Question

complementary and supplementary angles
assignment active
making conclusions
$angle fbc$ and $angle cbg$ are supplements, $angle dbg$ and $angle dbf$ are supplements, and $angle cbg cong angle dbf$.
by the congruent supplements theorem, what can you conclude?
$circ$ $angle cbg cong angle dbg$
$circ$ $angle fbc cong angle dbg$
$circ$ $angle cbg$ is supplementary to $angle dbf$.
$circ$ $angle fbc$ is supplementary to $angle dbg$.

Explanation:

Step1: Recall congruent supplements theorem

If two angles are supplements of congruent angles, then the two angles are congruent.

Step2: Define supplementary angle relationships

1. $\angle FBC + \angle CBG = 180^\circ$ (given as supplements)
2. $\angle DBG + \angle DBF = 180^\circ$ (given as supplements)

Step3: Substitute congruent angles

We know $\angle CBG \cong \angle DBF$, so substitute $\angle DBF$ with $\angle CBG$ in the second equation:
$\angle DBG + \angle CBG = 180^\circ$

Step4: Apply congruent supplements theorem

Both $\angle FBC$ and $\angle DBG$ are supplements of the congruent angle $\angle CBG$, so they are congruent.

Answer:

$\angle FBC \cong \angle DBG$