Supplementary angles have degree measures that total \(\displaystyle 180^{\circ }\). Since we have an \(\displaystyle 80^{\circ}\) angle, the supplementary angle would measure \(\displaystyle 180^{\circ}-80^{\circ }=100^{\circ}\) 

Complementary angles have degree measures that total \(\displaystyle 90^{\circ }\). Since we have an \(\displaystyle 18^{\circ}\) angle, the supplementary angle would measure \(\displaystyle 90^{\circ}-18^{\circ }=72^{\circ}\) 

Congruent angles have the same degree measure, so an angle congruent to a \(\displaystyle 119^{\circ}\) angle would also measure \(\displaystyle 119^{\circ}\). 

Two angles are supplementary if the total of their degree measures is \(\displaystyle 180^{\circ }\). Therefore, an angle supplementary to a \(\displaystyle 16^{\circ}\) angle measures \(\displaystyle 180^{\circ }-16^{\circ}=164^{\circ}\). 

Two angles are complementary if the total of their degree measures is \(\displaystyle 90^{\circ }\). Therefore, an angle complementary to a \(\displaystyle 70^{\circ}\) angle measures \(\displaystyle 90^{\circ }-70^{\circ}=20^{\circ}\). 

Two angles are congruent if they have the same degree measure. Therefore, an angle congruent to an \(\displaystyle 89^{\circ }\) angle also measures \(\displaystyle 89^{\circ }\). 

It is important to recall that all triangles add to 180 degrees and a right triangle contains one angle that is equal to 90 degrees. Therefore, in this particular problem we can write the following equation to solve for the missing variable.

\(\displaystyle \\90^\circ+50^\circ+2t=180^\circ \\140^\circ-2t=180^\circ \\2t=180^\circ-140^\circ \\2t=40^\circ \\t=20\)

The middle letter of an angle must be its vertex, so the vertex of \(\displaystyle \angle FBC\) is \(\displaystyle B\). 

An angle can be named after its vertex alone as long as it is the only angle with that vertex. This is not the case here, so \(\displaystyle \angle B\) cannot be correct. Also, \(\displaystyle \angle FCB\) and \(\displaystyle \angle BFG\) are incorrect since the middle letters are not \(\displaystyle B\).

The first and last letters of a three-letter angle name must be points on different sides of the angle. \(\displaystyle \angle CBD\) is incorrect since \(\displaystyle C\) and \(\displaystyle D\) are on the same side of the angle. The correct choice is therefore \(\displaystyle \angle DBF\), since \(\displaystyle D\) and \(\displaystyle F\) are on different sides of the angle.

A triangle can be named after its three vertices in any order. Since the vertices of the triangle are \(\displaystyle B,C,F\), any name that includes these three letters is valid. All of the choices fit this description.

An angle can be given a name with three letters if the middle letter is the name of the vertex and the other two letters denote points on different sides of the angle. All four of the three-letter choices fit this description.

An angle can be given a one-letter name if the letter is the name of the vertex and if it is the only angle shown in the diagram to have that vertex (thereby avoiding ambiguity). There are four angles in the diagram with vertex \(\displaystyle C\), so we cannot use \(\displaystyle \angle C\) to indicate any of them, including \(\displaystyle \angle ECA\). \(\displaystyle \angle C\) is the correct choice.