https://file.aiquickdraw.com/mathbot/file/1726012133022h2ml10i8.jpg

In this diagram, we are tasked with solving for the value of \( w \). The angles given are:

- \( (4w - 7)^\circ \)
- \( (7w - 2)^\circ \)
- A right angle, which is \( 90^\circ \).

Since these three angles form a straight line, they must add up to \( 180^\circ \). Thus, we can set up the following equation:

\[
(4w - 7)^\circ + (7w - 2)^\circ + 90^\circ = 180^\circ
\]

Now, let's solve for \( w \).

1. Combine all the terms:
\[
(4w - 7) + (7w - 2) + 90 = 180
\]
\[
4w + 7w - 7 - 2 + 90 = 180
\]
\[
11w + 81 = 180
\]

2. Subtract 81 from both sides:
\[
11w = 99
\]

3. Divide by 11:
\[
w = 9
\]

Thus, the value of \( w \) is \( \boxed{9} \).

Let me know if you want further details or have any questions!

Here are five related questions:
1. What is the value of \( 4w - 7 \) when \( w = 9 \)?
2. What is the value of \( 7w - 2 \) when \( w = 9 \)?
3. What other types of angle pairs sum to \( 180^\circ \) besides angles on a straight line?
4. How do complementary and supplementary angles differ?
5. What are alternate interior angles in parallel line geometry?

**Tip:** Always check your angle relationships (like supplementary, complementary, or angles in a triangle) when solving for unknowns in geometric diagrams.

Solve for the value of w. The angles given are (4w - 7)° and (7w - 2)°, and there is a right angle between them.

Learn how to solve for the value of w in an angle relationship problem where the given angles are (4w - 7)° and (7w - 2)°. The problem involves key concepts from geometry and algebra, suitable for students in Grades 8-10.

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