Similar Triangles

Similar Triangles are two triangles that are having congruent corresponding angles and the ratios of the corresponding sides are in proportion. This proportion is also called as similarity ratio. The similar triangles are also called as equiangular triangle. This is because in equilateral triangles, both the triangles have equal angles. The similar triangles have common shape but different sizes.

Similar Triangles

In the above figure, the two triangles are of same shape but different sizes. And so, the above two triangles are called as similar triangles.


Properties of Similar Triangles


Given below are some of the properties of similar triangles:

  • Corresponding Angles of both the triangles are equal.

       <P=<A, <Q=<B, <R=<C

  • The corresponding sides of the triangles are having same ratio.

       AB/PQ= BC/QR=AC/PR = X, where X is called as the similarity ratio


Rules of Similar Triangles


There are some rules to test the similarity of triangles. They are as follows:


  1. Angle Angle Angle (AAA)

  2. Side Side Side (SSS)

  3. Side Angle Side (SAS)

AAA Similarity: When three angles of the triangles are equal, we can say that the two triangles are similar triangles. That is, the corresponding angles are having equal measurement.

SSS Similarity: When three corresponding sides of the triangles are equal, we can say that the triangles are similar triangles.

SAS Similarity: When two sides in one triangle are in the same proportion to the corresponding sides of the other and the included angles are equal, we can say that both are similar triangle.


Solved Examples


 Example: The similar triangle ABC and PQR are shown below. Find the value of a and b in the triangle PQR.

Similar Triangles Problems


Solution: We know that the corresponding sides of the triangles are having same ratio,

That is, AB/PQ = BC/QR=AC/PR.

Consider, AB/PQ = BC/QR

8/6.4 = 7/a

Apply cross multiplication,

8a = 7(6.4)

8a = 44.8

a = 5.6

QR= a = 5.6

Consider BC/QR = AC/PR

7/5.6 = 6/b

7 b = 33.6

b = 33.6/7

b =  4.8

PR = b =4.8