https://file.aiquickdraw.com/mathbot/file/1727319368908ot1x7efr.jpg

Let's break down the information provided in the image and solve the problem.

We are given:
- Triangle \( VXT \) is equilateral, which means all its angles are 60°.
- \( WU = WY \), indicating that triangle \( WUY \) is isosceles.
- The measure of \( \angle VWU \) is twice the measure of \( \angle VUW \).

We are tasked with finding the measure of \( \angle TUY \).

### Step 1: Determine the angles in triangle \( VUW \)

- Since triangle \( VXT \) is equilateral, \( \angle VXT = 60^\circ \).
- Let \( \angle VUW = x \). According to the problem, \( \angle VWU = 2x \).
  
  In triangle \( VUW \), the sum of angles must equal 180°:
  \[
  x + 2x + 60^\circ = 180^\circ
  \]
  Simplifying:
  \[
  3x = 120^\circ \quad \Rightarrow \quad x = 40^\circ
  \]
  So, \( \angle VUW = 40^\circ \) and \( \angle VWU = 80^\circ \).

### Step 2: Use the isosceles property of triangle \( WUY \)

- Since \( WU = WY \), triangle \( WUY \) is isosceles. This means \( \angle UWY = \angle UYW \).
- We are given that \( \angle WUY = 50^\circ \).

  Using the sum of angles in triangle \( WUY \):
  \[
  \angle UWY + \angle UYW + \angle WUY = 180^\circ
  \]
  Let \( \angle UWY = \angle UYW = y \). Then:
  \[
  2y + 50^\circ = 180^\circ \quad \Rightarrow \quad 2y = 130^\circ \quad \Rightarrow \quad y = 65^\circ
  \]

So, \( \angle UWY = 65^\circ \).

### Step 3: Determine \( \angle TUY \)

Since \( \angle TUY \) is an exterior angle to triangle \( WUY \), we can use the exterior angle theorem. The exterior angle is equal to the sum of the opposite interior angles:
\[
\angle TUY = \angle UWY + \angle UYW = 65^\circ + 65^\circ = 130^\circ
\]

Thus, the measure of \( \angle TUY \) is **130°**.

---

Would you like further details or have any additional questions? Here are 5 related questions you can explore:

1. How do you prove that the sum of angles in any triangle is always 180°?
2. What is the exterior angle theorem, and how is it applied to triangles?
3. Can we use similar triangles in this problem? If so, how would they help?
4. What are some other properties of isosceles triangles and equilateral triangles?
5. How can you calculate angles in more complex polygon shapes using known triangles?

**Tip:** Always remember that the angles in an equilateral triangle are always 60°, and in an isosceles triangle, two angles are always equal!

In the figure at the left, triangle VXT is equilateral and WU = WY. If the measure of angle VWU is twice the measure of angle VUW, what is the measure of angle TUY?

Learn how to find the measure of angle TUY in a triangle involving both equilateral and isosceles properties. This problem includes key concepts like the Exterior Angle Theorem, angle sum property, and triangle angle relationships. Suitable for students in Grades 9-12.

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