Question

in the figure, $overrightarrow{ca}$ and $overrightarrow{ce}$ are opposite rays, $overrightarrow{ch}$ bisects $angle gcd$, and $overrightarrow{gc}$ bisects $angle bgd$. which of the following are linear pairs along the line containing $overrightarrow{ae}$? select all that apply. $squareangle bcg$ and $angle gcf$ $squareangle abg$ and $angle gbc$ $squareangle edf$ and $angle fdc$ $squareangle dcf$ and $angle gcb$ $squareangle gfc$ and $angle dfc$ $squareangle bcg$ and $angle gba$

Explanation:

Step1: Recall linear - pair definition

A linear pair of angles is a pair of adjacent angles whose non - common sides are opposite rays and whose sum is 180 degrees.

Step2: Analyze each option

• For ∠BCG and ∠GCF: They are adjacent but their non - common sides are not opposite rays along the line containing $\overrightarrow{AE}$, so they are not a linear pair.
• For ∠ABG and ∠GBC: Their non - common sides $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are opposite rays along the line containing $\overrightarrow{AE}$, and they are adjacent, so they are a linear pair.
• For ∠EDF and ∠FDC: Their non - common sides $\overrightarrow{DE}$ and $\overrightarrow{DC}$ are opposite rays along the line containing $\overrightarrow{AE}$, and they are adjacent, so they are a linear pair.
• For ∠DCF and ∠GCB: They are not adjacent, so they are not a linear pair.
• For ∠GFC and ∠DFC: Their non - common sides are opposite rays along the line containing $\overrightarrow{AE}$, and they are adjacent, so they are a linear pair.
• For ∠BCG and ∠GBA: They are not adjacent, so they are not a linear pair.

Answer:

∠ABG and ∠GBC, ∠EDF and ∠FDC, ∠GFC and ∠DFC