Question

how can you express each of the numbered angles in terms of x? ∠6 and ∠x, ∠8 and ∠x, ∠3 and ∠6, ∠4 and ∠7 ∠7 and ∠x ∠7 and ∠2 ∠3 and ∠7 ∠1 and ∠8

Explanation:

Step1: Identify angle - relationships

Assume the two lines \(l\) and \(m\) are parallel and \(n\) is a transversal.

Step2: Find \(\angle6\) in terms of \(\angle x\)

\(\angle6\) and \(\angle x\) are corresponding angles. So, \(\angle6=\angle x\).

Step3: Find \(\angle8\) in terms of \(\angle x\)

\(\angle8\) and \(\angle x\) are vertical - angles. So, \(\angle8 = \angle x\).

Step4: Find \(\angle3\) in terms of \(\angle6\)

\(\angle3\) and \(\angle6\) are alternate - interior angles. So, \(\angle3=\angle6\). Since \(\angle6 = \angle x\), \(\angle3=\angle x\).

Step5: Find \(\angle4\) in terms of \(\angle7\)

\(\angle4\) and \(\angle7\) are non - adjacent exterior angles. \(\angle4+\angle7 = 180^{\circ}\), so \(\angle4=180^{\circ}-\angle7\).

Step6: Find \(\angle7\) in terms of \(\angle x\)

\(\angle7\) and \(\angle x\) are supplementary (linear - pair). So, \(\angle7 = 180^{\circ}-\angle x\).

Step7: Find \(\angle7\) in terms of \(\angle2\)

\(\angle7\) and \(\angle2\) are corresponding angles. So, \(\angle7=\angle2\).

Step8: Find \(\angle3\) in terms of \(\angle7\)

\(\angle3\) and \(\angle7\) are supplementary (same - side interior angles). So, \(\angle3=180^{\circ}-\angle7\).

Step9: Find \(\angle1\) in terms of \(\angle8\)

\(\angle1\) and \(\angle8\) are alternate exterior angles. So, \(\angle1=\angle8\). Since \(\angle8=\angle x\), \(\angle1=\angle x\).

Answer:

\(\angle6=\angle x\), \(\angle8=\angle x\), \(\angle3=\angle x\), \(\angle4 = 180^{\circ}-\angle7\), \(\angle7=180^{\circ}-\angle x\), \(\angle7=\angle2\), \(\angle3=180^{\circ}-\angle7\), \(\angle1=\angle x\)