Question

Solve for the value of x​

Answer 1

Step-by-step explanation:

First consider triangle ABE. Since ∠ABE = 80° and ∠BAE = 70°, we can deduce ∠BEA = 30°.

Construct DF parallel to AB. The proof will proceed in three steps.

1. Prove ∠DFE = 80°

2. Prove FD = FE

3. Solve x = 20°.

Step 1: Prove ∠DFC = 80°

This step is quite easy!

By parallel lines, we will have ∠ABC = ∠DFE = 80° and ∠BAC = FDC = 80°. We also have ∠C = 20°.

Step 2: Prove FD = FE

This step is quite hard.

Construct AF and label its intersection with BD as G.

Consider triangle BDF. We are given ∠DBF = 20°. Then ∠BFD = 180° – ∠DFC = 180° – 80° = 100°. Thus we must have ∠BDF = 60°.

Then triangle AFD is congruent to BDF, so ∠AFD = 60°. The triangle GDF has two angles equal to 60° so it is an equilateral triangle. Consequently ∠DGF = 60°, and by vertical angles ∠AGB = 60°. Then triangle AGB has two angles equal to 60°, so it is equilateral and ∠BAG = 60°. Since ∠EAD = 10° was given information, we then have ∠FAE = 20° – 10° = 10°.

Triangle AFC is isosceles with two angles equal to 20°, so we have AF = FC.

Now construct CG which will bisect angle C. Then triangle GAC will be congruent to ECA by angle-side-angle (20° – side AC – 10°)

Now we can prove FG = FE because:

FG = FA – GA

FG = FC – GA (since FA = FC)

FG = FC – EC (since GA = EC)

FG = FE

Now recall triangle FGD is equilateral, so FG = FD. Thus we can conclude FD = FE.