Find the sum of two angles created by equilateral triangles on a line.

Question

Point A, C, E are on line G, in that order. Point B and D are on the same side of line G so that ABC and CDE are equilateral triangles. Find the sum the of <BAD and <BED.

I tried to solve this problem by making <DAE = x. However, this gets me nowhere because I have to introduce a second variable, y, no matter what I do. Even then, I can not find all the angles in terms of x and y.

Answer 1

It helps to draw a diagram.

Observe that $\triangle ACD \cong \triangle BCE$ (by SAS) and $AB //CD$.

This shows that $\angle BEC = \angle ADC = \angle BAD$, which is sufficient to solve your problem.

Answer 2

$$AC=BC$$

$$\angle ACD=\angle BCE=120$$

$$CD=CB$$

by S.A.S congruence theorem

$$\triangle ACD\cong \triangle BCE$$

$$\angle DAC=60-\angle BAD=\angle CBE=\angle BED$$

$$\angle BED+\angle BAD=60-\angle BAD+\angle BAD=60$$