Question

in this figure, ( mangle c = 86^{circ} ) and ( mangle dbc = 137^{circ} ).
what is ( mangle bac )?
51 degrees
53 degrees
50 degrees
52 degrees

Explanation:

Step1: Find the measure of $\angle ABC$

Since $\angle DBC$ and $\angle ABC$ are supplementary (they form a linear pair), we use the formula $\angle ABC = 180^{\circ}-\angle DBC$.
$\angle ABC=180^{\circ}- 137^{\circ}=43^{\circ}$

Step2: Use the triangle - angle sum theorem

In $\triangle ABC$, the sum of the interior angles is $180^{\circ}$, i.e., $\angle BAC+\angle ABC+\angle C = 180^{\circ}$.
We know $\angle ABC = 43^{\circ}$ and $\angle C=86^{\circ}$. Let $\angle BAC=x$. Then $x + 43^{\circ}+86^{\circ}=180^{\circ}$.
Simplify the left - hand side: $x+129^{\circ}=180^{\circ}$.
Subtract $129^{\circ}$ from both sides: $x=\angle BAC=180^{\circ}-129^{\circ}=51^{\circ}$

Answer:

51 degrees