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.