if line BD ⊥ line AC, angle DBE = 2x-1 and angle CBE = 5x-42, find the value of x

Given that line \( BD \perp AC \), this implies that angle \( DBE \) and angle \( CBE \) are adjacent angles that form a right angle at point \( B \). Thus, the sum of these two angles is \( 90^\circ \).

We have:
- Angle \( DBE = 2x - 1 \)
- Angle \( CBE = 5x - 42 \)

Since \( BD \perp AC \), we can set up the following equation:

\[
(2x - 1) + (5x - 42) = 90
\]

Let's solve for \( x \):

\[
7x - 43 = 90
\]

Now, add 43 to both sides:

\[
7x = 133
\]

Divide by 7:

\[
x = 19
\]

The value of \( x \) is **19**.

Would you like to see a detailed step-by-step solution or have any other questions? Here are a few related questions:

1. What are complementary angles and how are they used in geometry?
2. How do you solve equations involving angles formed by perpendicular lines?
3. What is the relationship between perpendicular lines and right angles?
4. How to verify if two lines are perpendicular given their angles?
5. How can trigonometry be used to find angles in perpendicular lines?

**Tip:** Remember that when two lines are perpendicular, the angles formed at their intersection are right angles (90 degrees).

Learn how to find the value of x when given angles formed by perpendicular lines. Solve the equation involving adjacent angles formed by line BD perpendicular to line AC. Suitable for high school geometry students.

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