Question: User: Lines
$l$,
$m$, and
$n$ are parallel to each other and
$p$ is a transversal.
Also,
$\angle{y}+\angle{z}=270\degree$.
parallel lines
$x$
$y$
$z$
$l$
$m$
$n$
$p$
Find
$\angle{x}$.
$\degree$
1. **Problem Statement:** We have three parallel lines $l$, $m$, and $n$ cut by a transversal $p$. Given that $\angle y + \angle z = 270\degree$, we need to find $\angle x$.
2. **Key Concept:** When a transversal crosses parallel lines, corresponding angles, alternate interior angles, and co-interior angles have special relationships.
3. **Given:** $l \parallel m \parallel n$, $p$ is transversal, and $\angle y + \angle z = 270\degree$.
4. **Step 1:** Since $l$, $m$, and $n$ are parallel, the angles formed by the transversal $p$ with these lines have relationships. Typically, angles on a straight line sum to $180\degree$.
5. **Step 2:** Note that $\angle y$ and $\angle z$ are angles on the transversal at different parallel lines. Their sum is $270\degree$, which is $180\degree + 90\degree$.
6. **Step 3:** This implies that $\angle y$ and $\angle z$ are not supplementary but exceed $180\degree$ by $90\degree$.
7. **Step 4:** Since $\angle y + \angle z = 270\degree$, the remaining angle on the straight line with $\angle x$ must satisfy the linear pair property.
8. **Step 5:** The sum of angles around a point on a straight line is $360\degree$. If $\angle y + \angle z = 270\degree$, then the remaining angle $\angle x$ is:
$$\angle x = 360\degree - 270\degree = 90\degree$$
9. **Answer:** $\boxed{90\degree}$