Points to Remember: Triangles Notes | EduRev

Mathematics Chart book, A Quick guide

Class 10 : Points to Remember: Triangles Notes | EduRev

 Page 1


Triangles 
Page 2


Triangles 
30
0
30
0
65
0
65
0
30
0
30
0
65
0
65
0
6cm
8cm
30
0
30
0
65
0
65
0
6cm 6cm
6cm
4.5cm
3cm
8cm
6cm
4cm
8cm
6cm
4cm
8cm
6cm
4cm
6cm
4.5cm
30
0
30
0
8cm
6cm
8cm
6cm
30
0
30
0
8cm
6cm
Angle - Angle (AA)
Only two pairs of corresponding angles of two triangles 
is enough to prove similarity, but the same is not enough
in the case of congruency.
Angle - Side - Angle (ASA)/(AAS/SAA)
Along with the 2 equal angles, a single side  gives us
the surety for congruency (equal side) of triangles 
& same is true for similarity (equal ratio) as well.
Not applicable
Side - Side - Side (SSS)
When all the 3 corresponding sides of two triangles are
in equal ratio, the triangles are similar and
when all the sides are equal (i.e.ratio = 1),
triangles are congruent.
Side - Angle - Side (SAS)
Two triangles can also be similar when any two
corresponding sides are in equal ratio & an angle between
them is equal whereas they will be congruent when two
corresponding sides as well as an angle between them
is equal.
SIMILARITY  Likeness CRITERIA CONGRUENCY  Equal
X
TRIANGLES 15
Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles.
Criteria for Similarity & Congruency of Triangles
Page 3


Triangles 
30
0
30
0
65
0
65
0
30
0
30
0
65
0
65
0
6cm
8cm
30
0
30
0
65
0
65
0
6cm 6cm
6cm
4.5cm
3cm
8cm
6cm
4cm
8cm
6cm
4cm
8cm
6cm
4cm
6cm
4.5cm
30
0
30
0
8cm
6cm
8cm
6cm
30
0
30
0
8cm
6cm
Angle - Angle (AA)
Only two pairs of corresponding angles of two triangles 
is enough to prove similarity, but the same is not enough
in the case of congruency.
Angle - Side - Angle (ASA)/(AAS/SAA)
Along with the 2 equal angles, a single side  gives us
the surety for congruency (equal side) of triangles 
& same is true for similarity (equal ratio) as well.
Not applicable
Side - Side - Side (SSS)
When all the 3 corresponding sides of two triangles are
in equal ratio, the triangles are similar and
when all the sides are equal (i.e.ratio = 1),
triangles are congruent.
Side - Angle - Side (SAS)
Two triangles can also be similar when any two
corresponding sides are in equal ratio & an angle between
them is equal whereas they will be congruent when two
corresponding sides as well as an angle between them
is equal.
SIMILARITY  Likeness CRITERIA CONGRUENCY  Equal
X
TRIANGLES 15
Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles.
Criteria for Similarity & Congruency of Triangles
Equilateral Triangle
Isosceles Triangle
Isosceles Right Angled
Triangle
Scalene Right Angled
Triangle
Scalene Triangle
Right Angled Triangle
A
B C
60°
60° 60°
A
B C
x x A
B
C
A
B C
A
B C
•    Three equal sides
•    Three equal angles
•    A triangle with one angle of 90°
•    Two equal sides
•    Two equal angles
•    Right triangle with
      two equal sides
•    Right triangle with
     no equal sides
•    All sides with di?erent lengths
•    All the angles with di?erent
     measurements
A
B C
a
b
c
A
B C
h
A
B C
Here are some of the ways to calculate area of a triangle
Area of a triangle (A)
S =
(Heron’s Formula)
A = Base x Height x
1
2
a + b + c
Sum of the angles
of a triangle is 180°
?A + ?B + ?C = 180°
2
A = s (s - a) (s - b) (s - c)
? ?
Triangles
A 2-dimensional ?gure with 3-sides and 3-angles is called a Triangle. This chart will help you learn the di?erent types of triangles based on
varying side lengths and angle measurement.
TRIANGLES 14
Page 4


