AAS(or SAA) test:

State whether the given statement is True or False.

Statement: The sum of the measures of angles in a triangle is $180^{0}$ . Therefore, if two corresponding pairs of angles in two triangles are congruent, then the remaining pair of angles is also congruent. Hence if two angles and a side adjacent to one of them are congruent with corresponding parts of the other triangle then the condition for ASA test is fulfilled. So, the triangles are congruent.

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True

False

answer is A.

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Detailed Solution

Let us take the two triangles ΔABC, ΔDEF as shown in below figure
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⇒ From the above triangles
⇒ Length of BC = a
⇒ Length of AC = b
⇒ Length of AB = c
⇒ Length of EF = d
⇒ Length of DF = e
⇒ Length of DE = f
⇒ The angle ∠BAC = α
⇒ The angle ∠ABC = β
⇒ The angle ∠ACB = γ
⇒ The angle ∠EDF = δ
⇒ The angle ∠DEF = ε
⇒ The angle ∠DFE = ζ
Given that, the sum of all the measures of the angles in a triangle is

180^{0}

, then
α + β + γ =

180^{0}

and δ + ε + ζ =

180^{0}

Now the sum of all the angles in the two triangles is
α + β + γ + δ + ε + ζ =

180^{0}

180^{0}

360^{0}

If α = δ = x and β = ε = y then the values of γ , ζ are
γ =

180^{0}

− (α + β) and ζ =

180^{0}

− (δ + ε)
Substituting the values of α,β,δ,ε in the above equations then we have
γ =

180^{0}

− (x + y) and
ζ =

180^{0}

− (x + y)
= γ
Now let us compare the two triangles
We have
α = δ β = ε
Then automatically γ = ζ
⇒ And AB = pDE [Where p is the constant]
⇒ We know that in two triangles the two angles and one corresponding side is equal/congruent to the corresponding angles and side in another triangle then the two triangles are congruent to each other on the Side Angle Angle rule.
⇒ So, the correct option is (1)
Hence, the given statement is true.

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