cbse class 10 mathematics geometry topic

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As originally posted on Edvie.com GEOMETRY Class Xth Notes Introduction: Observe the following figures By observation i), ii) are of the same shape but different in size iii), iv) are of the same shape and same size v), vi) are different in shape as well as size. Similar figures: Two figures are said to be similar if they have the same shapes although they may differ in sizes. The following pairs of figures are similar. Similar triangles: If two triangles are similar then their i) Corresponding angles are equal, and ii) Corresponding sides are in proportion. Here consider So, The symbol ‘~’ means “is similar to”. Corresponding sides: Sides opposite to equal angles in similar triangles are known as corresponding sides and they are proportional. . Therefore, AB and DE are corresponding sides as they are opposite to and respectively.

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Page 1: CBSE Class 10 Mathematics Geometry Topic

 

As originally posted on Edvie.com  

  

GEOMETRY  

 

Class Xth Notes 

 

Introduction: Observe the following figures 

 

By observation 

i), ii) are of the same shape but different in size iii), iv) are of the same shape and same size v), vi) are different in shape as well as size.  Similar figures: Two figures are said to be similar if they have the same shapes although they may differ in sizes. The following pairs of figures are similar. 

 

 

 

Similar triangles: If two triangles are similar then their 

i) Corresponding angles are equal,

and

ii) Corresponding sides are in

proportion.

 

Here consider  

 

So,  

 

The symbol ‘~’ means “is similar to”.  

Corresponding sides: Sides opposite to equal angles in similar triangles are known as corresponding sides and they are proportional.  

 

. Therefore, AB and DE are corresponding

sides as they are opposite to and respectively. 

Page 2: CBSE Class 10 Mathematics Geometry Topic

 

Similarly, BC and EF are a pair of corresponding sides. AC and DF are also a pair of

corresponding sides. 

Thus, , as corresponding sides of similar triangles are proportional. 

 Corresponding angles: Angles opposite to proportional sides in similar triangles are

known as corresponding angles.  

 

 

as they are opposite to corresponding sides BC and EF respectively. 

Similarly, . 

Congruency and similarity of triangles: Congruency is particular case of similarity.

In both the cases, three angles of one triangle are equal to the three corresponding

angles of the other triangle. But in similar triangles the corresponding sides are

proportional, while in congruent triangles the corresponding sides are equal. 

 

 

Where k is the constant of proportionality or the scale factor of size transformation. 

 

Page 3: CBSE Class 10 Mathematics Geometry Topic

 

Therefore, in congruent triangles the constant of proportionality between the

corresponding sides is equal to one. Thus, congruent triangles have the same shape

and size while similar triangles have the shape but not necessarily the same size. 

Congruent triangles are always similar, but similar triangles are not necessarily

congruent. 

Triangles that are similar to the same triangle are similar to each other. 

 

Here ΔABC ∼ ΔDEF and ΔLMN ∼ ΔDEF 

Therefore, ΔABC ∼ ΔLMN 

 Criteria of similarity between triangles: 

1. SAS criterion of Similarity: If two triangles have their corresponding angles equal

and the sides including between a pair of angles are proportional, then the triangles

are similar. 

 

 

Page 4: CBSE Class 10 Mathematics Geometry Topic

 

 2. AA or AAA Criterion of similarity: If two angles of one triangle are equal to two

corresponding angles of the other, then the triangles are similar. 

 If in two triangles, two angles of one are equal to two angles of the other, then the third

angle of the first triangle is also equal to the third angle of the other because the sum

of the three angles in a triangle is 180 0 . 

Thus, similar triangles are equiangular. 

 3. SSS criterion of similarity: If in two triangles, three sides of one are proportional to

the three sides of the other, then the triangles are similar. 

 

 Theorem: In a right-angled triangle, if a perpendicular is drawn from the right angled

vertex to the hypotenuse, the triangles on each side of it are similar to the whole

triangle and to one another. 

