Question: Find $m\angle U$ and $m\angle V$ in rhombus TUVW.
$3c-30^\circ$
$2c+18^\circ$
$m\angle U =$
$m\angle V =$
1. **State the problem:** We need to find the measures of angles $U$ and $V$ in rhombus $TUVW$ where $m\angle W = 3c - 30^\circ$ and $m\angle U = 2c + 18^\circ$.
2. **Recall properties of a rhombus:** Opposite angles in a rhombus are equal, and adjacent angles are supplementary (sum to $180^\circ$).
3. **Set up equations:** Since $W$ and $U$ are adjacent angles,
$$ (3c - 30) + (2c + 18) = 180 $$
4. **Simplify the equation:**
$$ 3c - 30 + 2c + 18 = 180 $$
$$ 5c - 12 = 180 $$
5. **Solve for $c$:**
$$ 5c = 180 + 12 $$
$$ 5c = 192 $$
$$ c = \frac{192}{5} $$
$$ c = 38.4 $$
6. **Find $m\angle U$:**
$$ m\angle U = 2c + 18 = 2(38.4) + 18 = 76.8 + 18 = 94.8^\circ $$
7. **Find $m\angle V$:** Since $V$ is opposite $T$ and $T$ is opposite $U$, $m\angle V = m\angle U = 94.8^\circ$.
**Final answers:**
$$ m\angle U = 94.8^\circ $$
$$ m\angle V = 94.8^\circ $$