Using the vertically opposite angles theorem we have proven that the sum of the labeled angles is 180°
Proof of angles
From the question, we are to prove that the sum of the labeled angles is 180°
Consider the horizontal line in the diagram
In the lower part of the line, two of the angles are labeled.
x is vertically opposite to the unlabeled angle.
By the vertically opposite angles theorem, the value of the unlabeled side is x
Also,
By the theorem for the sum of angles on a line segment, we can write that
z + x + y = 180°
This is because the sum of angles on a line segment is 180°'
Hence, we have proven that the sum of the labeled angles is 180°
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