Question

∠1 and angle, 2∠2 are vertical angles. If mangle, 1, equals, left parenthesis, 4, x, minus, 4, right parenthesis, degrees∠1=(4x−4)
∘
and mangle, 2, equals, left parenthesis, 5, x, minus, 29, right parenthesis, degrees∠2=(5x−29)
∘
, then find the measure of angle, 1∠1.

Answer 1

Since angles \( \angle 1 \) and \( \angle 2 \) are vertical angles, their measures are equal. This gives us the equation:

\[
m\angle 1 = m\angle 2
\]

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

\[
4x - 4 = 5x - 29
\]

Next, we will solve for \( x \). First, let's rearrange the equation by isolating \( x \):

1. Subtract \( 4x \) from both sides:

\[
-4 = 5x - 4x - 29
\]

This simplifies to:

\[
-4 = x - 29
\]

2. Now, add \( 29 \) to both sides:

\[
-4 + 29 = x
\]

This gives us:

\[
x = 25
\]

Now that we have the value of \( x \), we can find the measure of \( \angle 1 \):

\[
m\angle 1 = 4x - 4
\]

Substituting \( x = 25 \):

\[
m\angle 1 = 4(25) - 4 = 100 - 4 = 96
\]

Thus, the measure of \( \angle 1 \) is:

\[
\boxed{96} \text{ degrees}
\]