geometry

Step 1

​Let's draw the two crossing roads in the form of two lines intersecting with each other.

Step 2

​Now let's name the lines as AB and PQ. The lines intersect at the point O.

Step 3

​Now, we can say that
∠1 = ∠AOP,
∠2 =∠AOQ ,
∠3 = ∠BOQ,
∠4 =∠BOP ,

Step 4

​Here, ∠AOP and ∠AOQ are adjacent angles because they are next to each other and share a common arm and vertex.
They are also supplementary, because together they form a straight angle. 
∠AOP + ∠AOQ = 180°
Also,
∠1 + ∠2 = 180°
 

Step 5

​In the figure, ∠1 = x⁰ and ∠2 = (x - 60)⁰
Now by putting the values in step 4, we can write that:

∠1 + ∠2 = 180°
x⁰ + (x - 60)⁰ = 180⁰
x⁰ + x⁰ - 60⁰ = 180⁰
Now, lets eliminate 60⁰ from left side by adding 60⁰ on both sides
2x⁰ - 60⁰ = 180⁰ + 60⁰
2x⁰ = 240⁰
x⁰ = 120⁰
therefore, ∠1 = 120⁰

and (x - 60)⁰ = 120⁰ - 60⁰ = 60⁰
therefore, ∠2 = 60⁰

Step 6

​Vertically opposite angles always have the same measure.
Here ∠AOP and ∠BOQ are vertically opposite angles.
Also, ∠AOQ and ∠BOP are vertically opposite angles.
Therefore,
∠AOP = ∠BOQ i.e. ∠1 = ∠3
∠AOQ = ∠BOP i.e. ∠2 = ∠4


Now,
by putting the values of ∠1 and ∠2 from step 5,
we get the measure of :
∠3 = 120⁰
∠4 = 60⁰