Question

1. a fence on a hill uses vertical posts ( l ) and ( m ) to hold parallel rails ( n ) and ( p ). if ( mangle1 = 6x ) and ( mangle12 = 4x ), then what is the measure of ( angle5 )?

Explanation:

Step1: Use the property of parallel lines and transversal

Since \(L\) and \(M\) are vertical (parallel) and \(N\) and \(P\) are parallel, \(\angle1\) and \(\angle12\) are same - side interior angles. So \(m\angle1 + m\angle12=180^{\circ}\) (same - side interior angles are supplementary).
Given \(m\angle1 = 6x\) and \(m\angle12 = 4x\), we have \(6x+4x = 180\).

Step2: Solve for \(x\)

Combine like terms: \(10x=180\).
Divide both sides by \(10\): \(x = 18\).

Step3: Find \(m\angle12\)

Substitute \(x = 18\) into \(m\angle12=4x\). So \(m\angle12=4\times18 = 72^{\circ}\).

Step4: Use the property of vertical angles

\(\angle5\) and \(\angle12\) are vertical angles. Vertical angles are congruent. So \(m\angle5=m\angle12\).

Answer:

\(72^{\circ}\)