Question
In two triangles, two angles  and one side  of the first triangle are equal to the two angles  and one side of the second triangle. Will these two triangles always be congruent?[3 MARKS]
Answer:
:
Proof: 1 Mark
Steps: 2Marks
Consider two triangles, ΔABC andΔPQR in which,
∠ABC=∠PQR
∠ACB=∠PRQ
AB=PQ
We know that,
∠ABC+∠ACB+∠BAC=1800
∠BAC=1800−(∠ABC+∠ACB)....(i)
Similarly,
∠QPR=1800−(∠PQR+∠PRQ)......(ii)
From (i) and (ii),
∠BAC=∠QPR
Now, In ΔABC andΔPQR
∠ABC=∠PQR [Given]
∠BAC=∠QPR [ Proved above]
AB=PQ[Given]
⇒ΔABC≅ΔPQR [ ASA congruency rule]
⇒ These triangles are always congruent.
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:
Proof: 1 Mark
Steps: 2Marks
Consider two triangles, ΔABC andΔPQR in which,
∠ABC=∠PQR
∠ACB=∠PRQ
AB=PQ
We know that,
∠ABC+∠ACB+∠BAC=1800
∠BAC=1800−(∠ABC+∠ACB)....(i)
Similarly,
∠QPR=1800−(∠PQR+∠PRQ)......(ii)
From (i) and (ii),
∠BAC=∠QPR
Now, In ΔABC andΔPQR
∠ABC=∠PQR [Given]
∠BAC=∠QPR [ Proved above]
AB=PQ[Given]
⇒ΔABC≅ΔPQR [ ASA congruency rule]
⇒ These triangles are always congruent.
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