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 involving point $f$? select all that apply. a) $angle dfc$ and $angle cfg$ b) $angle dfh$ and $angle gfc$ c) $angle gfh$ and $angle gcf$ d) $angle hfd$ and $angle cfd$ e) $angle gfh$ and $angle gfc$ f) $angle hfg$ and $angle dfh$

Explanation:

Step1: Recall linear - pair definition

Two adjacent angles are a linear pair if they are supplementary (sum to 180°) and share a common side and a common vertex, and their non - common sides form a straight line.

Step2: Analyze option A

$\angle DFC$ and $\angle CFG$ share a common side $\overrightarrow{CF}$, a common vertex $F$, and their non - common sides $\overrightarrow{FD}$ and $\overrightarrow{FG}$ form a straight line. So they are a linear pair.

Step3: Analyze option B

$\angle DFH$ and $\angle GFC$ are not adjacent in the way required for a linear pair. Their non - common sides do not form a straight line.

Step4: Analyze option C

$\angle GFH$ and $\angle GCF$ are not adjacent angles with a common vertex at $F$ and non - common sides forming a straight line. $\angle GCF$ has vertex $C$ and $\angle GFH$ has vertex $F$.

Step5: Analyze option D

$\angle HFD$ and $\angle CFD$ share a common side $\overrightarrow{FD}$, a common vertex $F$, and their non - common sides $\overrightarrow{FH}$ and $\overrightarrow{FC}$ form a straight line. So they are a linear pair.

Step6: Analyze option E

$\angle GFH$ and $\angle GFC$ share a common side $\overrightarrow{GF}$, a common vertex $F$, and their non - common sides $\overrightarrow{FH}$ and $\overrightarrow{FC}$ form a straight line. So they are a linear pair.

Step7: Analyze option F

$\angle HFG$ and $\angle DFH$ share a common side $\overrightarrow{FH}$, a common vertex $F$, and their non - common sides $\overrightarrow{FG}$ and $\overrightarrow{FD}$ form a straight line. So they are a linear pair.

Answer:

A. $\angle DFC$ and $\angle CFG$, D. $\angle HFD$ and $\angle CFD$, E. $\angle GFH$ and $\angle GFC$, F. $\angle HFG$ and $\angle DFH$