Chapter 6

Introduction to Triangles

Similar Figures: Two figures are similar, if they have the same shape , but not necessarily the same size.

All congruent figures are similar but the similar figures need not be congruent.

Similar Triangles: Two triangles are similar, if

(i) Their corresponding angles are equal and

(ii) Their corresponding sides are in the same ratio.

For example: In triangles PQR and MNO,

If P = M, Q = N and R = O

And

Then, PQR  MNO

Theorem (Basic proportionality 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.

For example: In PQR,

If MN || QR,

Then,

Theorem (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.

For example: In PQR,

Then, MN || QR

Criteria for Similarity of Triangles:

(i) If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

(ii) If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. (AA)

For example: In triangles PQR and MNO,

If P = M and Q = N, then PQR  MNO.

Note: If two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angles will also be equal. (AA)

(iii) If in two triangles, sides of one triangle are in the same ratio of the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. (SSS)

For example: In triangles PQR and MNO,

If, then P = M, Q = N and R = O.

Therefore, PQR  MNO

(iv) 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. (SAS)

For example: In triangles PQR and MNO,

If and Q = N

Then, PQR  MNO

(v) If in two right triangles, hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other triangle, then the two triangles are similar. (RHS)

For example: In right angled triangles PQR and MNO,

If, then PQR  MNO

Theorem: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

For example: In triangles PQR and MNO,

If PQR  MNO,

Then,

Theorem: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Theorem (Pythagoras theorem): In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

For example: In PQR, Q is right angle.

Then by Pythagoras theorem,

PR² = PQ² + QR²

Theorem (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.