Question

If there are two supplementary angles, ∠D and ∠E , where m∠D=(4x+8)° and m∠E=(x+17)° , and ∠F is a vertical angle with ∠D , what is the measure of ∠F ? Please

Answer 1

Two angles are supplementary when the sum of their measures equals 180 degrees. Given the measures of angles \( \angle D \) and \( \angle E \):

\[
m\angle D = (4x + 8)^\circ
\]
\[
m\angle E = (x + 17)^\circ
\]

We can set up the equation for supplementary angles:

\[
m\angle D + m\angle E = 180^\circ
\]

Substituting in the expressions for \( m\angle D \) and \( m\angle E \):

\[
(4x + 8) + (x + 17) = 180
\]

Now, combine like terms:

\[
4x + x + 8 + 17 = 180
\]
\[
5x + 25 = 180
\]

Next, subtract 25 from both sides:

\[
5x = 155
\]

Now, divide by 5:

\[
x = 31
\]

Now that we have the value for \( x \), we can find the measures of \( \angle D \) and \( \angle E \):

\[
m\angle D = 4x + 8 = 4(31) + 8 = 124 + 8 = 132^\circ
\]
\[
m\angle E = x + 17 = 31 + 17 = 48^\circ
\]

Since \( \angle F \) is a vertical angle to \( \angle D \), its measure will be the same as \( m\angle D \):

\[
m\angle F = m\angle D = 132^\circ
\]

Thus, the measure of \( \angle F \) is:

\[
\boxed{132^\circ}
\]