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The problem involves finding the measures of angles given some algebraic expressions and a right angle. Let's break it down:

### Problem Statement:
Given:
- \( m \angle XZY = 3x + 6 \)
- \( m \angle ZYX = 6x - 15 \)
- \( \angle XZY \) is a right angle (i.e., \( m \angle XZY = 90^\circ \))

We are to find the measures of the angles \( m \angle XZY \), \( m \angle ZYX \), and \( m \angle ZXY \).

### Steps to Solve:

1. **Identify the right angle:**
   Since \( \angle XZY \) is a right angle, we know \( m \angle XZY = 90^\circ \).

2. **Set up the equation for \( m \angle XZY \):**
   \[
   3x + 6 = 90
   \]
   Solve for \( x \):
   \[
   3x = 90 - 6
   \]
   \[
   3x = 84
   \]
   \[
   x = \frac{84}{3} = 28
   \]

3. **Calculate the measure of \( m \angle ZYX \):**
   Substitute \( x = 28 \) into the expression for \( m \angle ZYX \):
   \[
   m \angle ZYX = 6x - 15 = 6(28) - 15 = 168 - 15 = 153^\circ
   \]

4. **Determine \( m \angle ZXY \):**
   Since \( \angle ZYX \) and \( \angle ZXY \) are complementary to \( \angle XZY \), we can find \( m \angle ZXY \) by subtracting the sum of the other two angles from \( 180^\circ \):
   \[
   m \angle ZXY = 180^\circ - 90^\circ - 153^\circ = -63^\circ
   \]
   However, this result does not make sense in the context of a triangle, suggesting a mistake or contradiction in the setup.

### Conclusion:
It seems there's a contradiction here because angles in a triangle cannot add up to more than \( 180^\circ \), and \( m \angle ZYX = 153^\circ \) is already too large when added to \( 90^\circ \). Therefore, the provided algebraic expressions may need revisiting or reevaluation.

Would you like to go over this or have any further questions? Let me know how you'd like to proceed! Here are some related questions:

1. Can you explain why the sum of angles in a triangle must be \( 180^\circ \)?
2. How do complementary angles relate to the concept of a right angle?
3. What are the consequences if angles in a triangle do not add up to \( 180^\circ \)?
4. Can you derive the formula for calculating the third angle in a triangle if two angles are known?
5. What steps can be taken if a mathematical problem leads to an illogical conclusion?

**Tip:** Always double-check your results when calculating angles in geometric problems, especially in triangles, as they must adhere to specific rules such as the sum of angles being \( 180^\circ \).

Find the measures of the angles XZY, ZYX, and ZXY given the expressions m∠XZY = 3x + 6, m∠ZYX = 6x - 15, and that ∠XZY is a right angle.

Solve for the angles XZY, ZYX, and ZXY using algebraic expressions and the theorem that the sum of angles in a triangle equals 180°. This problem explores geometry and algebra for students in Grades 8-10.

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