Geometry worksheet on parallel lines cut by a transversal, featuring angle calculations and relationship identification.

Let’s solve this step by step.

We are told that lines z and y are parallel, and they are cut by two transversals: line a and line b.

We’re given:
- At the top left intersection (line b crossing line y), one angle is 74°.
- At the bottom left intersection (line a crossing line z), one angle is 39°.

We need to find all missing angles from ∠1 to 18, and then name relationships for three angle pairs.

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Step 1: Understand vertical angles and linear pairs

When two lines cross, opposite angles (vertical angles) are equal. Adjacent angles on a straight line add up to 180° (linear pair).

Also, when parallel lines are cut by a transversal:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Same-side interior angles add to 180°.
- Vertical angles are always equal.
- Linear pairs add to 180°.

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Step 2: Start with known angles

#### Top-left intersection (line b crosses line y):

Given: one angle = 74° → let’s say it’s the top-left angle at that intersection.

Then:
- ∠1 = vertical angle to 74° → so ∠1 = 74°
- ∠2 = adjacent to 74° → 180° - 74° = 106°
- ∠3 = vertical to ∠2 → ∠3 = 106°

Wait — actually, looking at the diagram layout:

At the top-left intersection (line b and line y):

The 74° is labeled in the upper-left quadrant.

So:
- ∠1 is upper-right → adjacent to 74° → so ∠1 = 180° - 74° = 106°
- ∠2 is lower-left → vertical to 74° → ∠2 = 74°
- ∠3 is lower-right → vertical to ∠1 → ∠3 = 106°

Wait — let me double-check based on standard labeling.

Actually, in most diagrams like this, angles are labeled around the intersection point.

Assuming the 74° is the angle between line b and line y in the top-left corner.

Then:
- The angle directly opposite (vertical) is also 74° → that would be ∠2 if it's labeled in the bottom-left.
- The adjacent angles (top-right and bottom-left) would be 180° - 74° = 106°.

Looking at the labels:

∠1 is top-right → so ∠1 = 106°
∠2 is bottom-left → vertical to 74° → ∠2 = 74°
∠3 is bottom-right → vertical to ∠1 → ∠3 = 106°

But wait — the problem says “74°” is written near the top-left, and ∠1, ∠2, ∠3 are labeled around that intersection.

Typically, in such diagrams:

- ∠1 and ∠3 are vertical? No — usually ∠1 and ∠3 are adjacent or opposite depending on labeling.

Actually, let’s think differently.

In the top-left intersection (lines b and y):

There are four angles formed. One is given as 74°. Let’s assume that’s the angle in the northwest position.

Then:
- Northeast angle (∠1) = 180° - 74° = 106° (because they form a linear pair)
- Southwest angle (∠2) = 74° (vertical to given angle)
- Southeast angle (∠3) = 106° (vertical to ∠1)

Yes, that makes sense.

So:
→ ∠1 = 106°
→ ∠2 = 74°
→ ∠3 = 106°

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#### Bottom-left intersection (line a crosses line z):

Given: 39° in the bottom-left corner.

So:
- ∠12 is top-left → adjacent to 39° → 180° - 39° = 141°
- ∠13 is top-right → vertical to 39° → ∠13 = 39°
- ∠14 is bottom-right → vertical to ∠12 → ∠14 = 141°

Wait — again, check labeling.

If 39° is in the bottom-left, then:

- Top-left (∠12) = 180° - 39° = 141°
- Top-right (∠13) = vertical to 39° → 39°
- Bottom-right (∠14) = vertical to ∠12 → 141°

Yes.

So:
→ ∠12 = 141°
→ ∠13 = 39°
→ ∠14 = 141°

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Step 3: Use parallel lines to find other angles

Lines y and z are parallel.

Transversal a cuts them.

At top-right intersection (line a and line y):

We can use corresponding angles.

∠13 is 39° (at bottom-right of bottom intersection). Since line a is transversal, and y || z, then the corresponding angle at top-right (which is ∠4) should be equal to ∠13? Wait.

Actually, ∠13 is at the bottom intersection, on the right side, above line z.

Corresponding angle at top intersection would be ∠4 — which is on the right side, above line y.

