Parallel Lines cut by a transversal worksheet - Free Printable

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 two angles:
- At the top left (where line b crosses line y): 74°
- At the bottom left (where line a crosses line z): 39°

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

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Step 1: Understand vertical angles and linear pairs

When two lines cross, they form:
- Vertical angles — opposite each other, equal in measure.
- Linear pairs — adjacent angles that add up to 180°.

Also, when parallel lines are cut by a transversal:
- Corresponding angles are equal.
- Alternate interior/exterior angles are equal.
- Consecutive interior angles add to 180°.

But since we have two transversals, we’ll use basic angle rules first (vertical, linear) and then parallel line properties where needed.

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Step 2: Start with known angles

#### Top intersection (line b crossing line y):

Given: One angle is 74° (top-left).

→ Its vertical angle (∠3) is also 74°.

→ The adjacent angles (∠1 and ∠2) are linear pairs → 180° - 74° = 106°

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

#### Bottom intersection (line a crossing line z):

Given: One angle is 39° (bottom-left).

→ Its vertical angle (∠13) is also 39°

→ Adjacent angles (∠12 and ∠14) are linear pairs → 180° - 39° = 141°

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

Wait — actually, looking at the diagram:

The 39° is labeled at the bottom-left corner of the lower intersection (line a crossing line z). That would be ∠14? Let me check the labeling.

Looking at the diagram:

At the bottom-left intersection (line a and line z):

- ∠12 is top-left
- ∠13 is top-right
- ∠14 is bottom-left ← this is the 39° given
- ∠15 is bottom-right? Wait no — let's map it properly.

Actually, standard labeling:

For any intersection of two lines, angles are labeled around the point.

In the diagram:

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

- Top-left: 74°
- Top-right: ∠1
- Bottom-left: ∠2
- Bottom-right: ∠3

So yes, as above:

∠1 = 106°, ∠2 = 106°, ∠3 = 74°

At the bottom-left intersection (line a and line z):

- Top-left: ∠12
- Top-right: ∠13
- Bottom-left: 39° → this is ∠14
- Bottom-right: ∠15? Wait, no — looking again.

Actually, in the diagram, the 39° is written below line z, to the left of line a — so that’s ∠14.

Then:

- ∠14 = 39°
- Vertical angle to ∠14 is ∠13 → so ∠13 = 39°
- Linear pair with ∠14 is ∠12 → 180 - 39 = 141° → ∠12 = 141°
- Vertical to ∠12 is ∠15 → ∠15 = 141°? Wait, no.

Wait — let’s label clearly.

At the intersection of line a and line z (bottom-left):

Angles:

- Above line z, left of line a: ∠12
- Above line z, right of line a: ∠13
- Below line z, left of line a: ∠14 = 39°
- Below line z, right of line a: ∠15? But in the diagram, ∠15 is shown on the other side.

Actually, looking at the full diagram:

There are three intersections:

1. Line b crosses line y → angles ∠1, ∠2, ∠3, and the 74°
2. Line a and line b cross each other → angles ∠8, 9, ∠10, ∠11
3. Line a crosses line z → angles ∠12, ∠13, ∠14, and 39°
4. Line b crosses line z → angles ∠15, ∠16, ∠17, ∠18

Wait — actually, there are four intersections? No.

Lines:

- Two horizontal parallel lines: y (top), z (bottom)
- Two transversals: a (diagonal up-right), b (diagonal down-right)

So intersections:

A. Line b crosses line y → angles ∠1, ∠2, ∠3, and the 74° (which is not labeled with a number, but is adjacent to ∠1 and ∠2)

B. Line a crosses line y → angles ∠4, ∠5, ∠6, ∠7

C. Line a and line b cross each other → angles ∠8, ∠9, ∠10, ∠11

D. Line a crosses line z → angles ∠12, ∠13, ∠14, and the 39° (which is ∠14)

E. Line b crosses line z → angles ∠15, ∠16, ∠17, ∠18

Yes.

