Revision: Geometry >> Triangles Maths English Medium Class 10 CBSE

Two figures are similar if they have the same shape but may differ in size.

• Same shape means: Corresponding angles are equal

• May differ in size means: Corresponding sides are proportional

Two figures are congruent if they have the same shape and size.

Important relation:

• Congruent figures are always similar
• Similar figures are not necessarily congruent

In similar triangles, the angles opposite to proportional sides are the corresponding angles, and so, they are equal. 

• ∠A = ∠P

• ∠B = ∠Q

• ∠C = ∠R

In similar triangles, the sides opposite to equal angles are said to be the 
corresponding sides. 

ΔABC ∼ ΔPQR

\[\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}\]

Two triangles are similar if

• Their corresponding angles are equal, and
• Their corresponding sides are proportional.
• Symbolically:
  ΔABC ∼ ΔPQR (read as “ABC is similar to PQR”).

Statement:
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

To Prove:
$DE\parallel BC$

$Proof:$

1. Assume a line through point D parallel to BC meets AC at F.

2. By BPT, \[\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{\mathrm{AF}}{\mathrm{FC}}\]

3. Given, \[\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{\mathrm{AE}}{\mathrm{EC}}\]

4. Hence,\[\frac{AF}{FC}=\frac{AE}{EC}\]

5. ⇒ Points E and F coincide.

   Therefore,

   $DE\parallel BC$

Statement:
If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.

To Prove:
\[\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{\mathrm{AE}}{\mathrm{EC}}\]

Proof:

1. A line parallel to a side of a triangle forms equal corresponding angles.

2. Hence, the two triangles formed are similar (AAA similarity).

3. In similar triangles, corresponding sides are proportional.

Therefore, the line divides the two sides in the same ratio.

\[\frac{\mathrm{AD}}{\mathrm{DB}}=\frac{\mathrm{AE}}{\mathrm{EC}}\]

• AA / AAA → two angles equal

• SAS → included angle equal + sides proportional

• SSS → all sides proportional