Answer key for a worksheet on interior and exterior angles, displaying eight problems with geometric diagrams and solutions.

Looking at problem 8:

We have two parallel lines cut by a transversal. The angle labeled 73° and the angle labeled x are on the same side of the transversal, but one is inside the parallel lines (interior) and one is outside (exterior). Actually, let’s look more carefully.

The 73° angle is an interior angle on the bottom left. The angle x is an exterior angle on the top right — they are not directly corresponding or alternate.

Actually, notice that x and the 73° angle are consecutive interior angles? No — wait, x is above the top line, so it's an exterior angle. Let me think differently.

Better approach: Find the angle that is vertically opposite or supplementary to 73°, then relate it to x.

Actually, the 73° angle and the angle adjacent to x (on the same side of the transversal, between the parallels) are consecutive interior angles — which add up to 180°.

Wait — no. Let’s label mentally:

At the bottom intersection: the 73° angle is given. Its vertical angle is also 73°. The angle next to it (same side, linear pair) is 180° - 73° = 107°.

Now, that 107° angle is in the same position relative to the top line as angle x — meaning they are corresponding angles. Since the lines are parallel, corresponding angles are equal.

Therefore, x = 107°.

Alternatively: The 73° angle and x are on opposite sides of the transversal, but one is exterior and one is interior — actually, they form a “Z” shape? Not quite.

Another way: The angle adjacent to x (inside the parallel lines, on the same side as 73°) would be 180° - x. That angle and the 73° angle are consecutive interior angles → sum to 180°.

So: (180° - x) + 73° = 180°
→ 253° - x = 180°
→ x = 253° - 180° = 73°? Wait, that can’t be — because in the answer key it says 107°.

I made a mistake here.

Let me redraw mentally:

Top line: angle x is above the line, on the right side of the transversal.

Bottom line: 73° is below the line, on the left side of the transversal.

Actually, the 73° angle and the angle that corresponds to x are... let’s find the corresponding angle to x.

Angle x is at the top, above the line, on the right. The corresponding angle at the bottom would be below the bottom line, on the right — which is NOT the 73° angle. The 73° angle is on the left.

So instead, the 73° angle and the angle that is vertically opposite to the corresponding angle of x might help.

Easier: The 73° angle and the angle directly across from it (vertical angle) is also 73°. Now, that vertical angle is on the bottom left, inside the parallel lines.

The angle x is on the top right, outside. These two angles (73° and x) are on opposite sides of the transversal and both are "outside" relative to the space between the parallels? Not exactly.

Actually, they are alternate exterior angles? Let’s see:

Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines.

Here, 73° is below the bottom line (so exterior), on the left.

x is above the top line (exterior), on the right.

Yes! So they are alternate exterior angles. And for parallel lines, alternate exterior angles are equal.

But if that were true, x should be 73° — but the answer key says 107°. Contradiction.

Wait — I think I misidentified the positions.

Look again at the diagram description: In problem 8, the 73° is marked at the bottom intersection, on the lower-left side (between the transversal and the bottom line, below the parallel line). Angle x is at the top intersection, on the upper-right side (above the top line, to the right of the transversal).

Actually, these are NOT alternate exterior angles. Alternate exterior would be: one above top line on left, one below bottom line on right — or vice versa.

Here, 73° is below bottom line on left; x is above top line on right — so they are on the same side of the transversal? No — left vs right.

Transversal goes diagonally. Assume standard orientation: transversal slopes down to the right.

Then:

- At bottom intersection: 73° is the angle between the transversal and the bottom line, on the left side — so it’s the acute angle if 73° is acute.

- At top intersection: x is the angle between the transversal and the top line, on the right side — which would be the obtuse angle if the lines are slanted that way.

Actually, perhaps they are supplementary.

Note that the angle adjacent to 73° (on the same side, forming a straight line) is 180° - 73° = 107°. This 107° angle is now on the bottom right, inside the parallel lines.

This 107° angle and angle x are corresponding angles: both are on the right side of the transversal, and both are above their respective parallel lines? Wait, no — the 107° is below the top line? Let's clarify.

