Question

quadrilateral tuvw is a rhombus and $m\angle tvw = w - 3^\circ$. what is the value of w?
$w = \square^\circ$

Explanation:

Step1: Recall rhombus side properties

In a rhombus, all sides are equal, so $UV = UT$. This means $\triangle UVT$ is isosceles with $\angle UVT = \angle UTV$.

Step2: Calculate $\angle UVT$

The sum of angles in a triangle is $180^\circ$. For $\triangle UVT$:
$$\angle UVT = \frac{180^\circ - 62^\circ}{2} = 59^\circ$$

Step3: Use rhombus parallel sides property

$UV \parallel WT$, so $\angle TVW = \angle UVT$ (alternate interior angles are equal). We know $\angle TVW = w - 3^\circ$, so:
$$w - 3^\circ = 59^\circ$$

Step4: Solve for $w$

Add $3^\circ$ to both sides:
$$w = 59^\circ + 3^\circ = 62^\circ$$

Answer:

$w = 62$