Question

in the picture, line a is parallel to line b.
$m\angle1 = x^{2}+7x$ and $m\angle3 = 2x + 36$
$x =
$
$\angle1=
$
$\angle7=
$

Explanation:

Step1: Identify angle - relationship

Since line \(a\) is parallel to line \(b\), \(\angle1\) and \(\angle3\) are vertical - angles, so \(m\angle1 = m\angle3\).
\[x^{2}+7x=2x + 36\]

Step2: Rearrange to quadratic - form

Move all terms to one side to get a quadratic equation:
\[x^{2}+7x-2x - 36=0\]
\[x^{2}+5x - 36=0\]

Step3: Factor the quadratic equation

Factor \(x^{2}+5x - 36\):
\((x + 9)(x - 4)=0\)

Step4: Solve for \(x\)

Set each factor equal to zero:
If \(x+9 = 0\), then \(x=-9\); if \(x - 4=0\), then \(x = 4\).
We consider the non - negative value for the context of angle measures, so \(x = 4\).

Step5: Find \(m\angle1\)

Substitute \(x = 4\) into the expression for \(m\angle1\):
\[m\angle1=x^{2}+7x=4^{2}+7\times4=16 + 28=44^{\circ}\]

Step6: Find \(m\angle7\)

\(\angle1\) and \(\angle7\) are alternate exterior angles. Since \(a\parallel b\), \(m\angle1=m\angle7\), so \(m\angle7 = 44^{\circ}\)

Answer:

\(x = 4\)
\(\angle1=44^{\circ}\)
\(\angle7=44^{\circ}\)