A quadrilateral is any four-sided fi gure
In any quadrilateral the sum of the interior angles is
360c centre O and radius 6 cm has a
perpendicular line OC as shown 4 cm long.
A
B O
C 4 cm 6 cm
and BC , show that OC bisects the chord .
By proving congruent (b)
triangles, show that OC bisects the chord .
Proof
Draw in diagonal AC
180 ( )
( )
,
ADC DCA CAD
ABC BCA CAB
ADC DCA CAD ABC BCA CAB
ADC DCB CBA BAD
180
360 360 angle sum of similarly That is
`
c c
c c
+ + +
+ + +
+ + + + + +
+ + + +
D
+ + =
+ + =
+ + + + + =
+ + + =
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opposite sides
• of a parallelogram are equal
• opposite angles of a parallelogram are equal
• diagonals in a parallelogram bisect each other each diagonal bisects the parallelogram into two
•
congruent triangles
A quadrilateral is a parallelogram if:
both pairs of
• opposite sides are equal both pairs of
• opposite angles are equal one
• pair of sides is both equal and parallel the
• diagonals bisect each other
These properties can all be proven.
Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel
EXAMPLE
Find the value of i.
Solution
120 56 90 360 266 360 94
angle sum of quadrilateral i
i i
+ + + =
+ =
=
^ h
PROPERTIES
TESTS
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Rhombus
A rectangle is a parallelogram with one angle a right angle
the same as for a parallelogram, and also
•
diagonals are equal
•
A quadrilateral is a rectangle if its diagonals are equal
Application
Builders use the property of equal diagonals to check if a rectangle is accurate.
For example, a timber frame may look rectangular, but may be slightly slanting.
Checking the diagonals makes sure that a building does not end up like the Leaning Tower of Pisa!
It can be proved that all sides are equal.
If one angle is a right angle, then you can prove all angles are right angles.
A rhombus is a parallelogram with a pair of adjacent sides equal
the same as for parallelogram, and also
•
diagonals bisect at right angles
•
diagonals bisect the angles of the rhombus
•
Rectangle
PROPERTIES PROPERTIES
TEST
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Square
A square is a rectangle with a pair of adjacent sides equal
• the same as for rectangle, and also diagonals are perpendicular
•
diagonals make angles of
• 45c with the sides
Trapezium
A trapezium is a quadrilateral with one pair of sides parallel
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal
A quadrilateral is a rhombus if:
all sides are equal
•
diagonals bisect each other at right angles
•
TESTS
PROPERTIES
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EXAMPLES
1. Find the values of i, x and y , giving reasons.
Solution
( )
. ( )
. ( )
x y
83 6 7 2 3
opposite s in gram cm opposite sides in gram cm opposite sides in gram
c + <
<
<
i =
=
=
2. Find the length of AB in square ABCD as a surd in its simplest form if BD=6 .cm
Solution
( )
( )
AB x
ABCD AB AD x
A 90 Let
Since is a square, adjacent sides equal
Also,+ c by definition
=
= =
=
By Pythagoras’ theorem:
3
c a b
x x x x x 6 36 2 18
18 2 cm
2 2 2
2 2 2
2 2
`
= +
= +
=
=
=
=
CONTINUED
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1. Find the value of all pronumerals, giving reasons.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
4.7 Exercises
3.
Two equal circles have centres
(a) O and P respectively. Prove that OAPB
is a rhombus.
Hence, or otherwise, show that
(b) AB is the perpendicular bisector
of OP .
Solution
(a) ( )
( )
OA OB PA PB
OA OB PA PB
equal radii similarly Since the circles are equal,
=
=
= = =
` since all sides are equal, OAPB is a rhombus
The diagonals in any rhombus are perpendicular bisectors.
(b)
Since OAPB is a rhombus, with diagonals AB and OP , AB is the perpendicular bisector of OP .
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2. Given AB=AE, prove CD is perpendicular to AD .
3. (a) Show that + =C xc and ( ) .
B D 180 xc
+ =+ = -
Hence show that the sum of (b)
angles of ABCD is 360c.
4. Find the value of a and b .
5. Find the values of all
pronumerals, giving reasons.
(a)
(b)
(d)
(e)
7 3x y
x + 6
(f)
6. In the fi gure, BD bisects ADC.
+ Prove BD also bisects ABC.
+
7. Prove that each fi gure is a parallelogram.
(a)
(b)
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(c)
(d)
8. Evaluate all pronumerals.
(a)
(b)
ABCD is a kite
(c)
(d)
(e)
9. The diagonals of a rhombus are 8 cm and 10 cm long. Find the length of the sides of the rhombus.
10. ABCD is a rectangle with EBC 59c.
+ = Find +ECB,+EDC and +ADE.
11. The diagonals of a square are 8 cm long. Find the exact length of the side of the square.
12. In the rhombus, +ECB=33c. Find the value of x and y .
Polygons
A polygon is a closed plane fi gure with straight sides
A regular polygon has all sides and all interior angles equal
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Proof
Draw any n -sided polygon and divide it into n triangles as shown. Then the total sum of angles is n#180c or 180 .n But this sum includes all the angles at O . So the sum of interior angles is 180n-360 .c
That is, S n n
180 360 2 #180c
=
-=] - g
EXAMPLES
4-sided
(square) 3-sided
(equilateral triangle)
5-sided
(pentagon) 6-sided
(hexagon) 8-sided
(octagon) 10-sided (decagon)
DID YOU KNOW?
Carl Gauss (1777–1855) was a famous German mathematician, physicist and astronomer. When he was 19 years old, he showed that a 17-sided polygon could be constructed using a ruler and compasses. This was a major achievement in geometry.
Gauss made a huge contribution to the study of mathematics and science, including correctly calculating where the magnetic south pole is and designing a lens to correct astigmatism.
He was the director of the Göttingen Observatory for 40 years. It is said that he did not become a professor of mathematics because he did not like teaching.
The sum of the interior angles of an n -sided polygon is given by
( 2) 180
S n
S n
180 360
or # c
=
-= -
The sum of the exterior angles of any polygon is 360c
Proof
Draw any n -sided polygon. Then the sum of both the exterior and interior angles is n 180# c.
n
n n
n n
180
180 180 360 180 180 360 360
Sum of exterior angles # c sum of interior angles c
c c
=
-= -
-= - +
=
] g
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EXAMPLES
1. Find the sum of the interior angles of a regular polygon with 15 sides.
How large is each angle?
Solution
2. Find the number of sides in a regular polygon whose interior angles are 140c.
interior angle in a regular 7-sided polygon, to the nearest minute.