So let’s go one by one.

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Intersection A: Line b and line y

Given: angle in top-left = 74°

This angle is adjacent to ∠1 (top-right) and ∠2 (bottom-left)

So:

- ∠1 + 74° = 180° → ∠1 = 106°
- ∠2 + 74° = 180° → ∠2 = 106°
- ∠3 is vertical to the 74° angle → ∠3 = 74°

So:
∠1 = 106°
∠2 = 106°
∠3 = 74°

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Intersection D: Line a and line z

Given: ∠14 = 39° (bottom-left)

Then:

- ∠13 is vertical to ∠14 → ∠13 = 39°
- ∠12 is adjacent to ∠14 → ∠12 = 180° - 39° = 141°
- ∠15 is vertical to ∠12 → ∠15 = 141°? Wait, no.

At this intersection:

- ∠12: top-left
- ∠13: top-right
- ∠14: bottom-left = 39°
- ∠15: bottom-right? But in the diagram, ∠15 is shown on the right side of line b crossing line z.

I think I misassigned.

Looking back at the original diagram description:

At the bottom, line z has:

Left side: ∠12 = , ∠13 = , and below that 39° = ∠14

Then further right: ∠15 = , ∠16 = , ∠17 = , ∠18 =

And line b crosses line z at the right part.

So actually, the 39° is at the intersection of line a and line z, specifically ∠14.

Then:

- ∠14 = 39°
- Vertical angle is ∠13 → ∠13 = 39°
- Adjacent angles: ∠12 and the angle below ∠13? Let's define:

Standard: when two lines cross, four angles:

Label them clockwise:

Top-left: ∠12
Top-right: ∠13
Bottom-right: ∠15? No — in the diagram, after ∠13 comes ∠15? That doesn't make sense.

Perhaps the labeling is:

For line a crossing line z:

- Angles above line z: ∠12 (left), ∠13 (right)
- Angles below line z: ∠14 (left), and the one on the right is not labeled yet — but in the diagram, it's ∠15? No, ∠15 is at the next intersection.

I think there's confusion in labeling.

Let me reinterpret based on standard worksheet layout.

Typically, for such diagrams:

- The top horizontal line y has two transversals crossing it: line b on the left, line a on the right.
- The bottom horizontal line z has the same two transversals crossing it.
- Additionally, the two transversals cross each other in the middle.

So intersections:

1. Line b × line y → angles: let's say NW=74°, NE=∠1, SW=∠2, SE=∠3
2. Line a × line y → angles: NW=∠4, NE=∠5, SW=∠6, SE=∠7
3. Line a × line b → angles: NW=∠8, NE=∠9, SW=∠10, SE=∠11? Or depending on orientation.
4. Line a × line z → angles: NW=∠12, NE=∠13, SW=39°=∠14, SE=∠15? But in the diagram, ∠15 is at the next intersection.

Looking at the original text:

"∠12 = ∠13 = 39° ∠14 ="

It says "39°" is written, and then "∠14 =", so likely 39° is ∠14.

Similarly, at the top, "74°" is written, and then "∠1 =", so 74° is not numbered, but adjacent to ∠1.

To resolve this, let's assume:

At line a × line z:

- The angle labeled 39° is ∠14 (bottom-left)
- Then ∠13 is top-right, which is vertical to ∠14? No, vertical to ∠14 would be the top-left if it's a straight cross.

When two lines cross, vertical angles are opposite.

So if line a and line z cross, forming an X, then:

- The angle directly opposite to ∠14 is the one at the top-right, which should be ∠13.

Yes, so ∠13 = ∠14 = 39°? No, vertical angles are equal, so if ∠14 = 39°, then its vertical angle is ∠13, so ∠13 = 39°.

Then the adjacent angles are ∠12 and the angle at bottom-right.

Let’s call the bottom-right angle ∠X.

