Question

7 find the measure of each angle.
(7x+6)°
(9x+3)°

Explanation:

Step1: Identify Angle Relationship

The two angles \((7x + 6)^\circ\) and \((9x + 3)^\circ\) are alternate exterior angles (since lines \(m\) and \(n\) are parallel, cut by a transversal). Alternate exterior angles are equal. So, set the expressions equal: \(7x + 6 = 9x + 3\).

Step2: Solve for \(x\)

Subtract \(7x\) from both sides: \(6 = 2x + 3\).
Subtract \(3\) from both sides: \(3 = 2x\).
Divide by \(2\): \(x=\frac{3}{2}=1.5\)? Wait, no, wait—wait, maybe they are same - side exterior? Wait, no, the diagram: if lines \(m\) and \(n\) are parallel, and the transversal, maybe they are alternate interior? Wait, no, the angles given: \((7x + 6)\) and \((9x + 3)\) — maybe they are equal (alternate exterior). Wait, let's re - solve:

\(7x+6 = 9x + 3\)
\(6 - 3=9x - 7x\)
\(3 = 2x\)
\(x=\frac{3}{2}\)? That seems odd. Wait, maybe they are supplementary? Wait, maybe I misidentified the angle relationship. If the lines are parallel, and the angles are same - side exterior, they would be supplementary. Let's check:

If \((7x + 6)+(9x + 3)=180\)
\(16x+9 = 180\)
\(16x = 171\)
\(x=\frac{171}{16}\approx10.6875\). Wait, the diagram is a bit unclear, but assuming alternate exterior (equal) first:

Wait, maybe the angles are \((7x + 6)\) and \((9x + 3)\) as alternate exterior, so:

\(7x+6 = 9x + 3\)
\(9x-7x=6 - 3\)
\(2x = 3\)
\(x = 1.5\). Then angle 1: \(7(1.5)+6=10.5 + 6 = 16.5^\circ\), angle 2: \(9(1.5)+3 = 13.5+3 = 16.5^\circ\). But that seems too small. Maybe the angles are same - side interior? Wait, no, the problem says "find the measure of each angle". Wait, maybe the correct relationship is that they are equal (alternate exterior). Wait, perhaps I made a mistake in the relationship. Let's assume that the two angles are equal (alternate exterior angles):

Step1 (corrected): Set up equation

\(7x + 6=9x + 3\)

Step2 (corrected): Solve for \(x\)

\(9x - 7x=6 - 3\)
\(2x = 3\)
\(x=\frac{3}{2}=1.5\)

Step3: Find angle measures

For \((7x + 6)^\circ\): \(7\times1.5+6 = 10.5 + 6=16.5^\circ\)
For \((9x + 3)^\circ\): \(9\times1.5+3 = 13.5 + 3 = 16.5^\circ\)

But this seems unusual. Alternatively, if the angles are supplementary (same - side exterior angles):

Step1: Set up supplementary equation

\((7x + 6)+(9x + 3)=180\)

Step2: Solve for \(x\)

\(16x+9 = 180\)
\(16x=180 - 9 = 171\)
\(x=\frac{171}{16}=10.6875\)

Step3: Find angle measures

\(7x + 6=7\times10.6875+6 = 74.8125+6 = 80.8125^\circ\)
\(9x + 3=9\times10.6875+3 = 96.1875+3 = 99.1875^\circ\)
And \(80.8125+99.1875 = 180^\circ\), which makes sense for same - side exterior angles.

Wait, the confusion is from the diagram. Since the lines are parallel, alternate exterior angles are equal, same - side exterior are supplementary. Given that the angles are on the outside of the two parallel lines, if the transversal crosses them, alternate exterior are equal, same - side are supplementary. Let's check the problem again. The diagram shows two parallel lines \(m\) and \(n\), and a transversal creating two angles \((7x + 6)^\circ\) and \((9x + 3)^\circ\) on the outside. So, if they are on the same side of the transversal, they are supplementary; if on opposite sides, equal.

Assuming same - side exterior (supplementary) is more likely for non - trivial angles:

So, equation: \(7x + 6+9x + 3 = 180\)
\(16x+9 = 180\)
\(16x=171\)
\(x=\frac{171}{16}=10.6875\)

Then, angle 1: \(7\times10.6875 + 6=74.8125+6 = 80.8125^\circ\)
Angle 2: \(9\times10.6875+3 = 96.1875+3 = 99.1875^\circ\)

But maybe the correct relationship is equal (alternate exterior). Let's re - examine the diagram. If the transversal is crossing the two parallel lines, and the angles are on opposite sides of the transversal, outside the parallel lines, they are alternate exterior and equal.

So, \(7x + 6=9x + 3\)
\(2x = 3\)
\(x = 1.5\)

Then angles are \(16.5^\circ\) each.

Since the problem is about angle measures with parallel lines and transversal, the subfield is Geometry (Mathematics).

Answer:

If alternate exterior (equal): Angles are \(16.5^\circ\) and \(16.5^\circ\).
If same - side exterior (supplementary): Angles are \(80.8125^\circ\) and \(99.1875^\circ\).

(Assuming alternate exterior angles, the answer is \(16.5^\circ\) for each angle. But likely, the correct relationship is supplementary, so angles are \(80.8125^\circ\) and \(99.1875^\circ\))