectangle, Rhombus, Square)
Lesson 3

39. Proving theorems on the different kinds of
Parallelogram
Rectangle
Rhombus
Square
A rhombus is a special type of parallelogram with all sides are
equal.
Its opposite sides are parallel, and opposite angles are equal.

40. Proving theorems on the different kinds of
Parallelogram
Rectangle Square
A rectangle is not a square, but a square is a type of rectangle.
A rectangle has four right angles and two pairs of equal-length sides,
while a square is a specific type of rectangle where all four sides are

41. Proving Theorem on
Rectangle
Properties of Rectangle
A rectangle has four sides, four vertices and four angles.
Opposite sides are parallel.
Opposite sides are congruent.
Adjacent sides are perpendicular.

42. If a parallelogram has one right angle, then it has four
right angles and the parallelogram is a rectangle.
Theorem 1
Proving Theorem on
Rectangle

43. The diagonals of a rectangle
are congruent.
Theorem 2
Proving Theorem on
Rectangle

44. Congruent Triangles

45. “Corresponding Parts of Congruent Triangles are Congruent"
If two triangles have the same size and shape,
then all their matching sides and angles are
also equal in length and measurement.
CPCTC

46. Asynchronous Activity
November 11, 2025
Submission: Until 7:30 pm
(Corresponding Parts of Congruent Triangles are
Congruent)

48. SAS Postulates
Reflexive Property
and are right angles
All right angles are congruent

49. Proving Theorem on
Rhombus

51. The diagonals of a rhombus are perpendicular to each other.
Theorem 3
Proving Theorem on
Rhombus
Given: Rhombus WISE with diagonals
Prove:

52. STATEMENTS REASONS
1. Rhombus WISE w/
diagonals
1. Given
2. 2. Definition of a rhombus
3. and bisect each other at T. 3. Diagonals of a parallelogram
bisect each other.
4. 4. Definition of bisector
5. 5. Reflexive Property
6. 6. SSS Postulate
7. 7. CPCTC
8. form a linear pair 8. Definition of angles forming a linear pair
9. are supplementary 9. Linear Pair Postulate
10. are right angles 10. If two angles are both congruent and
supplementary, then they are right angles
11. 11. Definition of perpendicular lines

55. Each diagonal of a rhombus bisects its opposite angles.
Theorem 4
Proving Theorem on
Rhombus
Given: Rhombus MORE with diagonal
Prove:

56. STATEMENTS REASONS
1. MORE is a rhombus with
diagonal
1. Given
2. 2. All sides of a rhombus are
Definition of a rhombus.
3. 3. Opposite angles of parallelogram are
4. 4. SAS Postulate
5. 5. CPCTC
6. are isosceles triangles 6. Definition of isosceles triangle
7. ; 7. Base angles of isosceles triangle are congruent.
8. ; 8. Transitive Property
Given: Rhombus MORE with diagonal
Prove:
Each diagonal of a rhombus bisects its opposite angles.

57. EXAMPLE 1:
WIPE is a rhombus. Find the measures of the following angles if m = 96
Proving Theorem on
Rhombus
a.
Solution:
Diagonal is the bisector of WIP.
mWIP = 96
96 2
Answer: mWIE = 48

58. EXAMPLE 1:
WIPE is a rhombus. Find the measures of the following angles if m = 96
Proving Theorem on
Rhombus
b.
Solution:
Consecutive angles are supplementary
mWIP = 96
180 96
Answer: mE = 84

59. EXAMPLE 1:
WIPE is a rhombus. Find the measures of the following angles if m = 96
Proving Theorem on
Rhombus
c.
Solution:
Diagonal is the bisector of IWE.
mIWE = 84
84 2
Answer: mEWP = 42

60. EXAMPLE 1:
WIPE is a rhombus. Find the measures of the following angles if m = 96
Proving Theorem on
Rhombus
d.
Solution:
WPE
mEWP = 42
Answer: mWPE = 42

61. EXAMPLE 2:
Find the measure of each numbered angle in the rhombus.
Proving Theorem on
Rhombus

63. Activity 1

64. WARM is a square
Definition of square
SSS Postulate
WMR and RAW are isosceles triangle
WMR and RAW are isosceles right triangles

65. Because the diagonals RC and TA are perpendicular to each other
Because the diagonals of a square are congruent
45
90
25

66. A tessellation or tiling is the covering of a surface,
often a plane, using one or more geometric shapes,
called tiles, with no overlaps and no gaps.

67. Seatwork

68. MIDLINE
TRAPEZOID
KITE
Lesson 4

73. A kite is a quadrilateral with two distinct pairs of
consecutive sides that are congruent.
̅̅̅̅ ≅
≅

74. Theorem
If a quadrilateral is a kite, then the diagonals are perpendicular.
CA

75. Theorem
If a quadrilateral is a kite, then it has one pair of opposite
congruent angles.
≅

76. Theorem
If a quadrilateral is a kite, it has one diagonal that
bisects the other diagonal.
Diagonal 𝐵𝐷̅̅ bisects diagonal 𝐴𝐶̅̅̅

79. Given: 0KITE is a kite. Point O is the intersection of
diagonals KT and EI. Find the following: