Question

How many different (non-congruent) isosceles triangles can be drawn with one angle 80° and one side 8 centimetres.

Answer 1

Four different non-congruent isosceles triangles can be drawn with one angle 80° and one side 8 cm.

These triangles are as follows:

I . When AB = AC = 8 cm and ∠B = ∠C = 80°

In ΔABC, using angle sum property, we have:

∠A + ∠B + ∠C = 180°

⇒ ∠A + 80° + 80° = 180°

⇒ ∠A + 160° = 180°

⇒ ∠A = 180° − 160°

⇒ ∠A = 20°

II. When BC = 8 cm and ∠A = 80°

As ΔABC is an isosceles triangle, ∠B = ∠C.

In ΔABC, using angle sum property, we have:

∠A + ∠B + ∠C = 180°

⇒ 80° + 2∠B = 180°

⇒ 2∠B = 180° − 80°

⇒ 2∠B = 100°

⇒ ∠B = 50°

⇒ ∠C = 50°

III. When AB = AC = 8 cm and ∠A = 80°

As ΔABC is an isosceles triangle with AB = AC, ∠C = ∠B.

In ΔABC, using angle sum property, we have:

∠A + ∠B + ∠C = 180°

⇒ 80° + 2∠B = 180°

⇒ 2∠B = 180° − 80°

⇒ 2∠B = 100°

⇒ ∠B = 50°

⇒ ∠C = 50°

IV. When ∠B = ∠C = 80° and BC = 8 cm

As ΔABC is an isosceles triangle with ∠C = ∠B, AB = AC

In ΔABC, using angle sum property, we have:

∠A + ∠B + ∠C = 180°

⇒ ∠A + 2∠B = 180°

⇒ ∠A + 2 × 80° = 180°

⇒ ∠A + 160° = 180°

⇒ ∠A = 180° − 160°

⇒ ∠A = 20°

All the triangles shown above are non-congruent isosceles triangles.