Question

Using the definition of similarity prove that all the isosceles right angled triangles are similar.

Answer 1

To prove : all the isosceles right angled triangles are similar.

proof :- see figures, here given two isosceles right angled triangles .e.g., ∆XYZ and ∆ABC.

In ∆ABC, AB = BC and m∠B = 90
In ∆XYZ, XY = YZ and m∠Y = 90
i.e. ∆ABC and ∆XYZ are isosceles right angled triangles.
∆ABC and ∆XYZ are both isosceles right angled triangles in which m∠B = m∠Y = 90
Also AB = BC and XY = YZ, then
m∠A = m∠C = 45° and also m∠X = m∠Z = 45°
[ because AB = BC so, m∠A = m∠C but given m∠B = 90° so, m∠A = m∠C = 45°]

hence, m∠A = m∠X = 45°
m∠C = m∠Z = 45°
and m∠B = m∠Y = 90°
hence, from A - A - A rule of similarity
∆XYZ ~ ∆ABC .
hence proved