AA similarity Conditions under which two Triangles are Similar

Math Adda    Conditions under which two Triangles are Similar



Two triangles are similar when their corresponding angles are equal.

Let us prove these conditions as a theorem with the help of Thales Theorem.

Statement: Two triangles are similar if their corresponding angles are equal.

Proof: we are given D ABC and D PQR such that

< A = <P 
<B = <Q
<C = <R

We have to prove: DABC ~ DPQR

To prove this: On the sides PQ and PR of DPQR take points M and N such that 

AB = PM and AC = PN. Join MN.

Now, in DABC and DPMN

AB = PM [We have constructed] 
<A = <P [Given] 
AC = PN [We have constructed] 

Therefore DABC ~ DPMN [SAS congruency] 

This means
<PMN = <B [c. p. c. t]
But <B = <Q [Given] 

Therefore, 
<PMN = <Q
These are corresponding angles.

Thus, MN || QR

Now, 

PM / MQ = NR / PN {Thales theorom}

i.e. MQ / PM = NR / PN

i.e. MQ / PM + 1 = NR / PN +1

i.e. MQ + PM / PM = NR + PN/ PN

i.e. PQ / PM = PN / PR

i.e. PM / PQ = PR / PN

Since PM = AB and PN = AC

Therefore

AB / PQ = AC / PR

Similarly we can prove

AB / PQ = BC / QR

Hence AB / PQ = BC / QR = AC / PR, we already have been given that <A = <P, <B = <Q and <C = <F.

This way both conditions of similarity are fulfilled.

Thus, DABC ~ DPQR when their corresponding angles are equal.

We denote this condition as 

Angle - Angle - Angle similarity or AAA similarity.

Note: When two angles of a triangle are equal to two corresponding angles of another triangle then their remaining angles are also equal to each other. This AAA similarity is also written as AA similarity