Since y || z, and transversal a, then ∠4 corresponds to ∠13 → so ∠4 = ∠13 = 39°

Similarly:

∠5 is vertical to ∠4 → ∠5 = 39°
∠6 is adjacent to ∠4 → 180° - 39° = 141°
∠7 is vertical to ∠6 → ∠7 = 141°

Wait — let’s confirm:

At top-right intersection (line a and line y):

If ∠4 is top-right, and we said it corresponds to ∠13 (which is 39°), then yes.

But ∠13 is at the bottom, top-right of the bottom intersection? Actually, ∠13 is labeled at the bottom intersection, top-right.

Standard correspondence: when transversal crosses two parallel lines, corresponding angles are in same relative position.

So:

- ∠13 (bottom intersection, top-right) corresponds to ∠4 (top intersection, top-right) → so ∠4 = 39°

Then:
- ∠5 = vertical to ∠4 → 39°
- ∠6 = adjacent to ∠4 → 180° - 39° = 141°
- ∠7 = vertical to ∠6 → 141°

Good.

Now, transversal b cuts lines y and z.

At top-left, we have angles from before.

At bottom-right intersection (line b and line z):

We can use corresponding angles.

At top-left intersection (line b and y), we had:

∠2 = 74° (bottom-left)

Corresponding angle at bottom intersection would be ∠15? Let’s see.

∠2 is at top intersection, bottom-left.

Corresponding angle at bottom intersection would be bottom-left → which is ∠15? Looking at diagram, ∠15 is at bottom intersection, top-left? Wait.

Actually, at bottom intersection (line b and z), the angles are ∠15, ∠16, ∠17, ∠18.

Assuming ∠15 is top-left, ∠16 top-right, ∠17 bottom-left, ∠18 bottom-right.

Then, corresponding to ∠2 (which is bottom-left at top intersection) would be ∠17 (bottom-left at bottom intersection).

Since y || z, and transversal b, then ∠2 and ∠17 are corresponding → so ∠17 = ∠2 = 74°

Then:
- ∠15 = vertical to ∠17? No — ∠15 and ∠17 are adjacent? Wait.

At bottom intersection:

If ∠17 is bottom-left = 74°, then:

- ∠15 (top-left) = 180° - 74° = 106° (linear pair)
- ∠16 (top-right) = vertical to ∠17 → 74°
- ∠18 (bottom-right) = vertical to ∠15 → 106°

Wait — no:

Vertical angles: ∠15 and ∠18 are vertical? Or ∠15 and ∠17?

Standard: at an intersection, vertical angles are opposite.

So if ∠17 is bottom-left, then vertical is top-right → ∠16.

And ∠15 (top-left) and ∠18 (bottom-right) are vertical.

So:

Given ∠17 = 74° (corresponding to ∠2)

Then:
- ∠16 = vertical to ∠17 → 74°
- ∠15 = adjacent to ∠17 → 180° - 74° = 106°
- ∠18 = vertical to ∠15 → 106°

Yes.

So:
→ ∠15 = 106°
→ ∠16 = 74°
→ ∠17 = 74°
→ ∠18 = 106°

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Step 4: Find angles at the middle intersection (where lines a and b cross)

This is where lines a and b intersect each other, forming angles ∠8, ∠9, 10, ∠11.

Note: This intersection is not on the parallel lines, so we can’t directly use parallel line rules here. But we can use the fact that the angles around a point sum to 360°, and vertical angles are equal.

But we can also use the angles we already found on the transversals.

For example, consider line b: it goes from top-left to bottom-right.

At top intersection with y, we have ∠3 = 106° (bottom-right angle).

As we go down line b, when it crosses line a, the angle on the same side should relate.

Actually, since line b is a straight line, the angles it makes with other lines can help.

Consider triangle or just use vertical angles and straight lines.

At the middle intersection (a and b cross):

We can find some angles using the fact that on line b, the angles on one side must add appropriately.

For instance, at the top intersection, angle between line b and line y is 74° (given), and we found ∠3 = 106° on the other side.

But perhaps easier: use the angles we have on the transversals to find the angles at the crossing.

Note that ∠9 and ∠11 are vertical angles, so they are equal.

Similarly, ∠8 and ∠10 are vertical.

Also, ∠9 and ∠8 are adjacent, so they add to 180°.

But we need values.

Another way: consider the triangle formed or use alternate interior angles.

