Question: Note: Figure not drawn to scale.
Question 14
In the figure, parallel lines $r$ and $s$ are intersected by lines $t$, $u$, and $w$. If $d = 42$, $g = 143$, and $h > d$, which statement about $j$ and $p$ must be true?
A $j = p$
B $j + p = 90$
C $j > p$
D $j < p$
Lines, angles, and triangles-Angle Relationships Formed by Parallel Lines and Transversals
position_hint: centert
$u$
$w$
$r$
$s$
$j^6$
$g^6$
$d^6$
$h^6$
$p^6$
1. **State the problem:** We have two parallel lines $r$ and $s$ intersected by three transversals $t$, $u$, and $w$. Given angles $d = 42^\circ$, $g = 143^\circ$, and $h > d$, we want to determine the relationship between angles $j$ and $p$.
2. **Recall angle relationships:** When parallel lines are cut by transversals, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (sum to $180^\circ$).
3. **Analyze given angles:** Since $g = 143^\circ$ and $d = 42^\circ$, note that $g + d = 143 + 42 = 185^\circ$, which is more than $180^\circ$, so these are not supplementary angles on the same transversal.
4. **Given $h > d = 42^\circ$, so $h > 42^\circ$.
5. **Identify $j$ and $p$:** Angles $j$ and $p$ are formed by intersections of the transversals with the parallel lines. By the properties of parallel lines and transversals, angles $j$ and $p$ are corresponding angles or alternate interior angles.
6. **Use the fact that $j$ and $p$ are corresponding angles:** Corresponding angles are equal when lines are parallel.
7. **Therefore, $j = p$ must be true.**
**Final answer:** $\boxed{j = p}$ (Option A).