11 - Let's Recap - Triangles - Class 10 - Maths Class 10 Notes | EduRev

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Class 10 : 11 - Let's Recap - Triangles - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


TRIANGLES 
 
SIMILAR FIGURES: 
? Two figures having the same shape and not necessarily the same size are 
called similar figures. 
? Congruent figures are similar but similar figures need not be congruent. 
? Two polygons of the same number of sides are similar, if  
(1) Their corresponding angles are equal and  
(2) Their corresponding sides are in the same ratio (or proportion). 
? The sum of the ratio of the corresponding sides is referred to as the scale 
factor (or the representative fraction) for the polygons.  
? If one polygon is similar to another polygon and this second polygon is 
similar to a third polygon, then the first polygon is similar to the third 
polygon. 
SIMILARITY OF TRIANGLES: 
? Two triangles are similar, if  
(1) Their corresponding sides are in the same ratio (or proportion) and 
(2) Their corresponding angles are equal.  
? Basic Proportionality Theorem or Thales Theorem: If a line is drawn 
parallel to one side of a triangle to intersect the other two sides in distinct 
points, the other two sides are divided in the same ratio. 
If in Fig. 1, in  ABC, a line ‘l’ parallel to BC intersects AB at D and AC 
at E, then 
  
  
 
  
  
 
 
 
 
 
 
 
 
 
 
 
A 
 
A 
D 
E 
B 
C 
 
B 
D 
E 
B 
C 
 
C 
D 
E 
B 
C 
 
D 
D 
E 
B 
C 
 
E 
D 
E 
B 
C 
 
Fig. 1 
l 
Page 2


TRIANGLES 
 
SIMILAR FIGURES: 
? Two figures having the same shape and not necessarily the same size are 
called similar figures. 
? Congruent figures are similar but similar figures need not be congruent. 
? Two polygons of the same number of sides are similar, if  
(1) Their corresponding angles are equal and  
(2) Their corresponding sides are in the same ratio (or proportion). 
? The sum of the ratio of the corresponding sides is referred to as the scale 
factor (or the representative fraction) for the polygons.  
? If one polygon is similar to another polygon and this second polygon is 
similar to a third polygon, then the first polygon is similar to the third 
polygon. 
SIMILARITY OF TRIANGLES: 
? Two triangles are similar, if  
(1) Their corresponding sides are in the same ratio (or proportion) and 
(2) Their corresponding angles are equal.  
? Basic Proportionality Theorem or Thales Theorem: If a line is drawn 
parallel to one side of a triangle to intersect the other two sides in distinct 
points, the other two sides are divided in the same ratio. 
If in Fig. 1, in  ABC, a line ‘l’ parallel to BC intersects AB at D and AC 
at E, then 
  
  
 
  
  
 
 
 
 
 
 
 
 
 
 
 
A 
 
A 
D 
E 
B 
C 
 
B 
D 
E 
B 
C 
 
C 
D 
E 
B 
C 
 
D 
D 
E 
B 
C 
 
E 
D 
E 
B 
C 
 
Fig. 1 
l 
 
 
 
 
? Converse of Basic Proportionality Theorem: If a line divides any two 
sides of a triangle in the same ratio, then the line is parallel to the third 
side.  
If in Fig. 1, in  ABC, a line ‘l’ intersects AB at D and AC at E such that 
 
  
  
 
  
  
, then DE   BC. 
CRITERIA FOR SIMILARITY OF TRIANGLES: 
? If in two triangles, corresponding angles are equal, then their 
corresponding sides are in the same ratio (or proportion) and hence the 
two triangles are similar. This criterion is referred to as the AAA (Angle-
Angle-Angle) criterion of similarity of two triangles. 
Similarly, if two angles of one triangle are respectively equal to two 
angles of another triangle, (then by angle sum property their third angle 
will also be equal) the two triangles are similar. This is referred to as the 
AA (Angle-Angle) similarity criterion for two triangles. 
? If in two triangles, sides of one triangle are proportional to (i.e., in the 
same ratio of) the sides of the other triangle, then their corresponding 
angles are equal and hence the two triangles are similar. This criterion is 
referred to as the SSS (Side- Side - Side) criterion of similarity of two 
triangles. 
? If one angle of a triangle is equal to one angle of the other triangle  and 
the sides including these angles are proportional, then the two triangles 
are similar. This criterion is referred to as the SAS (Side- Angle - Side) 
criterion of similarity of two triangles. 
AREAS OF SIMILAR TRIANGLES: 
? Theorem Statement: The ratio of the areas of two similar triangles is 
equal to the ratio of squares of their corresponding sides. 
 