Actually, let’s use the fact that line a is straight, and we know angles on it.

For example, at the bottom intersection with z, we have ∠13 = 39° (top-right).

As we go up line a, when it crosses line b, the angle on the same side might be related.

Perhaps use the concept that the sum of angles in a triangle is 180°, but there’s no triangle labeled.

Wait — look at the angles around the middle intersection.

We can find ∠9 by considering the angles on line b.

At the top, line b makes an angle of 74° with line y (in the top-left).

Since line y is horizontal, and line b is going down to the right, the angle it makes with the horizontal is 74° on the left side.

When line b crosses line a, which is also going up to the right, we can find the angle between them.

Actually, line a has a slope such that at the bottom, it makes 39° with line z (horizontal).

So line a has an angle of 39° with horizontal.

Line b has an angle of 74° with horizontal? Not exactly, because at top, it's 74° from horizontal, but since it's a straight line, the angle with horizontal is constant.

Actually, the angle that line b makes with the horizontal line y is 74° on the left side, so on the right side it's 106°, but the acute angle is 74°.

Similarly, line a makes 39° with horizontal at the bottom.

So at their intersection, the angle between them can be found.

Specifically, the angle between two lines with inclinations θ1 and θ2 is |θ1 - θ2|.

Here, line b: if it makes 74° with horizontal (measured from positive x-axis, but in diagram, it's steep), actually from the given, at top-left, the angle between line b and line y is 74°, and since line y is horizontal, line b is at 74° from horizontal.

Similarly, line a at bottom makes 39° with horizontal, so its inclination is 39°.

So the angle between them is 74° - 39° = 35°.

Is that correct?

Let me think.

If both lines are measured from the same reference, say the positive x-axis.

Assume line y and z are horizontal.

Line b: at top-left, the angle between line b and line y is 74°, and since it's going down to the right, the angle from the positive x-axis would be 180° - 74° = 106°? Or 74° below?

Actually, in standard position, if line y is positive x-axis, then at the intersection, the angle in the second quadrant is 74°, so the direction of line b is 180° - 74° = 106° from positive x-axis.

Similarly, line a: at bottom, the angle with line z (horizontal) is 39°, and since it's going up to the right, its direction is 39° from positive x-axis.

So the angle between line a and line b is |106° - 39°| = 67°.

But that might not be the angle at the intersection.

The angle between two lines is the absolute difference of their inclinations.

Inclination of line a: 39°

Inclination of line b: since it goes from top-left to bottom-right, and at top-left it makes 74° with horizontal, that means its slope is negative, and the angle with positive x-axis is 180° - 74° = 106°.

So angle between them is |106° - 39°| = 67°.

Therefore, at the intersection, the acute angle is 67°, and the obtuse is 180° - 67° = 113°.

Now, looking at the diagram, ∠9 is likely the angle between them on one side.

From the labeling, ∠9 is probably the angle in the "north" direction at the intersection.

To determine which is which, let's use the angles we have.

Consider the path along line b.

From top to bottom, at the top intersection, the angle between line b and line y is 74° on the left.

As we move down line b, when we hit line a, the angle on the left side should be related.

Actually, we can use the fact that the sum of angles on a straight line is 180°.

For example, on line b, from top to bottom, the angles it makes with the transversals.

At the top, with line y, the angle on the "south" side is ∠3 = 106°.

At the bottom, with line z, the angle on the "north" side is ∠15 = 106° (we calculated earlier).

Since line b is straight, the total turn should be consistent.

But perhaps for the middle intersection, we can consider the triangle formed by the three lines, but it's messy.

Another approach: use vertical angles and the fact that around the point, angles sum to 360°.

But we need more info.

Let's list what we have so far:

From top-left:
∠1 = 106°
∠2 = 74°
∠3 = 106°

From top-right:
∠4 = 39°
∠5 = 39°
∠6 = 141°
∠7 = 141°

From bottom-left:
∠12 = 141°
∠13 = 39°
∠14 = 141°

From bottom-right:
∠15 = 106°
∠16 = 74°
∠17 = 74°
∠18 = 106°

Now for the middle intersection (a and b cross):

Angles ∠8, ∠9, 10, ∠11.

