```Name of Lecturer: Mr. J.Agius
Course: HVAC1
Lesson 46
Chapter 9: Angles and Shapes
A quadrilateral is any four-sided shape.
Any quadrilateral can be split up into two triangles by drawing in a diagonal, like this:
y
The sum of the four angles in any quadrilateral is 360
i.e.
o
x
w + x + y + z = 360o
w
z
You can find this by just splitting the quadrilateral into two triangles.
i.e.
ABC = 180o and
D
A
So the sum of the two triangles is equal to
180o + 180o = 360o
which is equal to the sum of the angles
of a quadrilateral since these two triangles
B
C
There are special types of quadrilateral :



the rectangle
the rhombus
the square
(those are all parallelograms), and there is also:


the trapezoid
the kite
If it isn't one of those it is an irregular quadrilateral.
9 Angles & Shapes
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Here are the details:
The Rectangle
means "right angle"
and
show equal sides
A rectangle is a four-sided shape where every angle is a right angle (90°).
Also opposite sides are parallel and of equal length.
The Rhombus
A rhombus is a four-sided shape where all sides have equal length.
Also opposite sides are parallel and opposite angles are equal.
Another interesting thing is that the diagonals (dashed lines in second figure) of a rhombus
bisect each other at right angles.
The Square
means "right angle"
show equal sides
A square has equal sides and every angle is a right angle (90°)
Also opposite sides are parallel.
A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all
sides are equal length).
9 Angles & Shapes
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
The Parallelogram
Opposite sides are parallel and equal in length, and
opposite angles are equal (angles "a" are the same,
and angles "b" are the same)
NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!
Example: if a parallelogram has all sides equal and angles "a" and "b" are right angles, then
it is also a square.
The Trapezoid (or Trapezium)
Trapezoid
Regular Trapezoid
A trapezoid (UK: trapezium) has one pair of opposite sides parallel.
It is called a regular trapezoid if the sides that aren't parallel are equal in length and both
angles coming from a parallel side are equal, as shown.
A trapezoid is not a parallelogram because only one pair of sides is parallel.
The Kite
Hey, it looks like a kite. It has two pairs of sides. Each pair is
made up of adjacent sides that are equal in length. The angles
are equal where the pairs meet. Diagonals (dashed lines) meet
at a right angle, and one of the diagonal bisects (cuts equally in
half) the other.
... and that's it for the special quadrilaterals; if it doesn't fit one
of those it is an Irregular Quadrilateral:
9 Angles & Shapes
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Any quadrilateral that doesn't match one of the previous types.
Polygon
A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided
polygon, a pentagon is a 5-sided polygon, and so on.
Example 1
Work out the value of x.
136o
Rule: The sum of interior angles of a
So
x + 136o + 45o + 100o = 360o
x + 281o = 360o
so
x = 360o – 281o = 79o
9 Angles & Shapes
45o
xo
100o
Page 4
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Exercise 1
Find the angles of each marked angle and give an appropriate reason.
1)
2)
112o
t
124o
y
80o
s
70
o
Ans. y = ___________
Ans. t = ___________ s = _____________
Reason:_________________________
Reason:_____________________________
Reason:_____________________________
3)
82o
135o
Ans. a = _____ Reason:________________
Ans. b = _____ Reason:________________
124o
4)
a
b
c
Ans. c = _____ Reason:________________
133o x
Ans. x = _____ Reason:________________
Ans. y = _____ Reason:________________
y
100o
5)
g
o
70
h
120o
Ans. g = _____ Reason:________________
Ans. h = _____ Reason:________________
j
i
Ans. i = _____ Reason:________________
Ans. j = _____ Reason:________________
9 Angles & Shapes
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
6)
Ans. c = _____ Reason:________________
c
f
Ans. d = _____ Reason:________________
e
Ans. e = _____ Reason:________________
105o
d
Ans. f = _____ Reason:________________
7)
116o
Ans. c = _____ Reason:________________
Ans. d = _____ Reason:________________
e
d
c
55o
8)
87o
k
Ans. e = _____ Reason:________________
Ans. j = _____ Reason:________________
l
Ans. k = _____ Reason:________________
58o
j
9 Angles & Shapes
73o
Ans. l = _____ Reason:________________
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Exercise 2
1)
1
2
3
4
6
5
a)
Write down the special name for each of these four-sided shapes.
b)
For each shape say which statements in the list below are true.
i.
ii.
iii.
iv.
v.
vi.
2.
The sides are equal
Both pairs of opposite sides are parallel
Only opposite angles are equal
There is just one pair of parallel sides
The four sides are made up of just two distinct lengths.
The diagonals make an angle of 90o where they meet.
In the following questions, some of the diagrams contain more than one quadrilateral.
Name each quadrilateral with the appropriate letters and find the size of each marked
angle.
a)
b)
s
o
86o
o
p
xo
o
35
ro
9 Angles & Shapes
to qo
zo
57o
yo
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
c)
d)
qo
o
66
po
zo
25o
xo
yo
3.
A
3xo
ro
B
4xo
2xo
D
a)
b)
50o
xo
C
Find the value of x.
Prove that ABCD is a trapezium.
4.
ABCD is a trapezium with AB parallel to DC. A = 127o and C =76o.
Work out B and D. (Hint: draw a diagram first.)
5.
JKLM is a rectangle. JK is the longer side. N is a point on JK such that KN = KL.
Given MNL = 79o work out JMN.
6.
PQRS is a rhombus with the diagonal QS equal to PQ.
Work out P.
7.
ABCD is a rectangle. E is a point on AB.
a)
What type of quadrilateral is BCDE?
b)
Given ADE = 50o work out DEB.
c)
If DE = EB work out EBD.
9 Angles & Shapes
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