 
 
Page 3


TRIANGLES 
 
SIMILAR FIGURES: 
? Two figures having the same shape and not necessarily the same size are 
called similar figures. 
? Congruent figures are similar but similar figures need not be congruent. 
? Two polygons of the same number of sides are similar, if  
(1) Their corresponding angles are equal and  
(2) Their corresponding sides are in the same ratio (or proportion). 
? The sum of the ratio of the corresponding sides is referred to as the scale 
factor (or the representative fraction) for the polygons.  
? If one polygon is similar to another polygon and this second polygon is 
similar to a third polygon, then the first polygon is similar to the third 
polygon. 
SIMILARITY OF TRIANGLES: 
? Two triangles are similar, if  
(1) Their corresponding sides are in the same ratio (or proportion) and 
(2) Their corresponding angles are equal.  
? Basic Proportionality Theorem or Thales Theorem: If a line is drawn 
parallel to one side of a triangle to intersect the other two sides in distinct 
points, the other two sides are divided in the same ratio. 
If in Fig. 1, in  ABC, a line ‘l’ parallel to BC intersects AB at D and AC 
at E, then 
  
  
 
  
  
 
 
 
 
 
 
 
 
 
 
 
A 
 
A 
D 
E 
B 
C 
 
B 
D 
E 
B 
C 
 
C 
D 
E 
B 
C 
 
D 
D 
E 
B 
C 
 
E 
D 
E 
B 
C 
 
Fig. 1 
l 
 
 
 
 
? Converse of Basic Proportionality Theorem: If a line divides any two 
sides of a triangle in the same ratio, then the line is parallel to the third 
side.  
If in Fig. 1, in  ABC, a line ‘l’ intersects AB at D and AC at E such that 
 
  
  
 
  
  
, then DE   BC. 
CRITERIA FOR SIMILARITY OF TRIANGLES: 
? If in two triangles, corresponding angles are equal, then their 
corresponding sides are in the same ratio (or proportion) and hence the 
two triangles are similar. This criterion is referred to as the AAA (Angle-
Angle-Angle) criterion of similarity of two triangles. 
Similarly, if two angles of one triangle are respectively equal to two 
angles of another triangle, (then by angle sum property their third angle 
will also be equal) the two triangles are similar. This is referred to as the 
AA (Angle-Angle) similarity criterion for two triangles. 
? If in two triangles, sides of one triangle are proportional to (i.e., in the 
same ratio of) the sides of the other triangle, then their corresponding 
angles are equal and hence the two triangles are similar. This criterion is 
referred to as the SSS (Side- Side - Side) criterion of similarity of two 
triangles. 
? If one angle of a triangle is equal to one angle of the other triangle  and 
the sides including these angles are proportional, then the two triangles 
are similar. This criterion is referred to as the SAS (Side- Angle - Side) 
criterion of similarity of two triangles. 
AREAS OF SIMILAR TRIANGLES: 
? Theorem Statement: The ratio of the areas of two similar triangles is 
equal to the ratio of squares of their corresponding sides. 
 
 
 
 
 
PYTHAGORAS THEOREM: 
? Theorem Statement: In a right triangle, the square of the hypotenuse is 
equal to the sum of the squares of the other two sides. 
? Converse of Pythagoras Theorem- In a triangle, if square of one side is 
equal to the sum of the squares of the other two sides, then the angle 
opposite the first side is a right angle. 
? If a perpendicular drawn from the hypotenuse to the vertex of a right 
triangle, the triangles on both sides of the perpendicular are similar to the 
whole triangle and to each other.  
INTERNAL BISECTOR OF AN ANGLE OF A TRIANGLE: 
? The internal bisector of an angle of a triangle divides the opposite side 
internally in the ratio of the sides containing the angle. 
? If a line segment drawn from the vertex of an angle of a triangle to its 
opposite side divides it in the ratio of the sides containing the angle, then 
the line segment bisects the angle. 
 
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