This document discusses different types of quadrilaterals including parallelograms, rectangles, squares, rhombuses, and trapezoids. It provides definitions and properties of each shape. Specifically for parallelograms, it outlines four key properties: opposite sides are parallel and congruent, opposite angles are congruent, and diagonals bisect each other. Examples are given to demonstrate using these properties to determine missing angle measures, side lengths, and whether a shape satisfies the definition of a parallelogram. Rectangles and squares are defined as special types of parallelograms with additional properties like right angles.

1. QUADRILATERAL
Lesson 6

2. • Any four-sided polygon is a quadrilateral.
• The sum of the interior angles is always 360°.

3. Parallelogram
LESSON 6
SLIDE
•Parallelogram
•Rectangle
•Square
•Rhombus
•Trapezoid
•Trapezium
•Isosceles Trapezoid
•Kite

4. Parallelogram
LESSON 6
SLIDE
A parallelogram
- is a quadrilateral whose opposite sides are parallel.
- Is a quadrilateral with two pairs of parallel sides.

5. Parallelogram
LESSON 6
SLIDE
• A parallelogram is named using all four vertices.
• You can start from any one vertex, but you must
continue in a clockwise or counterclockwise direction.
• For example, this can be either
ABCD or ADCB.
C
B
A D

6. Parallelogram
LESSON 6
SLIDE
• There are four basic properties of all parallelograms.
• These properties have to do with the angles, the sides
and the diagonals.

7. Parallelogram
LESSON 6
SLIDE
1. Sides
• If a quadrilateral is a
parallelogram, then its
opposite sides are congruent.
►PQ≅RS and SP≅QR
P
Q R
S

8. Parallelogram
LESSON 6
SLIDE
2. Angles
• If a quadrilateral is a
parallelogram, then its
opposite angles are congruent.
P ≅ R
and
Q ≅ S P
Q R
S

9. Parallelogram
LESSON 6
SLIDE
3. Consecutive Angles
• If a quadrilateral is a parallelogram,
then its consecutive angles are
supplementary (add up to 180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
mS + mP = 180°
P
Q R
S

10. Parallelogram
LESSON 6
SLIDE
4. Diagonals
• If a quadrilateral is a
parallelogram, then its
diagonals bisect each other.
QM ≅ SM
and
PM ≅ RM
Note: The diagonals are not congruent!
PR ≠ QS
P
Q R
S

11. Parallelogram
LESSON 6
SLIDE
• Determines the conditions that make a quadrilateral a
parallelogram
• Uses properties to find measures of angles, side and other
quantities involving parallelograms
• Proves theorems on the different kinds of parallelogram
(rectangle, square, rhombus)
• Solve problems involving parallelograms

12. • Any four-sided polygon is a quadrilateral.
• The sum of the interior angles is always 360°.

13. Parallelogram
LESSON 6
SLIDE
•Parallelogram
•Rectangle
•Square
•Rhombus
•Trapezoid
•Trapezium
•Isosceles Trapezoid
•Kite

14. Parallelogram
LESSON 6
SLIDE
1. ABCD
2. CDFH
3. AEDG
4. CDFG
5. BAFH
A
B C
D
E
F
G
H

15. Parallelogram
LESSON 6
SLIDE
1. ABCD - rectangle
2. CDFH - square
3. AEDG - rhombus
4. CDFG - trapezoid
5. BAFH - square
A
B C
D
E
F
G
H

16. Parallelogram
Lesson 6

17. Parallelogram
LESSON 6
SLIDE
A parallelogram
- is a quadrilateral whose opposite sides are parallel.
- Is a quadrilateral with two pairs of parallel sides.

18. Parallelogram
LESSON 6
SLIDE
• A parallelogram is named using all four vertices.
• You can start from any one vertex, but you must
continue in a clockwise or counterclockwise direction.
• For example, this can be either
ABCD or ADCB.
C
B
A D

19. Parallelogram
LESSON 6
SLIDE
• There are four basic properties of all parallelograms.
• These properties have to do with the angles, the sides
and the diagonals.

20. Parallelogram
LESSON 6
SLIDE
1. Sides
• If a quadrilateral is a
parallelogram, then its
opposite sides are congruent.
►PQ≅RS and SP≅QR
P
Q R
S

21. Parallelogram
LESSON 6
SLIDE
• FGHJ is a parallelogram. Find the
unknown length. Explain your
reasoning.
a. JH
SOLUTION:
a. JH = FG Opposite sides of a are ≅.
JH = 5 Substitute 5 for FG.
F G
J H
b.
5

22. Parallelogram
LESSON 6
SLIDE
2. Angles
• If a quadrilateral is a
parallelogram, then its
opposite angles are congruent.
P ≅ R
and
Q ≅ S P
Q R
S

23. Parallelogram
LESSON 6
SLIDE
PQRS is a parallelogram.
Find the angle measure.
a. mR
Solution:
a. mR = mP Opposite angles of a are ≅.
mR = 70° Substitute 70° for mP.
P
R
Q
70°
S

