If two angles are supplementary, that means the sum of their degrees of measure will add up to 180. In order to find the measure of angle B, subtract angle A from 180 like shown:

\(\displaystyle m\angle B=180^{o}-162^{o}=18^{o}\)

This gives us a final answer of 18 degrees for angle B.

If two angles are supplementary, that means the sum of their degrees of measure will add up to 180. In order to find the measure of angle B, subtract angle A from 180 like shown:

\(\displaystyle m\angle B=180^{o}-158^{o}=22^{o}\)

This gives us a final answer of 22 degrees for angle B.

The name of a ray includes two letters, so \(\displaystyle \overrightarrow{BCD}\) can be eliminated.

The first letter must be the endpoint. Since \(\displaystyle \overrightarrow{BD}\) is a name of the ray, the endpoint is \(\displaystyle B\), and any alternative name for the ray must begin with \(\displaystyle B\). This leaves only \(\displaystyle \overrightarrow{BC}\).

\(\displaystyle \angle CED\) and \(\displaystyle \angle ECD\) are two acute angles of a right triangle and are therefore complementary - that is, 

\(\displaystyle m \angle ECD + m \angle CED = 90^{\circ }\)

\(\displaystyle m \angle CED = 64^{\circ }\), so

\(\displaystyle m \angle ECD + 64^{\circ } = 90^{\circ }\)

\(\displaystyle m \angle ECD = 26^{\circ }\)

\(\displaystyle \angle ECD\) and \(\displaystyle \angle FEC\), being alternate interior angles formed by transversal \(\displaystyle \overrightarrow{CE}\) across parallel lines, are congruent, so \(\displaystyle m \angle FEC= 26^{\circ }\).

We now look at \(\displaystyle \Delta FEC\), whose interior angles must have degree measures totaling \(\displaystyle 180^{\circ }\), so

\(\displaystyle m\angle FCE + m \angle FEC + m\angle EFC = 180^{\circ }\)

\(\displaystyle m\angle FCE + 26 ^{\circ }+107^{\circ } = 180^{\circ }\)

\(\displaystyle m\angle FCE + 133^{\circ } = 180^{\circ }\)

\(\displaystyle m\angle FCE= 47^{\circ }\)

From the Parallel Postulate and its converse, as well as its various resulting theorems, two lines in a plane crossed by a transversal are parallel if any of the following happen:

Both lines are perpendicular to the same third line - this happens if \(\displaystyle \angle FED\) is a right angle, since, from this fact and the fact that \(\displaystyle \angle EDC\) is also right, both lines are perpendicular to \(\displaystyle \overline{ED}\).

Same-side interior angles are supplementary - this happens if \(\displaystyle \angle BCE\) and \(\displaystyle \angle FEC\) are supplementary, since they are same-side interior angles with respect to transversal \(\displaystyle \overrightarrow{CE}\).

Alternate interior angles are congruent - this happens if \(\displaystyle \angle AFC \cong \angle DCF\), since they are alternate interior angles with respect to transversal \(\displaystyle \overleftrightarrow{FC}\).

However, the fact that \(\displaystyle \overrightarrow{CE}\)  bisects \(\displaystyle \angle FCD\) has no bearing on whether \(\displaystyle \overleftrightarrow{AE} \parallel \overleftrightarrow{BD}\) is true or not, since it does not relate any two angles formed by a transversal. 

"\(\displaystyle \overrightarrow{CE}\)  bisects \(\displaystyle \angle FCD\)" is the correct choice.

In an intersecting line, vertical angles are equal to each other.

Set up an equation such that both angles are equal.

\(\displaystyle 2x-7 = 9x-14\)

Solve for \(\displaystyle x\).  Subtract \(\displaystyle 2x\) on both sides.

\(\displaystyle 2x-7 -(2x)= 9x-14 -(2x)\)

\(\displaystyle -7 = 7x-14\)

Add 14 on both sides.

\(\displaystyle -7 +14= 7x-14+14\)

\(\displaystyle 7= 7x\)

Divide by 7 on both sides.

\(\displaystyle \frac{7}{7}= \frac{7x}{7}\)

\(\displaystyle x=1\)

The answer is:  \(\displaystyle 1\)

Vertical angles are equal.

Set both angles equal and solve for x.

\(\displaystyle 2x-3 = 6x-8\)

Subtract \(\displaystyle 2x\) on both sides.

\(\displaystyle 2x-3 -2x= 6x-8-2x\)

\(\displaystyle -3 = 4x-8\)

Add 8 on both sides.

\(\displaystyle -3+8= 4x-8+8\)

\(\displaystyle 5=4x\)

Divide by 4 on both sides.

\(\displaystyle x=\frac{5}{4}\)

The answer is:  \(\displaystyle \frac{5}{4}\)

Vertical angles of intersecting lines must equal to each other.

Set up an equation such that both angle measures are equal.

\(\displaystyle 3x-3 = 33\)

Add three on both sides.

\(\displaystyle 3x-3 +3= 33+3\)

\(\displaystyle 3x =36\)

Divide by three on both sides.

\(\displaystyle \frac{3x }{3}=\frac{36}{3}\)

The answer is:  \(\displaystyle 12\)

Opposite angles equal.  Set up an equation such that both angle values are equal.

\(\displaystyle 5x-5 =75\)

Add 5 on both sides.

\(\displaystyle 5x-5 +5=75 +5\)

\(\displaystyle 5x = 80\)

Divide by 5 on both sides.

\(\displaystyle \frac{5x}{5} = \frac{80}{5}\)

\(\displaystyle x=16\)

\(\displaystyle 2x =32\)

The answer is:  \(\displaystyle 32\)

Opposite angles of two intersecting lines must equal to each other. Set up an equation such that both angle are equal.

\(\displaystyle 5x+1= 10x-9\)

Add 9 on both sides.

\(\displaystyle 5x+1+9= 10x-9+9\)

\(\displaystyle 5x+10= 10x\)

Subtract \(\displaystyle 5x\) on both sides.

\(\displaystyle 5x+10-5x= 10x-5x\)

\(\displaystyle 10 = 5x\)

\(\displaystyle x=2\)

This means that \(\displaystyle 7x\) equals \(\displaystyle 14\).