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We are given the following information from the image:

1. \( m \angle XYZ = 117^\circ \).
2. \( \angle XYW = (6x + 44)^\circ \).
3. \( \angle WYZ = (-10x + 65)^\circ \).

We need to find:
- \( m \angle XYW \)
- \( m \angle WYZ \).

### Step 1: Use the fact that the angles form a straight line.

Since the angles \( \angle XYW \), \( \angle WYZ \), and \( \angle XYZ \) are on a straight line, their sum should equal \( 180^\circ \):
\[
m \angle XYW + m \angle WYZ = 180^\circ - m \angle XYZ
\]
Substitute the given value \( m \angle XYZ = 117^\circ \):
\[
m \angle XYW + m \angle WYZ = 180^\circ - 117^\circ = 63^\circ
\]
Now we know that:
\[
(6x + 44)^\circ + (-10x + 65)^\circ = 63^\circ
\]

### Step 2: Simplify the equation.
Combine like terms:
\[
(6x - 10x) + (44 + 65) = 63
\]
\[
-4x + 109 = 63
\]

### Step 3: Solve for \( x \).
Subtract 109 from both sides:
\[
-4x = 63 - 109
\]
\[
-4x = -46
\]
Divide by -4:
\[
x = \frac{-46}{-4} = 11.5
\]

### Step 4: Find \( m \angle XYW \) and \( m \angle WYZ \).
Now substitute \( x = 11.5 \) into the expressions for \( \angle XYW \) and \( \angle WYZ \).

For \( m \angle XYW \):
\[
m \angle XYW = 6x + 44 = 6(11.5) + 44 = 69 + 44 = 113^\circ
\]

For \( m \angle WYZ \):
\[
m \angle WYZ = -10x + 65 = -10(11.5) + 65 = -115 + 65 = -50^\circ
\]

### Step 5: Verify the solution.
Check if the sum of \( m \angle XYW \) and \( m \angle WYZ \) equals 63°:
\[
113^\circ + (-50^\circ) = 63^\circ
\]
The solution checks out!

### Final Answers:
- \( m \angle XYW = 113^\circ \)
- \( m \angle WYZ = -50^\circ \)

Would you like any further explanation or details?

Here are 5 related questions:
1. How do we know that the sum of the angles equals \( 180^\circ \)?
2. Can we solve the same problem using a different method?
3. What happens if the angle \( m \angle WYZ \) turns out to be negative?
4. How do we interpret the negative value of \( m \angle WYZ \)?
5. What if the given angle \( m \angle XYZ \) was different? How would the equations change?

Tip: Always check the sum of angles to ensure your solution is correct when dealing with angles on a straight line.

Find m∠XYW and m∠WYZ given m∠XYZ = 117° and expressions for m∠XYW = (6x + 44)° and m∠WYZ = (-10x + 65)°.

In this problem, we calculate the values of m∠XYW and m∠WYZ using algebra and the concept that angles on a straight line sum to 180°. The expressions for the angles involve solving a linear equation, suitable for students in Grades 8-10.

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