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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Question : In a $\triangle ABC$, if $\angle A=90^{\circ}, AC=5 \mathrm{~cm}, BC=9 \mathrm{~cm}$ and in $\triangle PQR, \angle P=90^{\circ}, PR=3 \mathrm{~cm}, QR=8$ $\mathrm{cm}$, then:
Option 1: $\triangle ABC \cong \triangle PQR$
Option 2: $ar(\triangle ABC)\neq ar(\triangle PQR)$
Option 3: $ar(\triangle ABC) \leq ar(\triangle PQR)$
Option 4: $ar(\triangle ABC)=ar(\triangle PQR)$
Question : In $\triangle$ABC and $\triangle$PQR, AB = PQ and $\angle$B = $\angle$Q. The two triangles are congruent by SAS criteria if:
Option 1: BC = QR
Option 2: AC = PR
Option 3: AC = QR
Option 4: BC = PQ
Question : If $\triangle \mathrm{ABC} \cong \triangle \mathrm{PQR}, \mathrm{BC}=6 \mathrm{~cm}$, and $\angle \mathrm{A}=75^{\circ}$, then which one of the following is true?
Option 1: $\mathrm{QR}=6$ cm, $\angle \mathrm{R}=75^{\circ}$
Option 2: $\mathrm{QR}=6$ cm, $\angle \mathrm{Q}=75^{\circ}$
Option 3: $\mathrm{QR}=6$ cm, $\angle \mathrm{P}=75^{\circ}$
Option 4: $\mathrm{PR}=6$ cm, $\angle \mathrm{P}=75^{\circ}$
Question : It is given that ABC $\cong$ PQR, AB = 5 cm, $\angle$B = $40^{\circ}$, and $\angle$A = $80^{\circ}$. Which of the following options is true?
Option 1: PQ = 5 cm and $\angle$R = $60^{\circ}$
Option 2: QR = 5 cm and $\angle$R = $60^{\circ}$
Option 3: QR = 5 cm and $\angle$Q = $60^{\circ}$
Option 4: PQ = 5 cm and $\angle$P = $60^{\circ}$
Question : Let ABC and PQR be two congruent triangles such that $\angle $A = $\angle $P = $90^{\circ}$. If BC = 13 cm, PR = 5 cm, find AB.
Option 1: 12 cm
Option 2: 8 cm
Option 3: 10 cm
Option 4: 5 cm
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