Question

1. given: (l_1parallel l_2), (mangle1=(9x + 13)^{circ}), and (mangle6=(5x - 1)^{circ}). find the (mangle2).

a. solve for x.
b. (mangle2=)

1. find the length of the third side. if necessary, write your answer in the simplest radical form.

Explanation:

Step1: Use corresponding - angles property

Since \(l_1\parallel l_2\), \(\angle1\) and \(\angle5\) are corresponding angles and \(\angle5\) and \(\angle6\) are supplementary (\(\angle5+\angle6 = 180^{\circ}\)), also \(\angle1\) and \(\angle5\) are equal. So \(\angle1+\angle6=180^{\circ}\). We set up the equation \((9x + 13)+(5x-1)=180\).
\[9x + 13+5x-1=180\]

Step2: Combine like - terms

Combine the \(x\) terms and the constant terms: \((9x + 5x)+(13 - 1)=180\), which simplifies to \(14x+12 = 180\).
\[14x+12=180\]

Step3: Isolate the variable \(x\)

Subtract 12 from both sides: \(14x=180 - 12=168\). Then divide both sides by 14: \(x=\frac{168}{14}=12\).
\[x = 12\]

Step4: Find \(\angle1\)

Substitute \(x = 12\) into the expression for \(\angle1\): \(m\angle1=(9x + 13)^{\circ}=(9\times12 + 13)^{\circ}=(108+13)^{\circ}=121^{\circ}\).
\[m\angle1 = 121^{\circ}\]

Step5: Find \(\angle2\)

Since \(\angle1\) and \(\angle2\) are supplementary (\(\angle1+\angle2 = 180^{\circ}\)), then \(m\angle2=180 - m\angle1\). Substitute \(m\angle1 = 121^{\circ}\) to get \(m\angle2 = 59^{\circ}\).
\[m\angle2=59^{\circ}\]

Answer:

a. \(x = 12\)
b. \(m\angle2 = 59^{\circ}\)