Given: Let ABC be a triangle in which angle  

 

To prove:  

Proof: 

Page 5: CBSE Class 10 Mathematics Geometry Topic

 

(Given) 

(Common angle) 

Therefore ( By AA criterion of similarity) . . . . .(1)

 

(Given) 

(Common angle) 

Therefore ( By AA criterion of similarity) . .. .. .. (2) 

from 1, 2 . 

Hence Proved. 

Basic proportionality theorem: 

Theorem: A line drawn parallel to one side of a triangle divides the other two sides in

the same ratio (proportion). 

Given: In ΔABC,D and E are points on AB and AC respectively, such that DE ∥   BC

To prove:  Proof:  

In  

( Common angle) 

( Corresponding angles) 

( AA criterion of similarity) 

(Corresponding sides of similar triangles are proportional) 

( By subtracting 1 from both sides) 

 

 

Page 6: CBSE Class 10 Mathematics Geometry Topic

 

(by reciprocating term in above step).  

Hence Proved. 

Converse of Basic Proportionality Theorem: 

Theorem: The line dividing two sides of a triangle proportionally is parallel to the third side. 

Given: ΔABC, D and E are points on AB and AC respectively, such that  

To Prove: DE∥BC.    

Proof: 

(Given) 

(Taking reciprocals of both sides) 

(By adding 1 both sides) 

 

In  

(From statement 1) 

(Common angle) 

(By SAS criterion of similarity . 

(Corresponding angles of similar triangles are equal) 

(Corresponding angles are equal) 

Hence Proved. 

An important application of Basic Proportionality Theorem: The internal bisector of

an angle of a triangle divides the opposite side in the ratio of the sides containing the

angle. (Vertical angle bisector theorem). 

Given: AD is the internal bisector of  

Page 7: CBSE Class 10 Mathematics Geometry Topic

 

To prove:  

Construction:  

Proof: 

(Given) integral bisector . . . . (1) 

( ) 

( ) 

(from 1) 

AE = EC (opposite sides of equal angles of ) . . . (2) 

( ) 

(By statement 2) 

Hence Proved. 

 

 

 

The above proposition is true for external division also. 

 Converse of the above proposition is also true. So, if D is a point on BC such that

BD : DC = AB : AC then AD bisects the angle BAC internally or externally. 

 

Area of similar triangles: The ratio of the areas of two similar triangles has relation

with the ratio of the corresponding sides. The ratio of the areas of two similar triangles

is the square of the ratio of their corresponding sides. 

Theorem:  

Page 8: CBSE Class 10 Mathematics Geometry Topic

 

Statement: The ratio of the areas of two similar triangles is equal to the ratio of the

squares of their corresponding sides. 

Given: ΔABC ∼ ΔDEF 

So,  

Also,  

To prove:  

Construction: Through A draw AP BC and through D draw DQ EF.  

Proof:  

Thus,  

 

 

 

 

Putting the value of from (iv) in (i), we get 

 

Page 9: CBSE Class 10 Mathematics Geometry Topic

 

Similarly, it can also be prove that  

 

and  From (v), (vi) and (vii), we obtain 

  Example: Prove that the ratio of the areas of two similar triangles is equal to the ratio

the squares of their corresponding: i) altitudes ii) angle bisectors 

Solution:  

i) altitudes 

Given:  

To prove:  

Proof:  

 

ii) Angle bisectors 

Given:  

AM is the bisector of and DN is the

corresponding bisector of  

Page 10: CBSE Class 10 Mathematics Geometry Topic

 

To prove:  

Proof:  

 

 

 

Given below is the proof given in two ways. Let us check the relevancy of the proof for 

similarity. 

 Some students solve it as: The two triangles are similar as  

 

Page 11: CBSE Class 10 Mathematics Geometry Topic

 

Correct statements is:  Here   but including angles   and   are not equal. So 

the given triangles are not similar. 

 

 

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