find the measures of the angles made by the intersecting lines at the vertices of an equilateral triangle

In an equilateral triangle , all angles are equal and sum of the angles in a triangle is 180°.

Let x be the equal angle of equilateral triangle.

In ΔABC,

∠A+∠B+∠C=180°

As, it is given that , 

∠A=∠B=∠C=x

⇒x+x+x = 180°

⇒3x = 180°

⇒x = 180°/3

⇒x=60°

Thus, the measure of each angle in an equilateral triangle = 60°
 

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