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Class 9: Triangles – Lecture Notes

In this chapter we will learn about the concept of “Congruence” of triangles.

In simple, any two geometrical figures, exactly of the same size and shape are congruent.

Congruence of a line segment: Two line segment and are congruent if and only if their lengths are equal. i.e. if

Congruence of angles: Two angles are congruent if and only if their measures are equal. i.e.

Congruence of triangles:

Consider two and (as shown in the figure)

Two triangles are congruent if and only if one of them can be made to superpose on the other so as to cover it exactly.

Two triangles are congruent if and only if there exists a correspondence between their vertices such that the corresponding sides and the corresponding angles of the two triangles are equal. This means that if:

and

and

Also if   and 

Theorems of Congruence

Theorem 1: S.A.S (Side Angle Side): Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle.

Example 1: In the adjoining figure, is the mid point of and . Prove that (i) (ii) (iii)

Solution: Consider  and 

(Given)

(Given)

(vertically opposite angle)

Therefore  (By S.A.S theorem)

Since the triangles are congruent, (corresponding sides of congruent triangles are equal)

Also since the two triangles are congruent, (corresponding angles are also equal). Hence (alternate angles are equal)

Example 2: In the adjoining figure, it is given that and . Prove that .

Solution: is common segment of and

and

Therefore

Now consider  and 

(given)

(Proved above)

is common

Therefore 

Theorem 2: A.S.A (Angle Side Angle): Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Example 3: In the adjoining triangle, diagonal   is a quadrilateral bisects the and . Prove that and .

Solution:

Given: and 

Consider and

(given)

(given)

and is common.

Therefore  (A.S.A theorem)

Hence we can say that and

Example 4:    is a line segment, and are two equal line segments draw on opposite sides of line such that . If and intersect each other at , prove that (i) (ii) and bisect each other

Solution:

Given that  and is the transversal.

Therefore

Similarly,  and is the transversal.

Therefore

Now consider and

(given)

Therefore 

and

Theorem 3.1 A.A.S (Angle Angle Side): If any two angles and a not included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are congruent.

Solution / Proof: 

In the given and , we have , and .

Therefore

Hence we see that with and

Hence (A.S.A Theorem).

Theorem 3.2: If two angles of a triangle are equal, then sides opposite to them are also equal.

Solution:

Given:

Draw the bisector of . Now consider and

We have (angle bisector)

and is common or equal.

Hence by (A.A.S theorem)

Theorem 3.3: If the altitude of one vertex of a triangle bisects the opposite side, then the triangle is isosceles.

Solution:

In the is the altitude

(given)

and is common.

Therefore

Theorem 3.4: In an isosceles triangle altitude from the vertex bisects the base.

Solution:

In , given

(Altitude)

And is common.

Therefore (A.A.S theorem)

Theorem 3.5: If the bisector of the vertical angle of a triangle bisects the base of the triangle, then the triangle is isosceles.

Solution:

Extend to such that

Given: and

Now, consider and

Therefore (given) , (constructed)

and (vertically opposite angles)

Hence  (S.A.S theorem)

Therefore and

we know,

. Hence is an isosceles triangle.

Theorem 4: S.S.S (Side Side Side):  Two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides of the other triangle.

Given two and such that and

To Prove:

Solution:

In , draw such that and .

Join and

Consider, and We have

(given)

(by construction)

And (by construction)

Therefore (by S.A.S theorem)

and

Since and

In (angle opposite to the equal sides are equal)

Similarly, in In (angle opposite to the equal sides are equal)

Hence

Now consider and

(given)

(just proved above)

(given)

Therefore (by S.A.S theorem). Hence proven.

Theorem 5: R.H.S (Right Angle Hypotenuse Side): Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and the one side of the other triangle.

Given two right angled and . and

Also given, and

To Prove:

Solution: 

Consider and

(by construction)

(by construction)

(given)

Therefore  (By S.A.S theorem)

and 

(angle opposite to the equal sides are equal)

Therefore

In and , we have

Now consider and

(given)

(proved above)

(given)

Therefore  (by S.A.S theorem)

Inequality Relations in a Triangle

Theorem 1: If two sides of a triangle are unequal, the longer side has greater angle opposite to it.

Given in

Find a point on such that

… … … (i)

Consider

We know (exterior angle is the sum of other two angles)

or  … … … … … (ii)

Therefore from (i) and (ii) we get

and

… … … … … (iii)

From we get … … … … … (iv)

From (iii) and (iv) we get

. Hence proven.

Theorem 2: In a triangle the greater angle has the longer side opposite to it (converse of Theorem 1).

In , given

To Prove:

Solution: 

There are three possible cases (i) (ii)



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Class 9: Triangles – Lecture Notes

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