Question

using parallel lines to find angle measurement
in the figure, ( p parallel q ) with ( m angle 1 = 6 x + 39 ) and ( m angle 2 = 10 x - 3 ), first find ( x ) and then find ( m angle 2 ).
what is the relationship of ( angle 1 ) and ( angle 2 )?
which equation should you use to solve for ( x )?
( x = )
( m angle 2 = )

Explanation:

Step1: Determine the relationship between ∠1 and ∠2

Since \(p\parallel q\), ∠1 and ∠2 are alternate - interior angles. So, \(m\angle1 = m\angle2\).

Step2: Set up the equation

Given \(m\angle1=6x + 39\) and \(m\angle2 = 10x-3\).
Because \(m\angle1 = m\angle2\), we have the equation \(6x + 39=10x-3\).

Step3: Solve the equation for \(x\)

Subtract \(6x\) from both sides: \(39=10x - 3-6x\), which simplifies to \(39 = 4x-3\).
Add \(3\) to both sides: \(42=4x\).
Divide both sides by \(4\): \(x=\frac{42}{4}=\frac{21}{2}=10.5\).

Step4: Find \(m\angle2\)

Substitute \(x = 10.5\) into the formula for \(m\angle2\).
\(m\angle2=10x-3\), so \(m\angle2=10\times10.5-3\).
\(m\angle2 = 105 - 3=102\).

Answer:

The relationship between \(\angle1\) and \(\angle2\) is \(m\angle1 = m\angle2\). The equation to solve for \(x\) is \(6x + 39=10x-3\). \(x = 10.5\) and \(m\angle2=102\).