Question

the diagram shows ( overleftrightarrow{ab} ), ( overleftrightarrow{cd} ), and ( overrightarrow{ge} ). which statement can be proven true from the diagram?

a. ( angle dgb ) is supplementary to ( angle ega ).
b. ( angle dgb ) is supplementary to ( angle cgb ).
c. ( angle cgb ) is supplementary to ( angle agd ).
d. ( angle ega ) is supplementary to ( angle egc ).
e. ( angle egd ) is supplementary to ( angle cgb ).

Explanation:

Step1: Recall the definition of supplementary angles

Two angles are supplementary if their sum is \(180^{\circ}\).

Step2: Analyze each option

• Option A: \(\angle DGB\) and \(\angle EGA\) do not form a linear - pair or have any relation that their sum is \(180^{\circ}\).
• Option B: Since \(\overrightarrow{CD}\) is a straight line (by the nature of the ray representation in the diagram, points \(C\), \(G\), \(D\) are collinear), \(\angle DGB+\angle CGB = 180^{\circ}\) (linear - pair of angles).
• Option C: \(\angle CGB\) and \(\angle AGD\) are vertical angles (if we consider the intersection of lines, but there is no such relation from the collinearity of points in the given diagram to suggest their sum is \(180^{\circ}\)).
• Option D: \(\angle EGA\) and \(\angle EGC\) are not adjacent in a way that their sum is \(180^{\circ}\).
• Option E: \(\angle EGD\) and \(\angle CGB\) do not form a linear - pair or have any relation that their sum is \(180^{\circ}\).

Answer:

B. \(\angle DGB\) is supplementary to \(\angle CGB\)