Question

In ∆ABC and ∆PQR, it is given that AB = AC, ∠C= ∠P and ∠B= ∠Q. Then 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

Question:-

In ∆ABC and ∆PQR, it is given that AB = AC, ∠C= ∠P and ∠B= ∠Q. Then the two triangles are

a) Isosceles but not congruent

(b) Isosceles and congruent

(c) Congruent but not isosceles

(d) Neither congruent nor isosceles

Solution:-

Given that,

AB = AC, ∠C= ∠P and ∠B= ∠Q

We know that, If two sides of triangle are equal than there opposite angles are also equal.

So,

∠B = ∠C ---(i)

And

∠B = ∠Q

∠C = ∠P

∠P = ∠Q [ By equation (i) ]

Side opposite to ∠P and side opposite to ∠Q will be equal beacuse if two opposite angles of triangle equal then there opposite sides are also equal.

So,

PR = QR

And

PR = AC

AB = QR ----(ii)

Now, According to question:

In ∆ABC and ∆PQR

AB = QR [By equation (ii)]

∠B = ∠Q ( Given)

∠C = ∠P (Given)

By AAS congruency :

∆ABC ≌ ∆PQR

Therefore

Both triangles are Isosceles and congruent.