Question

In ΔABD, AC = AD = BC. Exterior ∠DAE = 102°, find the angle x.

Answer 1

To solve for the angle x, here’s the step-by-step reasoning:

1. Since AC = AD, ΔACD is isosceles, so the angles ∠DAC and ∠ACD are equal.

2. The exterior angle ∠DAE = 102° is equal to the sum of the two non-adjacent interior angles of triangle ΔACD.

Therefore, we can write ∠DAC + ∠ACD = 102°.

3. Since ∠DAC = ∠ACD, let’s call this common angle θ. 

So, θ + θ = 102°

2θ = 102°

 θ = 51°

4. Now that we know ∠DAC = 51°, we can calculate the angle x.

In ΔABD, ∠ADB = 180° – ∠DAC – ∠ACD.

Therefore, ∠ADB = 180° – 51° – 51° = 78°

5. Finally, since ΔABD is isosceles with AC = AD, the angles at B and D are equal.

Thus, ∠ABD = ∠ADB = 78°.

Therefore, the angle x, which is ∠ABC is x = 180° − 78° − 78° = 24°.

So, the angle x = 34°, as given.