\(\displaystyle \angle AFG\) and \(\displaystyle \angle GFE\) form a linear pair, so their measures add up to that of a straight angle, which is \(\displaystyle 180^{\circ }\). They must be supplementary (and cannot be complementary).

\(\displaystyle \angle AFC\) and \(\displaystyle \angle EFG\) are a pair of vertical angles - angles formed from a pair of intersecting lines which together have the lines as the union. They must be congruent, and need not be complementary.

\(\displaystyle \angle EFC\) and \(\displaystyle \angle FCD\) are a pair of same-side interior angles formed by transversal \(\displaystyle \overleftrightarrow{FC}\) across the parallel lines. They must be supplementary (and cannot be complementary).

\(\displaystyle \angle ECD\) and \(\displaystyle \angle DEC\) are the acute angles of right triangle \(\displaystyle \Delta DEC\). They must be complementary. This is the correct pair.

Since angles A and B are complementary, their measures add up to 90 degrees. Therefore we can set up our equation as such:

\(\displaystyle A + B = 90\)

-or-

\(\displaystyle 5x - 17 + 3x - 29 = 90\)

Combine like terms and solve for \(\displaystyle x\):

\(\displaystyle 8x - 46 = 90\)

\(\displaystyle 8x = 136\)

\(\displaystyle x = 17\)

The sum of complementary angles is equal to \(\displaystyle 90^o\). 

Set up the following equation and solve for \(\displaystyle x\):

\(\displaystyle 72+x=90\)

\(\displaystyle 72-72+x=90-72\)

\(\displaystyle x=18\)

Since angles A and B are complementary angles, their measurements add up to equal 90. Therefore, we need to set up our equation as follows:

\(\displaystyle A + B = 90\)

-or-

\(\displaystyle 4x + 25 + 6x - 15 = 90\)

Combine like terms:

\(\displaystyle 10x + 10 = 90\)

And solve for \(\displaystyle x\):

\(\displaystyle 10x = 80\)

\(\displaystyle x = 8\)

Since angles A and B are complementary, their measures add up to equal 90 degrees. Therefore we can set up an equation as follows:

\(\displaystyle A + B = 90\)

-or-

\(\displaystyle -2x + 50 + 5x - 20 = 90\)

Combine like terms and solve for \(\displaystyle x\):

\(\displaystyle 3x + 30 = 90\)

\(\displaystyle 3x = 60\)

\(\displaystyle x = 20\)

Now that we have found \(\displaystyle x\), we need to plug that back in to the two angle measurements

\(\displaystyle Angle A = -2(20) + 50\)

\(\displaystyle Angle A = -40 + 50\)

\(\displaystyle Angle A = 10\)

\(\displaystyle Angle B = 5(20) -20\)

\(\displaystyle Angle B = 100 - 20\)

\(\displaystyle Angle B = 80\)

The definition of a complementary angle means that two angles must add up to 90 degrees.

To find the angle, simply subtract 55 from 90.

\(\displaystyle 90-55 = 35\)

The answer is:  \(\displaystyle 35\)

Complementary angles must add up to ninety.

Subtract the given angle from 90 to find the other angle.

\(\displaystyle 90-10 = 80\)

The answer is:  \(\displaystyle 80\)

Set up the equation such that both angles sum up to 90.

\(\displaystyle (3x+2)+ (5x-7) = 90\)

\(\displaystyle 8x -5 = 90\)

Add 5 on both sides.

\(\displaystyle 8x -5 +5= 90+5\)

\(\displaystyle 8x=95\)

Divide both sides by 8.

\(\displaystyle \frac{8x}{8}=\frac{95}{8}\)

The value of \(\displaystyle 2x\) is:  \(\displaystyle 2(\frac{95}{8}) = \frac{95}{4}\)

The answer is:  \(\displaystyle \frac{95}{4}\)

Complementary angles must add up to 90 degrees.

Set up an equation so that the sum of two angles add up to 90.

\(\displaystyle (x+2)+(x+6) = 90\)

\(\displaystyle 2x+8 = 90\)

Subtract 8 on both sides.

\(\displaystyle 2x+8 -8= 90-8\)

Simplify both sides.

\(\displaystyle 2x = 82\)

Divide by two on both sides.

\(\displaystyle \frac{2x }{2}= \frac{82}{2}\)

\(\displaystyle x=41\)

The answer is:  \(\displaystyle 41\)

Note that this is in radians.  Recall that 2 complementary angles must add up to 90 degrees.

Since \(\displaystyle \pi\textup{ radians} = 180 \textup{ degrees}\),

\(\displaystyle \frac{\pi}{2}\textup{ radians} = 90 \textup{ degrees}\)

To find the complementary angle, subtract \(\displaystyle \frac{\pi}{3}\) from \(\displaystyle \frac{\pi}{2}\).

\(\displaystyle \frac{\pi}{2}-\frac{\pi}{3} = \frac{3\pi}{6} -\frac{2\pi}{6} = \frac{\pi}{6}\)

The answer is:  \(\displaystyle \frac{\pi}{6}\)