MATHS PPT FOR AREA OF TRIANGLES AND PARALLELOGRAMS

3. A triangle is one of the basic shapes of
geometry: a polygon with three corners or
vertices and three sides or edges which are
line segments.
In Euclidean geometry any three non-
collinear points joined together with three
line segments is called a triangle…
What is a Triangle?
A
C

4. Nature teaches us….
It is from nature that
man has learnt
everything and even
maths…. Here are
some of the triangles
that occur in nature….

5. But man has used his
intelligence to apply the
shapes of nature ….

6. In Euclidean geometry, a parallelogram is a convex
quadrilateral with two pairs of parallel sides. The
opposite or facing sides of a parallelogram are of
equal length and the opposite angles of a
parallelogram are of equal measure.
What is a parallelogram?

8. Area is a count of how many unit
squares fit inside a figure. To fully
understand this classic definition of area,
we need to picture the unit square. A
unit square is a square that is one unit
long by one unit wide. It can be 1'x1', 1
m x 1 m, 1 yd x 1 yd, 1" x 1", ...
1
1

9. Area of triangle is always calculated by
half its base multiplied to its corresponding
height…
Triangle
Area = ½b × h
b = base
h = vertical height

10. The area of the parallelogram is got by
multiplying its base to its
corresponding altitude (height)…
Parallelogram
Area = b × h
b = base
h = vertical height

11. If two figures have the same shape and the
same size, then they are said to be
congruent figures…
Congruent figures are exact duplicates of
each other. One could be fitted over the
other so that their corresponding parts
coincide.
The concept of congruent figures applies to
figures of any type.

12. As always , nature is the best teacher..
Nature teaches us congruence by means of
reflections and through BILATERAL
SYMMETRY etc…
Mostly, everything in this world is
bilaterally symmetrical especially we, the
humans.

13. If two figures have the same shape and
the same size, then they are said to be
congruent figures.
For example, rectangle ABCD and
rectangle PQRS are congruent
rectangles as they have the same shape
and the same size.
Side AB and side PQ are in the same
relative position in each of the figures.
We say that the side AB and side PQ
are corresponding sides.
D
C
BA
R
QP
S

14. Congruent triangles have the same size
and the same shape. The corresponding
sides and the corresponding angles of
congruent triangles are equal.
A B
C
D E
F
There are five
types of
congruence rules.
They are SSS, SAS,
AAS=SAA, ASA and
RHS congruence
Rules.

15. If two figures are congruent ,
they have equal areas.
But its converse IS NOT TRUE.
Two figures having equal
areas need not be congruent.
A B
Figures A and B
are congruent and
hence they have
equal areas.
D
C
Figures C
and D have
equal areas,
but they are
not
congruent.

16. THEOREM 1
Given: Two parallelograms ABCD and EFCD On the same base DC and between the same
parallels AF and DC.
TO PROVE: ar(ABCD)=ar(EFCD)
PROOF: In ADE and BCF,
Angle DAE= angle CBF (corresponding angles)
Angle AED= Angle BFC (corresponding angles)
Angle ADE= Angle BCF (angle sum property)
Also, AD =BC (opposite sides of the parallelogram ABCD)
Triangle ADE is congruent to triangle BCF (By ASA rule)
Therefore, ar(ADE)=ar(BCF) (congruent figures have equal areas)
Now, ar (ABCD) = ar(ADE) + ar(EDCB)
ar (ABCD) = ar (BCF)=ar(EDCB)
ar (ABCD) = ar (EFCD)
Hence they are equal in area
D C
A E B F

17. CONCLUSION OF THE THEOREM 1
 Parallelograms on the same base or equal bases and
lie between the same parallels are equal in area.
 Parallelograms on the same base or equal bases
that have equal areas lie between the same
parallels.
 Parallelograms which lie between same parallels
and have equal area lie on the same base.

18. THEOREM 2
GIVEN : ABCD is a parallelogram, AC is diagonal, Two
Triangles ADC and ABC on the same base AC and between the
Same parallels BC and AD
TO PROVE : ar (Δ ABC ) = ar (Δ CDA )
Proof: In triangles ABC , CDA
AB = CD (opposite sides of a parallelogram )
BC = DA (opposite sides of a parallelogram )
AC = AC (common )
Therefore Δ ABC Δ CDA (S-S-S congruency)
There fore Δ ABC = Δ CDA (if two triangles are congruent then their areas are equal ).
A B
CD

19. CONCLUSION OF THEOREM 2
 Triangles on the same base and between the same
parallels are equal in area.
 Triangles on the same base and having equal areas
lie between the same parallels.
 Triangles between the same parallels and have
equal area lie on the same base.

20. THEOREM 3
 IF A TRIANGLE AND A PARALLELOGRAM LIE ON THE SAME BASE
AND BETWEEN THE SAME PARALLELS, THEN THE AREA OF THE
TRIANGLE IS EQUAL TO HALF THE AREA OF THE
PARALLELOGRAM.
 Given: triangle ABP and parallelogram ABCD on the same base AB and between
the same parallels AB and PC.
 To prove: ar(PAB)=1/2 * ar (ABCD)
 Solution: Draw BQ//AP to get another parallelogram ABQP. Now parallelograms
ABQP and ABCD are on the same base AB and between the same parallels AB
and PC
 Therefore ar (ABQP) = ar (ABCD) [By theorem 1]……(1)
 But triangle PAB is congruent to triangle BQP (since diagnol PB divides
parallelogram ABQP into two congruent triangles)
 So, ar (PAB) = ar (BQP)…………(2)
 Therefore ar (PAB) = ½ * ar (ABQP) ………….(3)
 This gives ar (PAB) = ½ * ar (ABCD)…………..(FROM 1 AND 3)
P D Q C
A B