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In the given problem, we have triangle \( ZYX \) with \( YW \) as the bisector of \( \angle ZYX \). The angles and some expressions related to the angles are provided. Let’s break down the problem and solve it step-by-step.

### Step 1: Understanding the Angles
- \( \angle ZYW = 45^\circ \)
- \( \angle WYX = 45^\circ \)
- \( \angle ZXY = (3t + 14)^\circ \)
- \( \angle Z = (5t)^\circ \)

Since \( YW \) is the bisector of \( \angle ZYX \), we know that \( \angle ZYW = \angle WYX = 45^\circ \).

### Step 2: Sum of Interior Angles of a Triangle
The sum of the angles in triangle \( ZYX \) is always \( 180^\circ \). So, we can write the equation:
\[
\angle Z + \angle Y + \angle X = 180^\circ
\]
Substitute the given expressions:
\[
(5t) + (45 + 45) + (3t + 14) = 180^\circ
\]
Simplify this equation:
\[
5t + 90 + 3t + 14 = 180
\]
\[
8t + 104 = 180
\]
Solve for \( t \):
\[
8t = 180 - 104
\]
\[
8t = 76
\]
\[
t = \frac{76}{8} = 9.5
\]

### Step 3: Finding \( m \angle WYX \)
Now that we know \( t = 9.5 \), we can substitute this into the expression for \( \angle ZXY \):
\[
\angle ZXY = 3(9.5) + 14 = 28.5 + 14 = 42.5^\circ
\]
Thus, \( \angle ZXY = 42.5^\circ \).

Finally, \( m \angle WYX = 45^\circ \) since it's given that \( \angle YWZ = 45^\circ \) and \( YW \) bisects \( \angle ZYX \).

### Summary of Answers
- \( t = 9.5 \)
- \( m \angle WYX = 45^\circ \)

Would you like more details on any of these steps or further clarification?

### Related Questions:
1. What would happen to the angles if \( YW \) was not a bisector?
2. How does the property of an angle bisector help in solving geometric problems?
3. What is the general formula for solving angles in a triangle with an angle bisector?
4. Can you solve for the area of triangle \( ZYX \) given the sides?
5. How would you approach this problem if the triangle was not isosceles?

### Tip:
Always remember that the sum of the interior angles of a triangle is \( 180^\circ \), a key property in solving angle-related problems.

YW is a bisector of ∠ZYX, find the value of t and m∠WYX.

Learn how to solve a triangle geometry problem where YW bisects ∠ZYX. This step-by-step guide helps find the value of t and m∠WYX using angle bisector properties and the sum of triangle interior angles. Suitable for students in Grades 8-10.

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