Then ∠12 + ∠14 = 180° (linear pair) → ∠12 = 141°

∠X + ∠14 = 180° → ∠X = 141°

And ∠X is vertical to ∠12, so ∠X = ∠12 = 141°

In the diagram, what is labeled as ∠15? Looking at the text: "∠15 = " is listed after ∠13, and before ∠16.

Probably, at the intersection of line a and line z, the angles are:

- ∠12: top-left
- ∠13: top-right
- ∠14: bottom-left = 39°
- and the bottom-right is not labeled with a number yet — but in the diagram, it might be ∠15? No, because later ∠15 is at the other intersection.

I think I need to look at the sequence.

From the user's text:

"∠12 = ∠13 = 39° ∠14 ="

This suggests that 39° is associated with ∠14, and ∠12 and ∠13 are to be found.

Similarly, at the top: "74° ∠1 = ∠2 = ∠3 ="

So 74° is given, and ∠1, ∠2, ∠3 are to be found.

So for line a × line z:

- ∠14 = 39° (given)
- Then ∠13 is vertical to ∠14? Or adjacent?

In standard position, if you have two lines crossing, and you label the angles as:

Top-left: ∠12
Top-right: ∠13
Bottom-right: ∠15? But that's not consistent.

Perhaps the labeling is sequential around the intersection.

To avoid confusion, let's use geometry rules without relying on labels initially.

At the intersection of line a and line z:

One angle is 39°. Let's call this angle A.

Then:
- The vertical angle to A is also 39°.
- The two adjacent angles are each 180° - 39° = 141°.

Now, in the diagram, the 39° is at the bottom-left, so:
- Bottom-left: 39° = ∠14
- Top-right: vertical to it = ∠13 = 39°
- Top-left: adjacent = ∠12 = 141°
- Bottom-right: adjacent = let's call it ∠Y = 141°, and this is vertical to ∠12, so ∠Y = 141°

But in the diagram, what is ∠15? Looking at the text, after ∠13, it says "∠15 =", and then "∠16 =", etc.

Probably, ∠15 is at the intersection of line b and line z.

Similarly, for the top, after ∠3, it has ∠4, ∠5, etc., which are at the other intersection on line y.

So let's list all intersections and assign angles.

Define:

Intersection 1: Line b and line y
- Angles:
- Top-left: 74° (given)
- Top-right: ∠1
- Bottom-left: ∠2
- Bottom-right: ∠3

As before:
∠1 = 180° - 74° = 106°
∠2 = 180° - 74° = 106°
∠3 = 74° (vertical to given)

Intersection 2: Line a and line y
- Angles:
- Top-left: ∠4
- Top-right: ∠5
- Bottom-left: ∠6
- Bottom-right: ∠7

Since line y is straight, and line a crosses it, we can find these if we know corresponding or alternate angles, but we don't have direct info yet.

However, line a is a transversal, and we have information from the bottom.

At the bottom, line a and line z intersect, with ∠14 = 39°.

Since lines y and z are parallel, and line a is a transversal, then:

- Corresponding angles are equal.
- For example, the angle at line a × line y that corresponds to ∠14 would be ∠6 or ∠7?

Let's think.

Line a crosses parallel lines y and z.

At line z, ∠14 = 39° is at the bottom-left of the intersection.

At line y, the corresponding angle would be at the bottom-left of the intersection of line a and line y, which is ∠6.

Because both are on the same side of the transversal and both below the parallel lines? Let's see.

Standard corresponding angles:

If you have two parallel lines cut by a transversal, corresponding angles are in the same relative position.

So for transversal a:

- At line y: bottom-left angle is ∠6
- At line z: bottom-left angle is ∠14 = 39°

Since y || z, corresponding angles are equal, so ∠6 = ∠14 = 39°

Similarly, the top-left angle at line y would correspond to top-left at line z, which is ∠12.

At line z, top-left is ∠12 = 141° (as calculated earlier)

So ∠4 = ∠12 = 141° (corresponding angles)

Then, at line a × line y:

- ∠4 = 141°
- ∠6 = 39°
- ∠5 is vertical to ∠4? No.