Standard rule: When two parallel lines are cut by a transversal, corresponding angles are equal.

The angle that corresponds to x (which is above the top line, on the right) is the angle above the bottom line, on the right. But we don't have that labeled.

Instead, the 73° angle is below the bottom line, on the left. Its corresponding angle would be below the top line, on the left — which is not x.

Perhaps use the fact that the 73° angle and the angle that is vertically opposite to the co-interior angle with x.

Best method: The angle that is adjacent to x (on the same side, between the parallels) is supplementary to x. Call that angle y. So x + y = 180°.

Now, y and the 73° angle are consecutive interior angles (same side of transversal, between the parallels). For parallel lines, consecutive interior angles are supplementary: y + 73° = 180°.

Therefore, y = 180° - 73° = 107°.

Then, since x + y = 180°, x = 180° - y = 180° - 107° = 73°? Again 73°, but answer key says 107°.

I see the mistake! If y is the angle between the parallels on the same side as 73°, and 73° is also between the parallels? No.

Let's define clearly:

Assume the two horizontal lines are parallel, transversal cuts them.

At the bottom intersection:
- The 73° angle is shown. From the diagram description, it's likely the angle that is on the lower-left, meaning it's the angle between the transversal and the bottom line, measured from the bottom line up to the transversal on the left side. So this angle is inside the region between the two parallel lines? Or outside?

In standard notation, when we say "angle at intersection", if it's marked as 73° and it's in the corner, we need to know if it's interior or exterior.

From the answer key, x=107°, and 73°+107°=180°, so they are supplementary.

Why would they be supplementary? Because they are consecutive interior angles? But x is exterior.

Unless... in the diagram, the 73° angle is actually the interior angle, and x is the exterior angle on the same side, but for parallel lines, the exterior angle on one side equals the interior angle on the other side only if... I'm confusing myself.

Let me think of a different strategy. In such problems, often the angle x and the given angle are either equal or supplementary based on their positions.

Given that in the answer key it's 107°, and 180° - 73° = 107°, so likely x = 180° - 73°.

Why? Because the 73° angle and the angle that is vertically opposite to the corresponding angle of x are the same, but perhaps they are on a straight line.

Notice that the 73° angle and the angle that is adjacent to x on the top line might be related.

Actually, the correct reasoning is:

The 73° angle and the angle that is on the same side of the transversal but at the top intersection, inside the parallels, are consecutive interior angles, so they sum to 180°. So that interior angle at the top is 180° - 73° = 107°.

Now, angle x is vertically opposite to that 107° angle. Because vertically opposite angles are equal, so x = 107°.

Yes! That makes sense.

Confirming:

- At the bottom: 73° is given. Assume it's the interior angle on the left side.

- Then the consecutive interior angle at the top (also on the left side, between the parallels) is 180° - 73° = 107°.

- But angle x is not that angle; x is on the right side, above the top line.

The angle that is vertically opposite to the 107° interior angle at the top would be the angle below the top line, on the right side — which is not x.

Perhaps the 107° angle at the top (interior, left) has a corresponding angle at the bottom, but we already used that.

Let's consider the angle that is alternate interior to the 73° angle.

If 73° is interior on the bottom left, then the alternate interior angle is at the top right, inside the parallels. That angle would be equal to 73°.

Then, angle x is adjacent to that 73° angle on the top line, forming a straight line. So x + 73° = 180°, thus x = 107°.

Yes! That must be it.

Step-by-step:

1. Identify that the 73° angle is an interior angle on the bottom left.

2. The alternate interior angle to it is on the top right, inside the parallel lines. Since lines are parallel, alternate interior angles are equal, so that angle is also 73°.

3. Angle x is on the same vertex as that 73° angle, but on the other side of the transversal, and above the top line. Specifically, x and the 73° angle (alternate interior) are adjacent angles that form a straight line (linear pair).

4. Therefore, x + 73° = 180°.

5. So x = 180° - 73° = 107°.

Perfect. This matches the answer key.

Final Answer: 107°