##
**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 ~.

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

###
**Conditions for Similarity**

####
__Condition - 1. Angle-Angle-Angle (AAA)
axiom__

__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.

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.

####
__Condition - 2.__

__Condition - 2.__

When three sides of a triangle are proportional to the
corresponding three sides of another triangle, then they are similar.

In the given triangles ABC and PQR,

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

###
*Workout Examples*

*Workout Examples*

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

*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*

*Proved.*

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

*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*

*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*

*You can comment your questions or problems regarding similar triangles here.*

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