Confused about the angles: Help please

[mild river]

you can calculate v and x using law of cosines or law of sines

though you have to do it for each triangle

also notice that there's a > thing that's centered on the line

if two lines have that same thing they are considered parallel

[safe canopy]

So the angles you need to focus on will be the verticle angles at the center and angles formed by parallel lines cut by a transversal (specifically alternate interior angles)
From there, you just need to remember that similar triangles are proportional

[woeful hare]

If so, the diagram contains conflicting data.

[faint sun]

It is recommended to represent the points in the diagram using letters
|AC| || |BE|...Diagram
So, we use the properties of angles in parallel lines
@Pai mentioned them (though I wouldn't include "...at the center ... because the question did not state it")

Here are the explanations: Number (12.) on https://examssuccess.appspot.com/MATHEMATICS/Trigonometry.html#one-twenty

[woeful hare]

The answer says v=4, but it definitely contradicts the diagram. Because a triangle with sides 4, 5, 6 has no 35° angles.

[faint sun]

May you please tell me the theorem that states that a triangle with 35°, 65°, 80° cannot have 4m, 5m, 6m as the sides?

[woeful hare]

Yep, just apply the law of cosines, say find a cosine of the angle opposite to the side 6m. It equals 1/8. Tho cos(80°) is definitely not 1/8.

[faint sun]

May you show me your work on how you got 1/8? Please show all work.

[woeful hare]

While approximation to cos(80°) goes 0.173...

[faint sun]

I see your point.
Thank you for the observation.
I have removed the solution.

Cosine Law should work for all triangles.

[safe canopy]

I'm not sure why it would provide conflicting data. Maybe the angles arent the exact measures for the the given sides, but when proving similar triangles the most you'd need confirming 2 congruent angles. From there the triangles are proportional
Using laws of sine and cosine are fine, but its more exact this way in the sense you dont need to round decimals

[safe canopy]

Oh i see now
I think then the diagram is just incorrect