Question

In ΔABC, AC = BC. ∠BAC is bisected by AD and AD = AB. Find ∠ACB.

Answer 1

Given:

△ABC is isosceles with AC = BC.

∠BAC is bisected by AD, meaning ∠BAD = ∠DAC.

AD = AB.

We need to find ∠ACB.

Step 1: Label the angles and sides

Since △ABC is isosceles with AC = BC, we know:

∠ABC = ∠ACB   ...(Because in an isosceles triangle, base angles are equal)

Let ∠ABC = ∠ACB = x.

Step 2: Use the Angle Bisector Property

Since AD bisects ∠BAC, we have:

∠BAD = ∠DAC

Let ∠BAD = ∠DAC = y.

Thus, the total angle ∠BAC = 2y.

Step 3: Use the triangle angle sum property

The sum of angles in any triangle is 180°.

In △ABC, we have:

∠BAC + ∠ABC + ∠ACB = 180°

Substitute the known angles:

2y + x + x = 180°

2y + 2x = 180°

y + x = 90°  ...(Equation 1)

Step 4: Use the fact that AD = AB

Since AD = AB, ΔABD is isosceles,

So, ∠ABD = ∠ADB = z

Also, since ∠ADB + ∠BAD = 180° sum of angles on a straight line:

z + y = 180°

Thus, z = 180° – y.

Step 5: Relating x and y

In △ABC, the sum of the angles is 180°:

∠ABC + ∠ACB + ∠BAC = 180°

Since ∠ABC = ∠ACB = x and ∠BAC = 2y, we have:

2x + 2y = 180°

x + y = 90°

Thus, x = 36°. 

So, ∠ACB = 36°.