Question

In triangles ABC and PQR, AB = AC, $\angle$C = $\angle$P and $\angle$B = $\angle$Q. The two triangles are ___.

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

Answer 1

The correct option is A

isosceles but not congruent

$\angle$A+ $\angle$B+ $\angle$C = $\angle$P+ $\angle$Q+ $\angle$R

But, $\angle$C = $\angle$P and $\angle$B = $\angle$Q ----(1)

$\angle$A+ $\angle$B+ $\angle$C = $\angle$C+ $\angle$B+ $\angle$R

Therefore, $\angle$A = $\angle$R

Hence $△$ABC ≈ $△$PQR (AAA similarity)

Also, as AB = AC, $△$ABC is an isosceles triangle.

So, $\angle$B = $\angle$C (opposite angles of equal sides)

But from (1), $\angle$P = $\angle$Q

Therefore, $△$PQR is isosceles.

Since the relation between sides of the 2 triangles is not known, congruency between the 2 triangles either by SAS or ASA cannot be proved.

Hence, $△$ABC and $△$PQR are similar and isosceles triangles.