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This problem involves a quadrilateral \( TUVW \) with given angles, and we are asked to determine two missing angles.

### Step 1: Identify the Given Information
- Angle \( \angle TUV = 106^\circ \)
- Angle \( \angle VWU = 68^\circ \)

### Step 2: Use Properties of Quadrilaterals
In a quadrilateral, the sum of the interior angles is always \( 360^\circ \). Therefore, for quadrilateral \( TUVW \):

\[
\angle TUV + \angle VWT + \angle VWU + \angle T = 360^\circ
\]

### Step 3: Find Missing Angles

#### Part (a): Find angle \( \angle TUV \)
Angle \( TUV \) is already given as \( 106^\circ \), so no calculation is required.

#### Part (b): Find angle \( \angle VWT \)
We need to find the sum of all angles in the quadrilateral and subtract the given angles to find \( \angle VWT \).

First, sum the given angles:
\[
106^\circ + 68^\circ = 174^\circ
\]

Now subtract this from \( 360^\circ \):
\[
360^\circ - 174^\circ = 186^\circ
\]

This \( 186^\circ \) represents the sum of the remaining two angles \( \angle VWT \) and \( \angle T \). Since \( \angle T = 90^\circ \) (because it is marked with a square indicating a right angle), we can find \( \angle VWT \):

\[
186^\circ - 90^\circ = 96^\circ
\]

Thus, \( \angle VWT = 96^\circ \).

### Final Answers:
- \( \angle TUV = 106^\circ \) (already given).
- \( \angle VWT = 96^\circ \).

Let me know if you'd like more details or have questions!

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Here are 5 related questions:
1. How do we calculate the sum of interior angles in a polygon?
2. What is the sum of angles in a triangle?
3. How can we determine if a quadrilateral is cyclic?
4. What is the relationship between opposite angles in a cyclic quadrilateral?
5. How does the Pythagorean theorem relate to angles in right-angled triangles?

**Tip**: Always remember, the sum of angles in any quadrilateral is \( 360^\circ \), regardless of the shape.

For each of the following angles, work out its size: a) angle TUV, b) angle VWT. The angles in quadrilateral TUVW include 106° for angle TUV and 68° for angle VWU.

Learn how to calculate the missing angles in quadrilateral TUVW. Given angle TUV = 106° and angle VWU = 68°, this solution explains how to find angles TUV and VWT using the sum of angles in a quadrilateral. Suitable for students in Grades 7-9.

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