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Let's start by making sense of the relevant information. The statement
$∠ABC≅∠DEF$
tells us that the angle between $BA$ and $BC$ is congruent with the angle between $ED$ and $EF.$ This means that these angles are **equal in measure**. Let's mark these angles in blue.

We are also given $m∠ABC=114_{∘}.$ Therefore, so is $m∠DEF.$

$m∠ABC=114_{∘}=m∠DEF$

b

We are told that $BG$ bisects $∠ABC.$ This means that $∠ABC$ is divided into two equal angles. $∠ABG≅∠CBG$ Let's mark these angles in red in our diagram.

Since $m∠ABC=114_{∘},$ and $BG$ cuts this angle in half, we can conclude that $m∠ABG$ is half of $114_{∘}.$

$m∠ABG=2114_{∘} =57_{∘}$

c

Let's think back to Part B. We found that, because $BG$ is a bisector We also know that $∠ABG$ is congruent to $∠CBG.$ $∠ABG≅∠CBGandm∠ABG=57_{∘}$ Therefore, $m∠CBG=57_{∘}.$

d

We are told that $BG$ bisects $∠DEF.$ Therefore, the angle is divided into two equal angles. Since $∠ABC$ is congruent with $∠DEF,$ the following must be true. $∠ABG≅∠CBG≅∠DEG≅∠FEG$ Let's mark this in our diagram.

Therefore, $m∠DEG=57_{∘}.$