Note that ∠9 and ∠11 are vertical, so ∠9 = ∠11

∠8 and ∠10 are vertical, so ∠8 = ∠10

Also, ∠9 + ∠8 = 180° (adjacent on straight line)

Now, how to find the value.

Consider line a: it is a straight line, so the angles on one side should add up.

For example, at the bottom intersection with z, we have ∠13 = 39° (top-right).

As we go up line a, when it crosses line b, the angle on the same side.

Actually, the angle between line a and line b can be found from the angles they make with the parallel lines.

Specifically, at the bottom, line a makes 39° with line z.

At the top, line b makes 74° with line y.

Since y and z are parallel, the angle between line a and line b is the difference if they are on the same side, but let's think of the triangle formed by the three lines.

The three lines form a triangle in the middle.

The angles of the triangle can be found.

For example, at the top, the angle between line b and line y is 74°, but that's not inside the triangle.

Consider the triangle formed by lines a, b, and say, but they intersect at three points.

The three intersection points are: top-left (b and y), top-right (a and y), bottom-left (a and z), bottom-right (b and z), and middle (a and b).

So the middle intersection is separate.

Perhaps use the fact that the sum of angles around the middle point is 360°.

But we need relations.

Another idea: use alternate interior angles or corresponding angles involving the middle.

For example, consider transversal b cutting parallel lines y and z.

We have at top: ∠2 = 74° (bottom-left)

At bottom: ∠17 = 74° (bottom-left) — which we already used.

Now, for line a cutting y and z:

At top: ∠4 = 39° (top-right)

At bottom: ∠13 = 39° (top-right) — used.

Now, at the middle intersection, the angle ∠9 might be related to these.

Let's calculate the angle that line a makes with horizontal: 39° (since at bottom, it's 39° with z, and z is horizontal).

Line b makes with horizontal: at top, it's 74° with y, but since it's going down to the right, the angle with the positive x-axis is 180° - 74° = 106°, so the acute angle with horizontal is 74°, but the direction is 106°.

The angle between line a (39°) and line b (106°) is 106° - 39° = 67°.

So at the intersection, the smaller angle is 67°, larger is 113°.

Now, in the diagram, ∠9 is likely the angle in the "upper" part, which might be the larger one or smaller.

From the labeling, ∠9 is between the two lines in the north direction.

Given that line a is rising at 39°, line b is falling at 74° from horizontal, so at intersection, the angle above might be 180° - 67° = 113°, and below 67°.

Let me sketch mentally.

Line a: from bottom-left to top-right, slope positive, angle 39° with horizontal.

Line b: from top-left to bottom-right, slope negative, angle 74° with horizontal on the left, so on the right it's 106° from positive x-axis.

So when they cross, the angle between them in the upper half would be the supplement of the difference.

The difference in inclinations is 106° - 39° = 67°, so the acute angle is 67°, obtuse is 113°.

In the diagram, ∠9 is probably the obtuse angle, as it's labeled in the "open" space.

We can verify with the angles we have.

For example, consider the path from top to bottom along line b.

At the top, the angle between line b and line y is 74° on the left.

As we move down, when we hit line a, the angle on the left side of line b.

At the bottom, with line z, the angle on the left side is 106° (∠15).

Since line b is straight, the total change in angle should be consistent, but it's not direct.

Another way: the sum of angles in the quadrilateral or something.

Perhaps use the fact that for line a, the angles on one side.

Let's consider the angle at the middle intersection for line a.

On line a, from bottom to top, at the bottom intersection with z, the angle on the "north" side is ∠13 = 39°.

As we go up, when we cross line b, the angle on the north side of line a.

At the top intersection with y, the angle on the north side is ∠4 = 39°.

Since line a is straight, the angle it makes with any transversal should be consistent, but here the transversal is line b, which is different.

The angle between line a and line b is constant.

So at the middle intersection, the angle between them is 67° or 113°.

Now, to decide which is ∠9, let's look at the surrounding angles.

For example, ∠9 is adjacent to ∠8, and they are on a straight line with respect to line b or a.

Assume that ∠9 is the angle between the two lines in the region that is "between" the parallel lines.

In many diagrams, ∠9 is the angle that is vertically opposite to the angle formed by the extensions.

Perhaps calculate using the triangle formed by the three lines.

The three lines a, b, and say, but they form a triangle with the parallel lines, but it's complicated.

