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Sometimes houses are built on uneven surfaces, like a hill. What measure should the indicated angles be in order for the walls of this house to be vertical?

 

In this diagram, we see a house built on an uneven slope, and there are two angles marked as \( (5x + 30)^\circ \) and \( (11x - 10)^\circ \). These angles represent the angles between the walls of the house and the uneven surface.

For the walls of the house to be vertical, the two angles shown must sum to \( 90^\circ \), because the walls would form right angles with the ground.

We can set up the following equation to solve for \(x\):

\[
(5x + 30)^\circ + (11x - 10)^\circ = 90^\circ
\]

Now, let’s solve for \(x\):

1. Combine like terms:
\[
(5x + 11x) + (30 - 10) = 90
\]
\[
16x + 20 = 90
\]

2. Subtract 20 from both sides:
\[
16x = 70
\]

3. Divide both sides by 16:
\[
x = \frac{70}{16} = 4.375
\]

Now that we know \(x = 4.375\), we can find the measure of each angle:

1. \( 5x + 30 = 5(4.375) + 30 = 21.875 + 30 = 51.875^\circ \)
2. \( 11x - 10 = 11(4.375) - 10 = 48.125 - 10 = 38.125^\circ \)

Thus, the two angles are approximately:

- \( 51.875^\circ \)
- \( 38.125^\circ \)

These angles sum to \(90^\circ\), so the walls would indeed be vertical with these angle measures.

Do you have any questions or would you like further details?

Here are 5 related questions you might explore:
1. How would the angles change if the slope were steeper?
2. What would happen if the two angles didn't sum to 90°? Would the walls still be vertical?
3. Can you think of another situation where angles need to sum to 90° for vertical alignment?
4. How could you physically measure these angles on an actual building site?
5. Could the formula be adjusted if one angle is fixed and the other is variable?

**Tip:** When dealing with angles and slopes, always ensure that the sum of angles forms the required geometric configuration, like 90° for vertical alignment.

Sometimes houses are built on uneven surfaces, like a hill. What measure should the indicated angles be in order for the walls of this house to be vertical?

Explore how to calculate the measures of two angles formed by walls on a sloped surface to ensure the walls are vertical. This geometry problem involves solving for x using algebraic equations and angle sums. Suitable for Grades 7-10 students.

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