At this intersection:

- ∠4 and ∠5 are adjacent if they are on the same side, but typically:

Assume:
- ∠4: top-left
- ∠5: top-right
- ∠6: bottom-left
- ∠7: bottom-right

Then:
- ∠4 and ∠6 are on the left side, but not necessarily related directly.

Actually, ∠4 and ∠5 are adjacent along line y, so ∠4 + ∠5 = 180°? No, because they are on different sides of the transversal.

When two lines cross, the angles around the point sum to 360°, and vertical angles are equal.

So at line a × line y:

- ∠4 and ∠7 are vertical angles
- ∠5 and ∠6 are vertical angles

Is that correct?

If line a and line y cross, forming an X, then:

- The angle opposite to ∠4 is ∠7
- The angle opposite to ∠5 is ∠6

Yes.

So if ∠6 = 39°, then ∠5 = 39° (vertical)

If ∠4 = 141°, then ∠7 = 141° (vertical)

Also, ∠4 + ∠5 = 141° + 39° = 180°, which makes sense because they are adjacent on the straight line y? No, on the transversal.

Actually, at the intersection, adjacent angles sum to 180° if they form a linear pair.

For example, ∠4 and ∠5 are adjacent and form a linear pair along line y? No, along the transversal.

Let's clarify:

When two lines intersect, they form two pairs of vertical angles, and adjacent angles are supplementary.

So at line a × line y:

- ∠4 and ∠5 are adjacent if they share a ray, but in standard labeling, if ∠4 is top-left, ∠5 is top-right, then they are adjacent along the top, but since line y is horizontal, the angle between them is along line y, so yes, ∠4 and 5 are adjacent and their non-common rays are on line a, so they are not a linear pair; rather, ∠4 and the angle below it are linear pair.

Perhaps it's better to think:

The four angles at the intersection:

- Let’s say angle A = ∠4 (top-left)
- Angle B = ∠5 (top-right)
- Angle C = ∠6 (bottom-left)
- Angle D = ∠7 (bottom-right)

Then:
- A and C are vertical? No, A and D are vertical if it's a standard cross.

In a standard X, the vertical angles are:
- Top-left and bottom-right are vertical
- Top-right and bottom-left are vertical

So:
- ∠4 and ∠7 are vertical
- ∠5 and ∠6 are vertical

Yes.

So if ∠6 = 39°, then ∠5 = 39° (vertical)

If ∠4 = 141°, then ∠7 = 141° (vertical)

Also, ∠4 + ∠5 = 141° + 39° = 180°, which is correct because they are adjacent angles on a straight line? Not necessarily, but in this case, since line y is straight, the angles on one side of the transversal should add to 180° only if they are on the same side, but here ∠4 and 5 are on the same side of the transversal but on different sides of the parallel line.

Actually, at the intersection, the sum of angles around the point is 360°, and since vertical angles are equal, and adjacent angles sum to 180°.

For example, ∠4 and ∠5 are adjacent if they share the ray along line y, but in reality, the angle between ∠4 and 5 is the angle along line a, so they are not adjacent in the sense of sharing a common side that is part of the parallel line.

To simplify, once we have two angles, the others follow.

From above:
- ∠6 = 39° (corresponding to ∠14)
- ∠5 = ∠6 = 39° (vertical)
- ∠4 = 180° - ∠5 = 141°? Why? Because ∠4 and ∠5 are adjacent along the transversal? Let's see.

Actually, ∠4 and ∠5 are on the same side of the transversal, but on the parallel line, the angles on a straight line sum to 180°.

Specifically, along line y, the angles on the top side: ∠4 and ∠5 are not on the same straight line; rather, the straight line is y, so the angles on one side of the transversal on line y should be considered.

Perhaps it's easier to use the fact that at the intersection, the adjacent angles sum to 180°.

For example, ∠4 and the angle below it on the left are supplementary.