24. Parallelogram
LESSON 6
SLIDE
3. Consecutive Angles
• If a quadrilateral is a parallelogram,
then its consecutive angles are
supplementary (add up to 180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
mS + mP = 180°
P
Q R
S

25. Parallelogram
LESSON 6
SLIDE
PQRS is a parallelogram.
Find the angle measure.
a. mQ
P
R
Q
70°
S

26. Parallelogram
LESSON 6
SLIDE
4. Diagonals
• If a quadrilateral is a
parallelogram, then its
diagonals bisect each other.
QM ≅ SM
and
PM ≅ RM
Note: The diagonals are not congruent!
PR ≠ QS
P
Q R
S

27. Parallelogram
LESSON 6
SLIDE
• FGHJ is a parallelogram. Find the unknown
length. Explain your reasoning.
a. JH
SOLUTION:
JK = GK Diagonals of a bisect each other.
JK = 3 Substitute 3 for GK
F G
J H
K 3
b.

28. Parallelogram
LESSON 6
SLIDE
1. Show that both pairs of opposite sides are || . [definition]
2. Show that both pairs of opposite sides are  .
3. Show that one pair of opposite sides are both || and  .
4. Show that both pairs of opposite angles are  .
5. Show that the diagonals bisect each other .

29. Parallelogram
LESSON 6
SLIDE
Find the values of x and y that ensures the quadrilateral is a parallelogram.
6x = 4x + 8
2x = 8
x = 4
2y = y + 2
y = 2

30. Parallelogram
LESSON 6
SLIDE
Find the value of x and y that ensure the quadrilateral is a parallelogram.
2x + 8 = 120
2x = 112
x = 56
5y + 120 = 180
5y = 60
y = 12

31. Special Parallelogram
Lesson 6

32. Parallelogram
LESSON 6
SLIDE
A rectangle is a quadrilateral with four right angles.
• Is a rectangle is a parallelogram?
• Thus a rectangle has all the properties of a parallelogram.
 Yes, since opposite angles are congruent.

33. Properties of Rectangles
Parallelogram Properties:
 Opposite sides are parallel.
 Opposite sides are congruent.
 Opposite angles are congruent.
 Consecutive angles are supplementary.
 Diagonals bisect each other.
Plus:
 All angles are right angles.
 Diagonals are congruent.
 Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles
triangles

34. Parallelogram
LESSON 6
SLIDE
34
1. If AE = 3x +2 and BE = 29, find the value of x.
2. If AC = 21, then BE = _______.
6
5
4
3
2
1
E
D C
B
A

35. Parallelogram
LESSON 6
SLIDE
35
1. If AE = 3x +2 and BE = 29, find the value of x.
E
D C
B
A

36. Parallelogram
LESSON 6
SLIDE
36
2. If AC = 21, then BE = _______.
E
D C
B
A

37. Parallelogram
LESSON 6
SLIDE
A square is a quadrilateral with four congruent angles
and four congruent sides.
• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
Plus:
• Four right angles.
• Four congruent sides.
• Diagonals are congruent.
• Diagonals are perpendicular.
• Diagonals bisect opposite angles.
Since every square is a parallelogram as well as a rhombus and
rectangle, it has all the properties of these quadrilaterals.

38. Parallelogram
LESSON 6
SLIDE
Given: ABCD is a square. Complete the following.
1. If AB = 10, then AD = _____ and DC = _____.
2. If CE = 5, then DE = _____.
3. m<ABC = _____.
4. m<ACD = _____.
5. m<AED = _____.
8 7 6
5
4
3
2
1
E
D C
B
A
10 units 10 units
5 units
90°
45°
90°

39. Parallelogram
LESSON 6
SLIDE
A rhombus is a quadrilateral with four congruent sides.
• Since a rhombus is a parallelogram the following are true:
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
• Is a rhombus a parallelogram?
 Yes, since opposite sides are congruent.

40. Lesson 6-4: Rhombus & Square 40
Rhombus
Note: The four small triangles are congruent, by SSS.
This means the diagonals form
four angles that are congruent,
and must measure 90 degrees
each.
So the diagonals are perpendicular.
This also means the diagonals
bisect each of the four angles of
the rhombus
So the diagonals bisect opposite angles.

41. Properties of a Rhombus
.
Since a rhombus is a parallelogram the following are true:
• Opposite sides are parallel.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
Plus:
• All four sides are congruent.
• Diagonals are perpendicular.
• Diagonals bisect opposite angles.
• Also remember: the small triangles are RIGHT and CONGRUENT!

42. Parallelogram
LESSON 6
SLIDE
42
Given: ABCD is a rhombus. Complete the following.
1. If AB = 9, then AD = ______.
2. If m<1 = 65, the m<2 = _____.
3. m<3 = ______.
4. If m<ADC = 80, the m<DAB = ______.
5. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.
5
4
3
2
1
E
D C
B
A
9 units
65°
90°
100°
10

43. SLIDE
References
E-Math 9 - Work Text in Mathematics (Rex Book Store)
Math Ideas and Life Applications 9 - Second Edition (Abiva)
Spiral Math 9 – (Trinitas Publishing Inc.)
Parallelogram
LESSON 6