Question : $\triangle ABC$ and $\triangle PQR$ are two triangles. AB = PQ = 6 cm, BC = QR =10 cm, and AC = PR = 8 cm. If $\angle ABC = x$, then what is the value of $\angle PRQ$?
Option 1: $(180 ^{\circ}–x)$
Option 2: $x$
Option 3: $(90 ^{\circ}–x)$
Option 4: $(90 ^{\circ}+x)$
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Correct Answer: $(90 ^{\circ}–x)$
Solution :
Given: $\triangle ABC$ and $\triangle PQR$ are two triangles. AB = PQ = 6 cm, BC = QR =10 cm and AC = PR = 8 cm. Also, $\angle ABC = x$.
We know that the sum of all interior angles of the triangle is $180^{\circ}$.
By the SSS (side– side– side) rule, two triangles are congruent if all three of their sides are equal to the corresponding three of the second triangle.
In $\triangle ABC$ and $\triangle PQR$,
AB = PQ = 6 cm (given)
BC = QR =10 cm (given)
AC = PR = 8 cm (given)
So, $\triangle ABC\cong \triangle PQR$ (by SSS rule)
⇒ $\angle PRQ=\angle ACB$ (congruent parts of the congruent triangle are equal)
Now, $\angle ABC = x$
⇒ $\angle ACB= (90^{\circ}–x)=\angle PRQ$
Hence, the correct answer is $(90^{\circ}–x)$.
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