Similar Triangles and Similarity of Triangles

If two or more objects have the same shape but their sizes are different then such objects are called Similar. Consider a hula hoop and wheel of a cycle, the shapes of both these objects is similar to each other as their shapes are same. All equilateral triangles, squares of any side length are examples of similar objects.

We denote similarity of objects by ‘~‘ symbol.

In the figure given above, two circles \(C_1 \) and \( C_2 \) with radius R and r respectively are similar as they have the same shape, but necessarily not the same size. Thus, we can say that \( C_1\)~ \(C_2\).

It is to be noted that, two circles always have same shape, irrespective of their diameter. Thus, two circles are always Similar.

Triangle is the smallest three-sided polygon. For two triangles to be similar following two conditions must be satisfied: Two triangles are similar, if

i)        Corresponding angles of both the triangles are equal, and
ii)       Corresponding sides of both the triangles are in proportion to each other.

In the given figure, two triangles ΔABC and ΔXYZ are similar only if,

i)        ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
ii)       \( \frac {AB}{XY} \) = \( \frac {BC}{YZ} \) = \( \frac {AC}{XZ} \)

Hence if the above mentioned conditions are satisfied we can say that ΔABC ~ ΔXYZ

It is interesting to know that if the corresponding angles of two triangles are equal then such triangles are known as equiangular triangles.

For two equiangular triangles we state the Basic Proportionality Theorem (better known as Thales Theorem) as follows:

For two equiangular triangles the ratio of any two corresponding sides is always the same.

Similar triangles: criteria and conditions

i) AA (or AAA) criterion of similarity:

If any two angles of a triangle are equal to any two angles of another triangle then the two triangles are similar to each other.

From the figure given above, if \(\angle A = \angle X\) and \(\angle C = \angle Z\) then ΔABC ~ΔXYZ.

From the result obtained, we can easily say that,

\(\large \frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}\)

and \(\angle B = \angle Y\)

ii) SAS criterion of similarity:

If the two sides of a triangle is in the same proportion of the two angles of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangle are said to be similar.

Thus, if ∠A = ∠X and \( \frac{ AB}{XY} \) = \( \frac {AC}{DF} \) then ΔABC ~ΔXYZ.

From the congrunecy, we can state that,

\(\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}\)

and \(\angle B = \angle Y\) and \(\angle C = \angle Z\)

iii) SSS criterion for similarity:

If all the three sides of a triangle are in proportion to the three sides of another triangle then the two triangles are similar.

Thus, if \( \frac {AB}{XY} \) = \( \frac {BC}{YZ} \) = \( \frac {AC}{XZ} \) then ΔABC ~ΔXYZ.

From this result, we can infer that-

\(\angle A = \angle X\), \(\angle B = \angle Y\) and \(\angle C = \angle Z\)

Let us go through an example to understand it better.

Example: In theΔABC length of the sides are given as AP = 5 cm , PB = 10cm and BC = 20 cm.Also PQ||BC. Find PQ.

Solution: In ΔABC and ΔAPQ, ∠PAQ is common and ∠APQ = ∠ABC (corresponding angles)

⇒ ΔABC ~ ΔAPQ (AA criterion for similar triangles)

⇒ \( \frac {AP}{AB} \) =  \( \frac {PQ}{BC} \)

⇒ \( \frac {5}{15} \) =  \( \frac {PQ}{20} \)

⇒ PQ = \( \frac {20}{3} \)  cm