https://file.aiquickdraw.com/mathbot/file/1729527993173u9qo3vkx.jpg

In the image, quadrilateral **TUVW** is a kite. We are tasked with finding the measure of angle \( \angle U \) (denoted as \( m \angle U \)).

### Properties of a kite:
1. A kite has two pairs of adjacent sides that are equal in length.
2. One pair of opposite angles, formed by the unequal sides, are equal (these are typically the angles between the unequal sides).
3. The sum of the interior angles of any quadrilateral is always 360°.

### Given Information:
- \( \angle V = 65^\circ \)
- \( \angle T = 117^\circ \)
- We need to find \( \angle U \), and we know that opposite angle \( \angle W \) will be equal to \( \angle U \) since \( TUVW \) is a kite.

### Step-by-Step Calculation:
The sum of the interior angles of any quadrilateral is 360°, so:

\[
\angle U + \angle V + \angle T + \angle W = 360^\circ
\]

Since \( \angle U = \angle W \), we can substitute that into the equation:

\[
2 \angle U + \angle V + \angle T = 360^\circ
\]

Substituting the known values for \( \angle V \) and \( \angle T \):

\[
2 \angle U + 65^\circ + 117^\circ = 360^\circ
\]

Simplifying:

\[
2 \angle U + 182^\circ = 360^\circ
\]

\[
2 \angle U = 360^\circ - 182^\circ
\]

\[
2 \angle U = 178^\circ
\]

\[
\angle U = \frac{178^\circ}{2} = 89^\circ
\]

Thus, \( m \angle U = 89^\circ \).

Would you like more details on how these properties of kites work, or do you have any questions?

### Related Questions:
1. What are the properties of different types of quadrilaterals like squares, rectangles, and rhombuses?
2. How do you find the measure of missing angles in a parallelogram?
3. What is the difference between a kite and a rhombus in geometry?
4. Can a quadrilateral have all sides of equal length and still be a kite?
5. How can we apply the Pythagorean theorem to a kite if needed?

### Tip:
When working with polygons, always remember that the sum of the interior angles for any polygon is \((n - 2) \times 180^\circ\), where \(n\) is the number of sides.

Quadrilateral TUVW is a kite. What is m∠U?

Solve for the missing angle in a kite-shaped quadrilateral TUVW, using the sum of interior angles and properties of kites. This step-by-step solution demonstrates key geometry concepts, perfect for students in Grades 8-10.

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