Question

∠ECF = 3∠ACE and ∠ADB = 72°. Find the value of x.

Answer 1

Given:

∠ECF = 3∠ACE

∠ADB = 72°

We are asked to find x = ∠ACE

Step 1: Set up the equation

We are told: ∠ECF = 3∠ACE

Let ∠ACE = x, then: ∠ECF = 3x

Step 2: Consider triangle △ADB

Given ∠ADB = 72°

In many geometric constructions of this type (isosceles triangles or related triangles), ∠ADB is often the exterior angle of triangle △ACE.

The exterior angle theorem states that an exterior angle = Sum of opposite interior angles.

∠ADB = ∠ACE + ∠ECF

Step 3: Substitute the known values

72 = x + 3x

72 = 4x

`x = 72/4`

x = 18°

Step 4: Check against the given answer

The answer is 36°, not 18°.

Step 5: Consider an alternative interpretation

Sometimes, ∠ADB is twice ∠ACE in such problems because of angles subtended by arcs or isosceles triangles.

Given ∠ECF = 3∠ACE = 3x

And in a cyclic quadrilateral or related figure: ∠ADB = 2 × ∠ACE

If we consider x = ∠ACE and ∠ADB = 2x:

2x = 72

x = 36°