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The diagram shows a quadrilateral \( ZWXY \) with three right angles at \( Z \), \( W \), and \( X \). The two marked angles in the triangle are \( (5t)^\circ \) at \( Y \) and \( (3t + 14)^\circ \) at \( X \), while two sides of the quadrilateral, \( ZW \) and \( WX \), are labeled with 45° angles.

### Step-by-step solution:

#### Step 1: Understanding the total angle sum of a quadrilateral
In any quadrilateral, the sum of all interior angles is 360°.

In this case:
- Three angles are right angles (90° each at \( Z \), \( W \), and \( X \)).
- The fourth angle is at \( Y \), which is expressed as \( 5t \) and \( 3t + 14 \).

The equation representing the angle sum is:
\[
90^\circ + 90^\circ + 90^\circ + \left(5t + (3t + 14)\right)^\circ = 360^\circ
\]

#### Step 2: Simplifying the equation
First, simplify the expression inside the parentheses:
\[
5t + (3t + 14) = 8t + 14
\]
Now, substitute this into the equation:
\[
90 + 90 + 90 + (8t + 14) = 360
\]
Simplify:
\[
270 + 8t + 14 = 360
\]
\[
8t + 284 = 360
\]

#### Step 3: Solving for \( t \)
Subtract 284 from both sides:
\[
8t = 360 - 284
\]
\[
8t = 76
\]
Now, divide by 8:
\[
t = \frac{76}{8} = 9.5
\]

Thus, \( t = 9.5 \).

### Final Answer:
\[
t = 9.5
\]

Would you like more details or explanations on any part of the solution?

Here are five related questions to deepen understanding:
1. How do you calculate the sum of interior angles in any polygon?
2. Why do right angles add up to 90°?
3. How does the angle sum in quadrilaterals change when a figure is not a rectangle?
4. What is the formula to find the sum of interior angles of polygons in general?
5. How would the solution change if one of the marked angles were different?

**Tip:** Always remember that the sum of the interior angles of any quadrilateral is 360°, regardless of the shape.

In quadrilateral ZWXY, find the value of t, where there are three right angles at Z, W, and X. The angles are labeled (5t)° at Y and (3t + 14)° at X, and sides ZW and WX are both 45°.

Learn how to find the value of t in quadrilateral ZWXY, where three right angles are present, and angles (5t)° and (3t + 14)° form part of the solution. This problem covers geometry and algebra concepts suitable for students in Grades 8-10.

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