Which of the following inequalities holds true with respect to the given figure?

Applying the angle sum property of triangles in ΔBDC:

∠CBD = 180° – (90° + 60°) = 30°

Applying the angle sum property of triangles in ΔAEC:

∠EAC = 180° – (90° + 60°) = 30°

Also, ∠A = 70°

∴ ∠BAO = ÐA – ÐEAC = 70° – 30° = 40°

Applying the angle sum property of triangles in ΔABD:

∠ABD = 180° – (90° + 70°) = 20°

Applying the angle sum property of triangles in ΔOBE:

∠BOE = 180° – (90° + 30°) = 60°

We know that in a triangle, the side opposite to the larger angle is longer.

Now let us check each option one by one.

Option:

Consider ΔBOE

Here, ∠EOB = 60° and ∠EBO = 30°

⇒ EO < EB

Option:

Consider ΔOAB

∠OAB = 40° and ∠OBA = 20°

⇒ OB > OA

Option :

Consider ΔODA

Here, ∠ODA = 90° and ∠OAD = 30°

⇒ OA > OD

Option :

Consider ΔABC

Here, ∠ABC = 50° and ∠ACB = 60°

⇒ AC < AB

Hence, the relation OB > OA is the correct alternative.