Question

1. if ( overline{bd} perp overline{ac} ), ( mangle dbe = (2x - 1)^circ ), and ( mangle cbe = (5x - 42)^circ ), find the value of ( x ).

Explanation:

Step1: Identify angle relationship

Since \( BD \perp AC \), \( \angle CBD = 90^\circ \). Also, \( \angle CBE=\angle CBD + \angle DBE \), so \( m\angle CBE = 90^\circ+m\angle DBE \).
Substitute \( m\angle DBE=(2x - 1)^\circ \) and \( m\angle CBE=(5x - 42)^\circ \):
\( 5x - 42=90+(2x - 1) \)

Step2: Solve the equation

Simplify the right - hand side: \( 5x - 42=90 + 2x-1=89 + 2x \)
Subtract \( 2x \) from both sides: \( 5x-2x - 42=89+2x - 2x \), which gives \( 3x - 42 = 89 \)
Add 42 to both sides: \( 3x-42 + 42=89 + 42 \), so \( 3x=131 \)? Wait, no, wait. Wait, maybe I made a mistake in the angle relationship. Wait, looking at the diagram, \( \angle CBE\) and \( \angle DBE\): since \( BD\perp AC \), \( \angle CBD = 90^\circ \), and \( \angle CBE+\angle DBE = 90^\circ \)? Wait, no, the diagram: points A, B, C are colinear, D and E are on the other lines. Wait, actually, \( BD\perp AC \), so \( \angle ABD=\angle CBD = 90^\circ \). And \( \angle DBE\) and \( \angle CBE \): let's re - examine. The sum of \( \angle DBE\) and \( \angle CBE \) should be equal to \( \angle CBD \)? No, wait, maybe \( \angle CBE+\angle DBE = 90^\circ \)? Wait, no, the correct relationship: since \( BD\perp AC \), \( \angle CBD = 90^\circ \), and \( \angle CBE\) and \( \angle DBE \) are angles such that \( \angle CBE+\angle DBE=\angle CBD = 90^\circ \)? Wait, no, the original problem: \( m\angle DBE=(2x - 1)^\circ \), \( m\angle CBE=(5x - 42)^\circ \), and \( BD\perp AC \), so \( \angle CBE+\angle DBE = 90^\circ \)? Wait, no, maybe I had the relationship wrong. Wait, let's start over.

Since \( BD\perp AC \), \( \angle CBD = 90^\circ \). Also, \( \angle CBE\) and \( \angle DBE \): if we look at the lines, \( \angle CBE=\angle CBD+\angle DBE \)? No, that would be more than 90. Wait, no, maybe \( \angle CBE-\angle DBE = 90^\circ \)? Wait, the correct approach: from the diagram, \( BD\perp AC \), so \( \angle ABD=\angle CBD = 90^\circ \). The angle \( \angle CBE \) and \( \angle DBE \): let's assume that \( \angle CBE = 90^\circ+\angle DBE \) is wrong. Wait, let's use the fact that \( \angle CBE\) and \( \angle DBE \) are related such that \( \angle CBE+\angle DBE = 90^\circ \)? No, that can't be. Wait, the problem says \( BD\perp AC \), so \( \angle CBD = 90^\circ \). Let's write the equation correctly.

Wait, the correct equation: \( m\angle CBE=m\angle CBD + m\angle DBE \), but \( m\angle CBD = 90^\circ \), so \( 5x-42=90+(2x - 1) \) is wrong. Wait, no, maybe \( \angle CBE\) and \( \angle DBE \) are complementary to the right angle? Wait, no, let's look at the diagram again. The lines: AC is a straight line, BD is perpendicular to AC, so BD is vertical (assuming AC is horizontal). Then BE is a line that makes angles \( \angle DBE=(2x - 1)^\circ \) and \( \angle CBE=(5x - 42)^\circ \) with BD and BC respectively. So \( \angle CBE+\angle DBE = \angle CBD = 90^\circ \)? Wait, that would mean \( (5x - 42)+(2x - 1)=90 \)

Let's solve that:

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

Combine like terms: \( 7x-43 = 90 \)

Add 43 to both sides: \( 7x=90 + 43=133 \)

Divide by 7: \( x = 19 \)

Wait, that makes sense. I made a mistake in the angle relationship earlier. The sum of \( \angle CBE\) and \( \angle DBE \) is equal to \( \angle CBD \) which is \( 90^\circ \) because \( BD\perp AC \). So the correct equation is \( (5x - 42)+(2x - 1)=90 \)

Step1: Set up the equation

Since \( BD\perp AC \), \( \angle CBD = 90^\circ \). And \( \angle CBE+\angle DBE=\angle CBD \), so:
\( (5x - 42)+(2x - 1)=90 \)

Step2: Simplify the left - hand side

Combine like terms: \( 5x+2x-42 - 1=90 \), which is \( 7x-43 = 90 \)

Step3: Solve for x

Add 43 to both sides: \( 7x=90 + 43=133 \)
Divide both sides by 7: \( x=\frac{133}{7}=19 \)

Answer:

\( x = 19 \)