Let's use the following: the angle ∠9 can be found as the supplement of the sum of other angles in a triangle, but there's no triangle labeled.

Another idea: use the fact that the alternate interior angles or corresponding angles can be used for the middle.

For example, consider transversal b cutting the parallel lines, but the middle is not on the parallel lines.

Perhaps the angle ∠9 is equal to the difference of the angles.

Let's calculate the angle that line b makes with line a.

From the bottom, line a makes 39° with horizontal.

Line b, at the bottom, makes with line z: we have ∠15 = 106° (top-left), which is the angle between line b and line z on the top-left side.

Since line z is horizontal, the angle that line b makes with horizontal at the bottom is 106° from the positive x-axis? Let's see.

At bottom-right intersection, line b and line z.

∠15 is top-left, which is the angle between line b and line z in the northwest direction.

Since line z is horizontal, and line b is going down to the right, the angle from the positive x-axis to line b is 180° - 106° = 74°? No.

If ∠15 = 106° is the angle in the top-left quadrant at that intersection, that means from the positive x-axis (line z to the right), the line b is at 180° - 106° = 74° above the negative x-axis, so from positive x-axis, it's 180° - 74° = 106°? I'm confusing myself.

Let's define:

At any intersection, the angle between the transversal and the parallel line.

For line b at bottom intersection with z:

The angle in the top-left is ∠15 = 106°.

This means that from the direction of line z (positive x-axis), turning to line b, in the counterclockwise direction, it's 106° to the top-left, so the direction of line b is 106° from positive x-axis.

Similarly, at top intersection with y, the angle in the top-left is 74°, so from positive x-axis, line b is at 180° - 74° = 106°? No.

At top-left intersection, the 74° is given, and it's the angle between line b and line y in the top-left, so if line y is positive x-axis, then line b is at 180° - 74° = 106° from positive x-axis.

At bottom-right intersection, ∠15 = 106° is the angle in the top-left, so from positive x-axis, line b is at 180° - 106° = 74°? That can't be, because it should be the same line.

Mistake here.

Line b is a straight line, so its direction is constant.

At top-left intersection, the angle between line b and line y (horizontal) is 74° in the second quadrant, so the slope is such that the angle with positive x-axis is 180° - 74° = 106°.

At bottom-right intersection, the angle between line b and line z (horizontal) should be the same, because it's the same line.

But we have ∠15 = 106° at bottom, which is the angle in the top-left quadrant.

If line b is at 106° from positive x-axis, then at any point, the angle with the horizontal line (positive x-axis) is 106°, so the acute angle is 74°, but the actual angle from positive x-axis is 106°.

So at bottom intersection, the angle in the top-left quadrant should be the angle from the positive x-axis to line b, which is 106°, but that would mean the angle between them is 106°, which matches ∠15 = 106°.

Yes, so at bottom, ∠15 = 106° is consistent with line b at 106° from positive x-axis.

Similarly, at top, the given 74° is the angle in the top-left, which is the supplement, but in terms of direction, it's the same.

Now for line a: at bottom-left intersection, the given 39° is in the bottom-left quadrant.

So from positive x-axis, line a is at 39° (since it's going up to the right).

At top-right intersection, ∠4 = 39° is in the top-right, which is consistent with 39° from positive x-axis.

So line a is at 39° from positive x-axis.

Line b is at 106° from positive x-axis.

So the angle between them is |106° - 39°| = 67°.

Therefore, at the intersection, the smaller angle is 67°, larger is 113°.

Now, in the diagram, ∠9 is likely the larger angle, as it's labeled in the "wide" part.

Moreover, from the labeling, ∠9 is between the two lines in the north, which would be the obtuse angle if the lines are crossing with acute angle below.

So let's assume ∠9 = 113°, then ∠11 = 113° (vertical), and ∠8 = 180° - 113° = 67°, ∠10 = 67° (vertical).

But let's verify with another method.

Consider the triangle formed by the three lines, but perhaps use the sum of angles around the point.

Or use the fact that for line a, the angle it makes with line b.

At the bottom, line a makes 39° with horizontal, line b makes 106° with horizontal, so the angle between them is 67°.

At the middle, same thing.

Now, to confirm, let's see if we can find ∠9 using the angles on the parallel lines.

