Two intersecting lines create four angles, numbered 1, 2, 3, and 4, where ∠4 is adjacent to ∠1 and ∠3, and m∠2=110°. Which other angle has a measure that is also equal to 110°?(1 point)
In a diagram, we can represent the situation as follows:
We are given that $\angle 2$ has a measure of $110^\circ.$ Since lines $m$ and $n$ are intersecting, we know that angles $3$ and $4$ are supplementary (opposite angles). This means that $m\angle 3 = 180^\circ - m\angle 4.$
Since $\angle 4$ is adjacent to $\angle 1$ and $\angle 3,$ we can conclude that $\angle 4 + \angle 1 + \angle 3 = 180^\circ,$ which is an example of the Angle Addition Postulate.
Substituting the given equation, we have: \begin{align*}
110^\circ + \angle 1 + (180^\circ - \angle 4) &= 180^\circ\\
290^\circ + \angle 1 - \angle 4 &= 180^\circ.
\end{align*} Simplifying, we find $\angle 1 - \angle 4 = -110^\circ.$
To solve for $\angle 1,$ note $\angle 1 + \angle 3 = \angle 3 + \angle 4 = \angle 3 + 110^\circ = 180^\circ.$ Therefore, $\angle 3 = 70^\circ.$
Since $\angle 1$ and $\angle 3$ are supplementary, $\angle 1 = 110^\circ.$
Since we found $\angle 1 = 110^\circ,$ which is equal to the given angle measure of $110^\circ,$ the answer is $\boxed{\text{Angle 1}}.$
huh?
I apologize for the confusion. Let me explain it again.
We are given that $\angle 2$ has a measure of $110^\circ.$
Since lines $m$ and $n$ are intersecting, we have vertical angles. This means that angles 1 and 2 are congruent, and angles 3 and 4 are congruent.
So, $m\angle 1 = m\angle 2 = 110^\circ.$
Hence, the other angle that has a measure equal to $110^\circ$ is $\boxed{\text{Angle 1}}.$