Question

the diagram shows $overleftrightarrow{ab}$, $overleftrightarrow{cd}$, and $overleftrightarrow{ge}$. which statement can be proven true from the diagram? a. $angle dgb$ is complementary to $angle ega$. b. $angle dgb$ is supplementary to $angle cgb$. c. $angle cgb$ is complementary to $angle agd$. d. $angle ega$ is complementary to $angle egb$. e. $angle egd$ is supplementary to $angle cgb$

Explanation:

Step1: Recall angle - relationship definitions

Complementary angles add up to 90°, and supplementary angles add up to 180°.

Step2: Analyze linear - pair angles

\(\overrightarrow{AB}\) is a straight - line. \(\angle DGB\) and \(\angle AGB\) form a linear pair, and \(\angle AGB = 180^{\circ}\). \(\angle DGB\) and \(\angle CGB\) are adjacent angles that form a linear pair on the straight - line \(\overrightarrow{CD}\) passing through point \(G\). So \(\angle DGB+\angle CGB = 180^{\circ}\), which means \(\angle DGB\) is supplementary to \(\angle CGB\).

Step3: Analyze other angle pairs

• Option A: There is no indication that \(\angle DGB\) and \(\angle EGA\) add up to 90°.
• Option C: \(\angle CGB\) and \(\angle AGD\) are vertical angles, not complementary.
• Option D: \(\angle EGA\) and \(\angle EGB\) form a linear pair, so they are supplementary, not complementary.
• Option E: There is no relationship that makes \(\angle EGD\) and \(\angle CGB\) supplementary.

Answer:

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