Question

In $∆ABC\text{and}∆PQR$, it is given that $AB=AC,\angle C=\angle P\text{and}\angle B=\angle Q$. Then, the two triangles are:

• A. isosceles but not congruent
• B. neither congruent nor isosceles
• C. isosceles and congruent
• D. congruent but not isosceles

Answer 1

The correct option is A

isosceles but not congruent

Explanation for the correct option

Option A: isosceles but not congruent

Considering $∆ABC\text{and}∆PQR$,

$$\begin{gathered} \angle C=\angle P \\ \angle B=\angle Q\left[Given\right] \\ \angle A=\angle R[Thirdangle] \end{gathered}$$

So, $∆\mathrm{ABC}~∆\mathrm{PQR}$. Then,

$$\begin{gathered} AB=AC\left[Given\right] \\ \Rightarrow \angle B=\angle C[Anglesoppositetoequalsides] \end{gathered}$$

$$\begin{gathered} As\angle \mathrm{B}=\angle \mathrm{Q}\text{and}\angle \mathrm{C}=\angle \mathrm{P} \\ \Rightarrow \angle \mathrm{Q}=\angle \mathrm{P} \\ \Rightarrow \mathrm{PR}=\mathrm{QR}[\mathrm{Sides}\mathrm{opposite}\mathrm{to}\mathrm{equal}\mathrm{angles}] \end{gathered}$$

Therefore, the triangles are isosceles but cannot be congruent.

Step 2: Explanation for the incorrect options

Option B: neither congruent nor isosceles

Since the triangles are isosceles as proved above, so this option is incorrect.

Option C: isosceles and congruent

Since the triangles are isosceles but not congruent as proved above, so this option is incorrect.

Option D: congruent but not isosceles

Since the triangles are not congruent but are isosceles as proved above, so this option is incorrect.

Hence, the correct option is A.