Question

∠fbc and ∠cbg are supplements, ∠dbg and ∠dbf
are supplements, and ∠cbg ≅ ∠dbf.
by the congruent supplements theorem, what can you
conclude?
∠cbg ≅ ∠dbg
∠fbc ≅ ∠dbg
∠cbg is supplementary to ∠dbf
∠fbc is supplementary to ∠dbg

Explanation:

Step1: Recall the congruent supplements theorem

If two angles are supplementary to the same angle (or to congruent angles), then the two angles are congruent.
We know that \(\angle FBC\) and \(\angle CBG\) are supplements (\(\angle FBC+\angle CBG = 180^{\circ}\)), \(\angle DBG\) and \(\angle DBF\) are supplements (\(\angle DBG+\angle DBF=180^{\circ}\)), and \(\angle CBG\cong\angle DBF\).

Step2: Apply the congruent supplements theorem

Let \(\angle A=\angle FBC\), \(\angle B = \angle CBG\), \(\angle C=\angle DBG\), \(\angle D=\angle DBF\).
Since \(\angle A+\angle B=180^{\circ}\), \(\angle C+\angle D = 180^{\circ}\) and \(\angle B\cong\angle D\).
By the congruent - supplements theorem, \(\angle A\cong\angle C\), that is \(\angle FBC\cong\angle DBG\).

Answer:

\(\angle FBC\cong\angle DBG\) (the second option)