Question

∠1 and angle, 2∠2 are vertical angles. If mangle, 1, equals, left parenthesis, x, plus, 12, right parenthesis, degrees∠1=(x+12)
∘
and mangle, 2, equals, left parenthesis, 6, x, minus, 8, right parenthesis, degrees∠2=(6x−8)
∘
, then find the measure of angle, 2∠2.

Answer 1

Since angles \( \angle 1 \) and \( \angle 2 \) are vertical angles, they are equal. This means we can set up the equation:

\[
\angle 1 = \angle 2
\]

Substituting the expressions for \( \angle 1 \) and \( \angle 2 \):

\[
x + 12 = 6x - 8
\]

Now, we'll solve for \( x \):

1. Subtract \( x \) from both sides:

\[
12 = 6x - x - 8
\]

This simplifies to:

\[
12 = 5x - 8
\]

2. Next, add 8 to both sides:

\[
12 + 8 = 5x
\]

\[
20 = 5x
\]

3. Now, divide by 5:

\[
x = \frac{20}{5} = 4
\]

With \( x = 4 \), we can find the measure of \( \angle 2 \) by substituting \( x \) back into the expression for \( \angle 2 \):

\[
\angle 2 = 6x - 8
\]

Substitute \( x = 4 \):

\[
\angle 2 = 6(4) - 8 = 24 - 8 = 16
\]

Thus, the measure of \( \angle 2 \) is

\[
\boxed{16^\circ}
\]