Similar Triangles

Similar Triangles

Similar Triangles

Two triangles are said to be similar if they are exactly same by their shape. Two triangles of the same shape is mean three angles of one triangle are correspondingly equal to the three angles of the other triangle. The symbol for similarity is ~.
ΔABC ~ ΔPQR
For example: In the given triangles ABC and PQR, A = P, B = Q and C = R. So the triangles ABC and PQR are similar. It is written as ΔABC ~ ΔPQR.

When two triangles are similar then their corresponding sides are in proportion i.e
Corresponding sides are in proportion.                                                               

Conditions for Similarity

Condition - 1. Angle-Angle-Angle (AAA) axiom

When three angles of any triangle are equal to the corresponding three angles of another triangle, then they will be similar by AAA axiom.
Angle-Angle-Angle (AAA) axiom
In the given triangles ABC and PQR, A = P (A), B = Q (A), C = R (A). So, ΔABC ~ ΔPQR by AAA axiom. Corresponding sides of similar triangles are proportional, i.e.
Corresponding sides of similar triangles are proportional.

Condition - 2.

When three sides of a triangle are proportional to the corresponding three sides of another triangle, then they are similar.
Triangles ABC and PQR
In the given triangles ABC and PQR,    
Corresponding sides are in proportion.

So, ΔABC ~ ΔPQR. Corresponding angles of similar triangles equal, i.e. A=P, B=Q and C=R.

Workout Examples

Example 1: In ΔABC, XY//BC. Prove that:
Example 1: ΔABC

Solution: Here,
                      In ΔAXY and ΔABC,
    i.       AXY = ABC (A) -----------> corresponding angle
    ii.     AYX = ACB (A) -----------> corresponding angles
    iii.    XAY = BAC (A) ------------> common angle
  ΔAXY ~ ΔABC ----------------> by AAA axiom
Corresponding sides of similar triangles.
                                                                                 Proved.


Example 2: In the adjoining figure DE//BC. Find the values of x and y.
Example 2: ΔABC

Solution: Here,
                      In ΔADE and ΔABC,
    i.        DAE = BAC (A) -----------> common angle
    ii.      DEA = ACB (A) -----------> corresponding angles
    iii.    ADE = ABC (A) ------------> remaining angles
  ΔADE ~ ΔABC ----------------> by AAA axiom
Ratios corresponding sides of similar triangles.
or,  3x = x + 18
or,  3x – x = 18
or,  2x = 18
or,  x = 18/2
or, x = 9
Again, taking second and third ratios,
or,  y = 18
  x = 9cm and y = 18cm


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