Triangles 
30
0
30
0
65
0
65
0
30
0
30
0
65
0
65
0
6cm
8cm
30
0
30
0
65
0
65
0
6cm 6cm
6cm
4.5cm
3cm
8cm
6cm
4cm
8cm
6cm
4cm
8cm
6cm
4cm
6cm
4.5cm
30
0
30
0
8cm
6cm
8cm
6cm
30
0
30
0
8cm
6cm
Angle - Angle (AA)
Only two pairs of corresponding angles of two triangles 
is enough to prove similarity, but the same is not enough
in the case of congruency.
Angle - Side - Angle (ASA)/(AAS/SAA)
Along with the 2 equal angles, a single side  gives us
the surety for congruency (equal side) of triangles 
& same is true for similarity (equal ratio) as well.
Not applicable
Side - Side - Side (SSS)
When all the 3 corresponding sides of two triangles are
in equal ratio, the triangles are similar and
when all the sides are equal (i.e.ratio = 1),
triangles are congruent.
Side - Angle - Side (SAS)
Two triangles can also be similar when any two
corresponding sides are in equal ratio & an angle between
them is equal whereas they will be congruent when two
corresponding sides as well as an angle between them
is equal.
SIMILARITY  Likeness CRITERIA CONGRUENCY  Equal
X
TRIANGLES 15
Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles.
Criteria for Similarity & Congruency of Triangles
Equilateral Triangle
Isosceles Triangle
Isosceles Right Angled
Triangle
Scalene Right Angled
Triangle
Scalene Triangle
Right Angled Triangle
A
B C
60°
60° 60°
A
B C
x x A
B
C
A
B C
A
B C
•    Three equal sides
•    Three equal angles
•    A triangle with one angle of 90°
•    Two equal sides
•    Two equal angles
•    Right triangle with
      two equal sides
•    Right triangle with
     no equal sides
•    All sides with di?erent lengths
•    All the angles with di?erent
     measurements
A
B C
a
b
c
A
B C
h
A
B C
Here are some of the ways to calculate area of a triangle
Area of a triangle (A)
S =
(Heron’s Formula)
A = Base x Height x
1
2
a + b + c
Sum of the angles
of a triangle is 180°
?A + ?B + ?C = 180°
2
A = s (s - a) (s - b) (s - c)
? ?
Triangles
A 2-dimensional ?gure with 3-sides and 3-angles is called a Triangle. This chart will help you learn the di?erent types of triangles based on
varying side lengths and angle measurement.
TRIANGLES 14
Theorem1 : A line parallel to one side of a triangle divides the 
other two sides in equal proportion
Given:
DE || BC
A
B
C
D
<
E
>
AD
DB
=
AE
EC
AD
DB
=
AE
EC
AD
BD
........ (1)
To Prove:
Property
Proof:
1
st
1
st
Property 2
nd
Step
2
nd
Step
3
rd
Step
=
=
? (    ADE)
A
? (    BDE) A
? (    BDE) A
[ Triangles with same height ]
[ Triangles with same height ]
[ Triangles with equal base & height ]
AE
EC
........ (2)
........ (3)
{ From........(1),(2) & (3)}
=
? (    ADE)
A
? (    CDE) A
? (    CDE) A
--------------
-------------------
Triangles drawn between two parallel
lines with same base have equal areas
h
b1
b1 b1
b2
b2
b2
Ratio of areas of triangles with equal 
height is equal to the ratio of their bases
Base
Height
= =
Basic Proportionality Theorem
TRIANGLES 16
Learn these properties to prove BPT
B =     D = 90
o
Given: ?ABC is a right angled triangle
To prove: (Hyp)
2
 = (Side1)
2
 + (Side2)
2
  (AC)
2
 = (AB)
2
 + (BC)
2
Theorem 2 : For any right angled triangle, square of hypotenuse is 
equal to the sum of squares of the other two sides.
B =    D = 90
o
Draw a perpendicular line from the right angle to its opposite
side. 
Proof: We’ll justify the original triangle is similar to the 2
newly formed right angled triangles.
AB
AD
AC
AB
=
=
BC
DB
A is common
In ?ABC & ?ADB      
?ABC      ?ADB 
AB
2
 = AD x AC 
   .......(1)
AB
BD
BC
DC
= =
AC
BC
C is common
BC
2
 = DC x AC
   ........(2)
Construction:
A
B
C
A
B
C
D
A
B
C A B
D
A
B
C B C
D
AB
2
 + BC
2
AB
2 
+ BC
2 
= AC
2
On adding equation
(1) & (2) we get
= AD x AC + DC x AC
= AC (AD+DC)
= AC x AC
= AC
2 
Pythagoras Theorem
In ?ABC & ?BDC
. .
.
?ABC      ?BDC 
. .
.
Page 5


