Question

ABCD and LMNO are squares. Angle CBL = x. Prove that triangles ADN and CBL are congruent.

Answer 1

Given:

ABCD and LMNO are squares

Angle CBL = x

To find:

Prove that triangles ADN and CBL are congruent.

Solution:

According to the properties of a square

→ All four angles of a square are right angles

i.e., ∠A = ∠B = ∠C = ∠D = ∠L = ∠M = ∠N = ∠O = 90°

In BLC, from angle sum property of a triangle, we get

∠BCL = 180° - x° - 90° = (90 - x)° ..... (i)

Also, ∠ABO = (90 - x)°

In Δ ABO, from angle sum property of a triangle, we get

∠BAO = 180° - (90 - x)° - 90° = 180° - 90° + x° - 90° = x°

∴ ∠DAN = 90° - ∠BAO = (90 - x)° ...... (ii)

In Δ ADN and Δ CBL, we have

∠BLC = ∠AND = 90° ..... [Linear Pairs of angle ∠L & ∠N]

∠BCL = ∠DAN ..... [from (i) & (ii)]

AD = BC .... [sides of a square ABCD]

∴ Δ ADN ≅ Δ CBL ...... [By AAS congruency]

Hence proved

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