For example, consider the alternate interior angles for transversal a or b, but the middle is not on the parallel lines.

Another way: the angle ∠9 can be found as the sum of the remote interior angles or something, but let's calculate the angle in the triangle formed by the three intersection points.

The three lines form a triangle with vertices at: P1 = intersection of a and y, P2 = intersection of a and z, P3 = intersection of b and y, P4 = intersection of b and z, and P5 = intersection of a and b.

So the triangle is P1, P2, P5 or something.

Perhaps the triangle is formed by P3 (b and y), P4 (b and z), and P5 (a and b), but P3 and P4 are on line b, so not a triangle.

The triangle is formed by the three lines: vertices at P1 (a and y), P2 (a and z), and P5 (a and b)? No, P1, P2, P5 are not all connected.

Actually, the three lines intersect at three points: let's call A = a ∩ y, B = a ∩ z, C = b ∩ y, D = b ∩ z, E = a ∩ b.

Then the triangle is A, B, E or C, D, E, but A and B are on a, C and D on b, so the triangle is A, C, E or something.

Perhaps it's easier to accept that the angle between a and b is 67°, and from the diagram, ∠9 is the obtuse angle, so 113°.

Moreover, in many such problems, the angle in the middle is the supplement.

Let's calculate the angle at E for the triangle A-C-E or something.

Consider triangle A-C-E, where A = a∩y, C = b∩y, E = a∩b.

At A, the angle between a and y is 39° (since ∠4 = 39°, and it's the angle between a and y).

At C, the angle between b and y is 74° (given).

Then in triangle A-C-E, the angle at A is the angle between a and y, which is 39°, but that's not the angle of the triangle.

The triangle A-C-E has vertices at A, C, E.

At A, the angle of the triangle is the angle between lines a and the line from A to C, but A and C are both on line y, so the line AC is along y.

So at A, the angle of the triangle is the angle between line a and line y, which is 39°.

Similarly, at C, the angle between line b and line y is 74°.

Then in triangle A-C-E, the sum of angles is 180°, so angle at E is 180° - 39° - 74° = 67°.

This angle at E is the angle between lines a and b in the triangle, which is the acute angle, 67°.

But in the diagram, ∠9 is likely the other angle, the reflex or the obtuse one.

In the diagram, ∠9 is labeled in the region that is not the triangle, so probably the obtuse angle.

Since the acute angle is 67°, the obtuse is 113°, and ∠9 is probably 113°.

Moreover, in the labeling, ∠9 is between the two lines in the north, which would be the larger angle if the lines are crossing with the acute angle south.

So I'll go with ∠9 = 113°, ∠11 = 113°, ∠8 = 67°, ∠10 = 67°.

To confirm, let's see if it makes sense with the surrounding.

For example, on line a, from bottom to top, at bottom, angle with z is 39°, at top with y is 39°, and at middle with b is 67° or 113°.

The angle that line a makes with line b is 67°, so when they cross, the angle on one side is 67°, on the other 113°.

In the diagram, ∠8 is probably the acute angle, ∠9 the obtuse.

Looking at the labels: ∠8 is on the left, ∠9 on the top, etc.

Typically, ∠9 is the angle in the "top" position, which might be the obtuse one.

I think it's safe to set:

∠8 = 67°
∠9 = 113°
∠10 = 67°
∠11 = 113°

Now, let's list all angles:

From top-left:
∠1 = 106°
∠2 = 74°
∠3 = 106°

From top-right:
∠4 = 39°
∠5 = 39°
∠6 = 141°
∠7 = 141°

From middle:
∠8 = 67°
∠9 = 113°
∠10 = 67°
∠11 = 113°

From bottom-left:
∠12 = 141°
∠13 = 39°
∠14 = 141°

From bottom-right:
∠15 = 106°
∠16 = 74°
∠17 = 74°
∠18 = 106°

Now, for the last part: name the relationship between angle pairs.

a. ∠9 & ∠11

These are vertical angles, so they are vertical angles.

b. ∠15 & ∠3

∠15 is at bottom-right, top-left angle = 106°

∠3 is at top-left, bottom-right angle = 106°

Lines y and z are parallel, transversal is line b.

∠3 is at top intersection, bottom-right.

∠15 is at bottom intersection, top-left.

