Let's break down the problem:

We are given:

1. $m\angle WXZ = (5x + 3)^\circ$
2. $m\angle ZXY = (8x - 4)^\circ$
3. $\angle WXY$ is a right angle, meaning $m\angle WXY = 90^\circ$.

The diagram shows that:

• $\angle WXY$ is the sum of $\angle WXZ$ and $\angle ZXY$, since they form a straight line at $X$.

So, we can set up the equation: $m\angle WXZ + m\angle ZXY = m\angle WXY$ Substitute the expressions: $(5x + 3) + (8x - 4) = 90$ Now, let's solve for $x$.

1. Combine like terms: $5x + 8x + 3 - 4 = 90$ $13x - 1 = 90$

2. Add 1 to both sides: $13x = 91$

3. Solve for $x$: $x = \frac{91}{13} = 7$

Now that we have $x = 7$, we can find the measures of $\angle WXZ$ and $\angle ZXY$.

• $m\angle WXZ = 5x + 3 = 5(7) + 3 = 35 + 3 = 38^\circ$
• $m\angle ZXY = 8x - 4 = 8(7) - 4 = 56 - 4 = 52^\circ$

Therefore:

• $x = 7$
• $m\angle WXZ = 38^\circ$
• $m\angle ZXY = 52^\circ$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How can you verify the solution using the sum of angles?
2. What happens if $\angle WXY$ was not a right angle?
3. Can you find the value of $x$ if $\angle WXZ$ and $\angle ZXY$ were given as actual measures?
4. What is the relationship between adjacent angles that form a straight line?
5. How would the problem change if a different expression for $m\angle ZXY$ were given?

Tip: When dealing with angles in geometric figures, remember that angles on a straight line always add up to 180°, and angles in a triangle sum to 180°.