Triangles 
30
0
30
0
65
0
65
0
30
0
30
0
65
0
65
0
6cm
8cm
30
0
30
0
65
0
65
0
6cm 6cm
6cm
4.5cm
3cm
8cm
6cm
4cm
8cm
6cm
4cm
8cm
6cm
4cm
6cm
4.5cm
30
0
30
0
8cm
6cm
8cm
6cm
30
0
30
0
8cm
6cm
Angle - Angle (AA)
Only two pairs of corresponding angles of two triangles 
is enough to prove similarity, but the same is not enough
in the case of congruency.
Angle - Side - Angle (ASA)/(AAS/SAA)
Along with the 2 equal angles, a single side  gives us
the surety for congruency (equal side) of triangles 
& same is true for similarity (equal ratio) as well.
Not applicable
Side - Side - Side (SSS)
When all the 3 corresponding sides of two triangles are
in equal ratio, the triangles are similar and
when all the sides are equal (i.e.ratio = 1),
triangles are congruent.
Side - Angle - Side (SAS)
Two triangles can also be similar when any two
corresponding sides are in equal ratio & an angle between
them is equal whereas they will be congruent when two
corresponding sides as well as an angle between them
is equal.
SIMILARITY  Likeness CRITERIA CONGRUENCY  Equal
X
TRIANGLES 15
Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles.
Criteria for Similarity & Congruency of Triangles
Equilateral Triangle
Isosceles Triangle
Isosceles Right Angled
Triangle
Scalene Right Angled
Triangle
Scalene Triangle
Right Angled Triangle
A
B C
60°
60° 60°
A
B C
x x A
B
C
A
B C
A
B C
•    Three equal sides
•    Three equal angles
•    A triangle with one angle of 90°
•    Two equal sides
•    Two equal angles
•    Right triangle with
      two equal sides
•    Right triangle with
     no equal sides
•    All sides with di?erent lengths
•    All the angles with di?erent
     measurements
A
B C
a
b
c
A
B C
h
A
B C
Here are some of the ways to calculate area of a triangle
Area of a triangle (A)
S =
(Heron’s Formula)
A = Base x Height x
1
2
a + b + c
Sum of the angles
of a triangle is 180°
?A + ?B + ?C = 180°
2
A = s (s - a) (s - b) (s - c)
? ?
Triangles
A 2-dimensional ?gure with 3-sides and 3-angles is called a Triangle. This chart will help you learn the di?erent types of triangles based on
varying side lengths and angle measurement.
TRIANGLES 14
Theorem1 : A line parallel to one side of a triangle divides the 
other two sides in equal proportion
Given:
DE || BC
A
B
C
D
<
E
>
AD
DB
=
AE
EC
AD
DB
=
AE
EC
AD
BD
........ (1)
To Prove:
Property
Proof:
1
st
1
st
Property 2
nd
Step
2
nd
Step
3
rd
Step
=
=
? (    ADE)
A
? (    BDE) A
? (    BDE) A
[ Triangles with same height ]
[ Triangles with same height ]
[ Triangles with equal base & height ]
AE
EC
........ (2)
........ (3)
{ From........(1),(2) & (3)}
=
? (    ADE)
A
? (    CDE) A
? (    CDE) A
--------------
-------------------
Triangles drawn between two parallel
lines with same base have equal areas
h
b1
b1 b1
b2
b2
b2
Ratio of areas of triangles with equal 