Are they corresponding? Corresponding would be same relative position.

∠3 is bottom-right at top, ∠15 is top-left at bottom — not corresponding.

Alternate interior: interior angles are between the parallel lines.

∠3 is below line y, above the space, but since y is top, z is bottom, interior is between them.

∠3 is at top intersection, on the bottom side, so it's below line y, which is the interior side if we consider the region between y and z.

Similarly, ∠15 is at bottom intersection, on the top side, so above line z, which is also interior.

Now, are they on opposite sides of the transversal?

Transversal is line b.

∠3 is on the right side of line b (since at top, bottom-right).

∠15 is on the left side of line b (at bottom, top-left).

So they are on opposite sides of the transversal, and both interior, so they are alternate interior angles.

And since lines are parallel, they should be equal, which they are: both 106°.

So relationship is alternate interior angles.

c. ∠14 & ∠5

∠14 is at bottom-left, bottom-right angle = 141°

∠5 is at top-right, top-right angle = 39°

Lines y and z parallel, transversal is line a.

∠14 is at bottom intersection, bottom-right.

∠5 is at top intersection, top-right.

Both on the right side of transversal a.

∠14 is below line z, ∠5 is above line y.

Since y and z are parallel, and transversal a, then ∠5 and ∠14 are on the same side of the transversal, but one is exterior, one is exterior?

∠5 is above line y, so exterior.

∠14 is below line z, so exterior.

And both on the same side (right side) of transversal a.

So they are corresponding angles? Corresponding angles are in the same relative position.

∠5 is top-right at top intersection.

∠14 is bottom-right at bottom intersection.

So yes, they are corresponding angles.

But are they equal? ∠5 = 39°, ∠14 = 141°, not equal.

Corresponding angles should be equal if lines are parallel.

Mistake.

For corresponding angles, when transversal cuts two parallel lines, corresponding angles are equal.

Here, ∠5 is at top, top-right.

The corresponding angle at bottom would be bottom-right, which is ∠14.

But ∠5 = 39°, ∠14 = 141°, not equal.

Why? Because ∠14 is not the corresponding angle.

Let's think.

At top intersection, line a and y, ∠5 is the angle in the top-right quadrant.

At bottom intersection, line a and z, the corresponding angle would be the angle in the top-right quadrant, which is ∠13 = 39°.

Yes! I forgot.

∠13 is top-right at bottom, which corresponds to ∠5 top-right at top.

So ∠5 and ∠13 are corresponding, both 39°.

But the pair is ∠14 and ∠5.

∠14 is bottom-right at bottom.

What is the corresponding angle to ∠14? At top, bottom-right, which is ∠6 = 141°.

And ∠6 = 141°, ∠14 = 141°, so they are corresponding.

But the pair is ∠14 and ∠5, which are not corresponding.

∠5 is top-right, ∠14 is bottom-right.

So they are both on the right side, but one is above the top line, one is below the bottom line.

So they are both exterior angles, on the same side of the transversal.

So they are consecutive exterior angles or same-side exterior angles.

And since lines are parallel, same-side exterior angles are supplementary.

Check: ∠5 = 39°, ∠14 = 141°, sum = 180°, yes.

So relationship is same-side exterior angles.

Sometimes called consecutive exterior angles.

So for c, it's same-side exterior angles.

Now, to summarize:

Final answers for angles:

∠1 = 106°
∠2 = 74°
∠3 = 106°
∠4 = 39°
∠5 = 39°
∠6 = 141°
∠7 = 141°
∠8 = 67°
∠9 = 113°
∠10 = 67°
∠11 = 113°
∠12 = 141°
∠13 = 39°
∠14 = 141°
∠15 = 106°
∠16 = 74°
∠17 = 74°
∠18 = 106°

For relationships:

a. ∠9 & ∠11: vertical angles
b. ∠15 & ∠3: alternate interior angles
c. ∠14 & ∠5: same-side exterior angles

Now, for the final answer, since the user asked to solve the problem, and the problem has many parts, but the instruction is to provide the final answer after reasoning.

But the final answer should include all the angle measures and the relationships.

Since it's a worksheet, probably list all.

But in the response, we can box the key parts.

Given the format, I'll provide the final answer as the completed values.

But to be precise, let's write it clearly.