In ΔWXY,  start overline, W, Y, end overline 
WY
  is extended through point Y to point Z,  m, angle, Y, W, X, equals, left parenthesis, 2, x, plus, 9, right parenthesis, degreesm∠YWX=(2x+9) 
∘
 ,  m, angle, W, X, Y, equals, left parenthesis, 2, x, plus, 16, right parenthesis, degreesm∠WXY=(2x+16) 
∘
 ,   and m, angle, X, Y, Z, equals, left parenthesis, 8, x, plus, 1, right parenthesis, degreesm∠XYZ=(8x+1) 
∘
 .   Find m, angle, W, X, Y, .m∠WXY.


We are given the triangle \( \Delta WXY \) with the following information:

- \( m\angle YWX = (2x + 9)^\circ \),
- \( m\angle WXY = (2x + 16)^\circ \),
- \( m\angle XYZ = (8x + 1)^\circ \), where \( WY \) is extended to \( Z \).

We need to find \( m\angle WXY \), which is already given as \( (2x + 16)^\circ \). However, before we can directly determine the value of \( m\angle WXY \), we need to solve for \( x \).

### Step-by-step solution:

1. **Exterior Angle Theorem:**
   The angle \( \angle XYZ \) is an exterior angle of \( \Delta WXY \). By the Exterior Angle Theorem, we know that:
   \[
   m\angle XYZ = m\angle YWX + m\angle WXY.
   \]
   Substituting the given expressions:
   \[
   (8x + 1) = (2x + 9) + (2x + 16).
   \]
   
2. **Simplify the equation:**
   Combine like terms on the right-hand side:
   \[
   8x + 1 = 2x + 9 + 2x + 16,
   \]
   \[
   8x + 1 = 4x + 25.
   \]
   
3. **Solve for \( x \):**
   Subtract \( 4x \) from both sides:
   \[
   8x - 4x + 1 = 25,
   \]
   \[
   4x + 1 = 25.
   \]
   Subtract 1 from both sides:
   \[
   4x = 24.
   \]
   Divide by 4:
   \[
   x = 6.
   \]

4. **Find \( m\angle WXY \):**
   Now that we have \( x = 6 \), substitute it back into the expression for \( m\angle WXY \):
   \[
   m\angle WXY = 2x + 16 = 2(6) + 16 = 12 + 16 = 28^\circ.
   \]

Thus, \( m\angle WXY = 28^\circ \).

---

Would you like further details or have any questions?

Here are five related questions to extend the topic:
1. How do you apply the Exterior Angle Theorem in different triangle problems?
2. What are the key differences between interior and exterior angles of triangles?
3. Can the sum of the angles in any triangle ever exceed 180°?
4. How can you use algebra to solve problems involving angles in polygons?
5. How do you classify triangles based on their angles?

**Tip:** Always verify your results by substituting your solution back into the original problem to ensure accuracy!

In ΔWXY, WY is extended through point Y to point Z, m∠YWX = (2x + 9)°, m∠WXY = (2x + 16)°, and m∠XYZ = (8x + 1)°. Find m∠WXY.

Learn how to apply the Exterior Angle Theorem to solve for m∠WXY in triangle ΔWXY. This problem covers key geometric concepts and involves algebraic expressions for angles. Suitable for students in Grades 8-10.

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