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.