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Chapter 5

Introduction to Deductive Method
5.1 Intuitive Method and Deductive Method
Intuitive method

the way of arriving at conclusions through
observations
and
experiments
.
Example
Ancient people observed that the sun rose in the east everyday.
Conclusion: ‘the sun rises in the east’
This method

not reliable

the cases we observed are limited
may be wrong

Request students to give another example.
To disprove a conclusions drawn by intuitive method
counter example is needed.
Example
Fact: Sparrow is a bird and it can fly.
Parrot is a bird and it can fly.
Conclusion: All birds can fly.
Group discussion: Try to find an counter example to disprove conclusion
Another method:
Deductive method
Example
Deductive me
thod is better than intuitive method.
5.2 The origin of deductive geometry
Three elements of deductive geometry:
I.
Definitions

give everyone a common understanding of geometrical objects.
Example
1. A
point
only shows a location & has no size.
2. A line h
as length but no width.
3. The ends of a
line segment
are points.
4. A surface only has length and width.
II.
Axioms
(Request students to present the 5 Axioms to the whole class.)

are some generally accepted and correct geometrical facts.
It is the common k
nowledge
of people that can be understood without proofs.

are the foundations of deductive proofs of geometry.
5 axioms chosed by Euclid (a Greek mathematician who put geometrical theorems with
proofs in a logical order.):
wrong
Given conditions
Logical reasoning
Conclusion
Given conditions
Logical reasoning
a)
TYT students are good students.
b)
_______ is TYT students.
_______ is a good student.
Conclusion
a)
Al
l the positive numbers are
greater than 0.
b)
5 is a positive number.
5 is greater than 0.
The study of
geometry in a
deductive way
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Axiom 1: Only one st. line can
be drawn between two points
Axiom 2: A st. line can be extended towards two ends infinitely.
Axiom 3: A circle can be fromed by taking any point as the center and any length as the radius.
Axiom 4: All right angles are equal.
Axiom 5: When two st. line ar
e cut by a transversal on the same plane, if the sum of 2 interior
angles on the same side is smaller than 2 right angles (i.e.
), then the 2 lines must
meet when they are produced on that side.
III.
Theorems
Euclid used the deductive met
hod to deduce
more than 460 theorems about geometry.
Ask students to prepare 5.3 in advance.
5.3 Theorems related to straight lines
(Presented by students)
Theorem 1:
If the sum of two adjacent angles
and
is
, the AOB is a straight line.
i.e. If
then AOB is a straight line
[adj.
supp.]
Note: “
and
are supplementary” means that
Also: “
and
are complementary” means that
Example
1
In the figure,
, prove that AOB is a straight line.
AOB is a straight line [adj.
supp.]
O
B
A
b
a
O
B
A
D
C
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Theorem 2:
The sum of all angles at a point is
i.e.
at a pt.]
Example
2
Find x in the figure.
at a pt.)
Theorem 3:
If two straight lines intersect, the vertically opposite angles are equal.
i.e. If AOB and COD are two st. lines,
then a = b, and c = d (vert. opp.
)
Example
In the
figure, AOB, COD and EOF are st. lines, prove that
[vert. opp.
[vert. opp.
]
[vert. opp.
at a pt.)
Classwork: (Ex5A,no.5 and 8)
Solution: No.5
No.8
(vert. opp.
)
A
B
D
C
a
b
c
d
x
A
C
D
B
c
a
d
b
E
D
B
F
C
A
O
z
y
x
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5.4 Theorems related to triangles
A.
Theorems for congruent triangles
1.
SAS
If AB = PQ, BC = QR and
,
Then
[S.A.S.]
2.
S.S.S
If AB = PQ, AC = PR and BC = QR,
Then
(S.S.S)
3.
A.S.A
If
and BC = QR
Then
(A.S.A)
4.
R.H.S
If AB = PQ, AC =
PR and
Then
(R.H.S)
* Note:
When 2 triangles are congruent, then
a)
the corresponding angles are equal. [corr.
]
b)
the corresponding sides are equal. [corr. sides,
]
Should be an included angle.
A
C
B
P
R
Q
A
C
B
P
R
Q
A
C
B
P
R
Q
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Example
Refer to the above No.3,
,
If
,
Then
[corr.
]
* Distribute worksheet 5.1 and ask students to complete it within 10 minutes.
Answers f
or worksheet 5.1
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Example
In the figure, AX = AC and AB = AY
a)
Prove that
b)
If
and
, find
Solution:
a)
Consider
and
A
B = AY (given)
(common angle)
AC = AX (given)
(S.A.S)
b)
sum of
)
Or common
C
Y
X
B
A
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B.
Theorems for isosceles triangles
Basic knowledge:
When two sides of a triangle are equal, the triangle is called an
isosceles triangle
.
Theorem of isosceles triangles:
Theorem 4:
The angles at the base (or base angl
es) of an isosceles triangle are equal.
i.e. In
if AB = AC
then
Theorem 5:
If two angles of a triangles are equal, then their opposite sides are equal.
i.e. In
if
then AB = AC (sides opp. equal
)
Example
In the figure, ABD is a straight line.
and AB = BC. Find a, b and c.
Solution:
on st. line)
AB =BC(given)
i.e.
c = a (proved)
A
B
C
A
C
B
C
A
B
D
b
a
c
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Example
In the figure, BDEC is a straight line. BD = EC and
. Show that
is an isosceles triangle.
(sides opp. equal
)
on st. line)
on st. line)
But
BD = EC (given)
is an isosceles triangle.
Classwork
(textbook P.187,Ex5B No.6)
Solution:
a)
Consider
and
CB = BC (common side)
AC = AB (given)
and AY = AX (given)
=
BX
b)
(proved in (a))
(sides opp. equal
)
is an isosceles triangle.
Appendix E5
Construction
I.
Construction of angle bisector of an angle
Step 1
Take B as the center.
Choose an appropriate radius,
and draw an arc to intersect
BA and BC at P and Q
re
spectively.
A
B
C
D
E
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Step 2
Take P and Q as centers, select
a radius longer than
PQ and
draw two arcs such that these
two arcs intersect at D.
Step 3
Join BD. BD is the angle
bisector of
II.
Construction of perpend
icular bisector of a line segment
Step 1
Take A as the center. Choose a
radius longer than
and
draw an arc above and below
the line segment AB.
Step 2
Take B as the center. Use the
same radius and draw another
two ones to intersec
t the arcs at
C and D.
Step 3
Join CD. CD intersects AB at
M, M is the mid

point of AB.
CD is the perpendicular
bisector of AB
III.
Construction of a line passing through a given point on a line segment and perpendicular to
that line segment
Step 1
Take
P as the center. Choose
an appropriate radius and draw
an arc to intersect AB at H and
K.
Step 2
Take H and K as centers. Use a
radius longer than
and
draw two arcs such that they
meet at Q.
PLK Tang Yuk Tien College/Maths/F.3/Teaching Not
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Step 3
Join PQ. Then
.
IV.
Construction of a line passing through a point lying outside a line segment and
perpendicular to that line segment
Step 1
Take P as the center. Choose
an appropriate radius and draw
an arc to intersect AB at H and
K.
Step 2
Take H and K as
centers.
Choose a radius longer than
and draw two arcs such
that the two arcs meet at Q.
Step 3
Join PQ. The line PQ
intersects AB at M, and
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