height is equal to the ratio of their bases
Base
Height
= =
Basic Proportionality Theorem
TRIANGLES 16
Learn these properties to prove BPT
B =     D = 90
o
Given: ?ABC is a right angled triangle
To prove: (Hyp)
2
 = (Side1)
2
 + (Side2)
2
  (AC)
2
 = (AB)
2
 + (BC)
2
Theorem 2 : For any right angled triangle, square of hypotenuse is 
equal to the sum of squares of the other two sides.
B =    D = 90
o
Draw a perpendicular line from the right angle to its opposite
side. 
Proof: We’ll justify the original triangle is similar to the 2
newly formed right angled triangles.
AB
AD
AC
AB
=
=
BC
DB
A is common
In ?ABC & ?ADB      
?ABC      ?ADB 
AB
2
 = AD x AC 
   .......(1)
AB
BD
BC
DC
= =
AC
BC
C is common
BC
2
 = DC x AC
   ........(2)
Construction:
A
B
C
A
B
C
D
A
B
C A B
D
A
B
C B C
D
AB
2
 + BC
2
AB
2 
+ BC
2 
= AC
2
On adding equation
(1) & (2) we get
= AD x AC + DC x AC
= AC (AD+DC)
= AC x AC
= AC
2 
Pythagoras Theorem
In ?ABC & ?BDC
. .
.
?ABC      ?BDC 
. .
.
TRIANGLES 17
Converse of Basic Proportionality Theorem Converse of Pythagoras Theorem
Theorem: If a line divides any two sides of a triangle in a same 
ratio, then the lines is parallel to the third side.
< >
A
D E
B C
< >
A
D E
F
B C
Given :- 
AD
DB
AE
EC
=
AD
DB
AF
FC
=
AD
DB
AE
EC
=
AF
FC
AE
EC
=
AC
FC
AC
EC
=
AC
FC
AC
EC
_ 
= 1
_ 
1
To Prove :- DE || BC
Proof : 
Assume that DE is not parallel to BC, then we draw a line DF || BC
So,
But
by equation (1) and (2)
But AF = AC - FC and 
        AE = AC - EC
}       .............(4 )
AC - FC AC - EC
FC EC
=
FC = EC
It is only possible when F and E coincide each other which
So DF is nothing but DE
So DE is Parallel to BC i.e. DE || BC
by 3 and 4
{by BPT}.............(1)
{Given}.............(2)
.............(3)
.............(5)
In a triangle, if the square of one side is equal to the sum of the 
squares of the other two sides, then the angle opposite to the 
Given :- ? ABC such that AB
2
 + BC
2
 = AC
2
To prove :- ? ABC is right angled at angle B
Construction :- Construct a right angle ? PQR, right angle at Q 
                               such that PQ = AB and QR = BC
Proof :- In ? PQR, Since     Q = 90
0
PQ + QR = PR  .......... (Pythagoras therom)
or AB + BC = PR ........ (Construction).........(1)
But AB + BC = AC ......(Given).........(2)
From equations (1) and (2) , we get
PR = AC i.e. PR = AC ...... (3)
In ?PQR and  ?ABC,
 PQ = AB ........ (Construction)
 QR = BC ......... (Construction) 
 AC = PR ........ (From equation 3)
   ? ABC      ? PQR .......(by SSS criterion for Congruency)
       B =     Q .........................(Corresponding angle of congruent triangle) 
~
=
But     Q = 90
0
  (by construction)
B = 90
Hence, ? ABC is a right angle triangle, right angled at B
P
Q
R
A
B
C
.
.
.
.
.
.
Construction
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