But in our case, we have ∠4 and ∠6 on the left side.

∠4 and ∠6 are on the same side of the transversal, but on different parallel lines.

At the single intersection, the angles that are adjacent (share a common ray) sum to 180°.

So for line a × line y:

- ∠4 and ∠5 are adjacent if they share the ray along line a, but typically, ∠4 and the angle below it (which is ∠6) are not adjacent; rather, ∠4 and the angle to its right (∠5) are adjacent if they share the ray along line y.

I think I have it:

When two lines intersect, each pair of adjacent angles (sharing a common side) sums to 180°.

So at line a × line y:

- ∠4 and ∠5 are adjacent (share the ray along line y to the right) — no.

Let's define the rays.

Suppose line y is horizontal, line a is diagonal.

At the intersection, there are four rays: left along y, right along y, up along a, down along a.

Then the angles are between these rays.

So:
- Angle between left-y and up-a: this is ∠4 (top-left)
- Angle between up-a and right-y: this is ∠5 (top-right)
- Angle between right-y and down-a: this is ∠7 (bottom-right)
- Angle between down-a and left-y: this is ∠6 (bottom-left)

Then, adjacent angles:
- ∠4 and ∠5 are adjacent, sharing the up-a ray, and their non-common rays are left-y and right-y, which are opposite, so ∠4 + ∠5 = 180° because they form a straight line along y? No, the sum of ∠4 and ∠5 is the angle from left-y to right-y passing through up-a, which is not 180° unless a is perpendicular.

Actually, the sum of all four angles is 360°, and vertical angles are equal.

Specifically:
- ∠4 and ∠7 are vertical angles
- ∠5 and ∠6 are vertical angles

And adjacent angles like ∠4 and ∠5 are not necessarily supplementary; only if they are on a straight line.

In this case, ∠4 and ∠5 are on the same side of the transversal, but on the parallel line, the key is that the angle between the two rays on the parallel line is 180°, so for example, the angle between left-y and right-y is 180°, which is composed of ∠4 + ∠5 if a is between them, but in this case, since a is crossing, the angle from left-y to right-y via the top is ∠4 + ∠5, and this must be 180° because left-y and right-y are opposite rays on the straight line y.

Yes! That's it.

On the straight line y, the total angle on one side is 180°.

So for the top side of line y, the angle from the left ray to the right ray along the top is 180°, and this is split by the transversal a into ∠4 and ∠5.

So ∠4 + ∠5 = 180°

Similarly, on the bottom side, ∠6 + ∠7 = 180°

Also, vertical angles: ∠4 = ∠7, ∠5 = ∠6

From earlier, we have ∠6 = 39° (corresponding to ∠14)

Then ∠5 = ∠6 = 39° (vertical)

Then ∠4 = 180° - ∠5 = 180° - 39° = 141°

Then ∠7 = ∠4 = 141° (vertical)

Perfect.

So for intersection 2 (line a × line y):
- ∠4 = 141°
- ∠5 = 39°
- ∠6 = 39°
- ∠7 = 141°

Now, intersection 3: line a and line b cross each other.

This is in the middle.

Angles: ∠8, ∠9, ∠10, ∠11

We need to find these.

We can use the fact that we have angles from other intersections.

For example, at line b × line y, we have ∠3 = 74° (bottom-right)

At line a × line y, we have ∠6 = 39° (bottom-left)

Now, the angle between line a and line b at the top can be found.

Consider the triangle or the angles around the point.

At the intersection of a and b, the angles can be found using the angles from the parallel lines.

Specifically, the angle between line a and line b can be found from the difference or sum of the angles they make with the parallel lines.

For example, at line y, line b makes an angle of 74° with y (since ∠3 = 74°, which is the angle between line b and line y on the bottom-right).

Similarly, line a makes an angle of 39° with line y on the bottom-left (∠6 = 39°).

Since both are measured from line y, and they are on the same side, the angle between line a and line b at the top intersection can be found.

At the top, near line y, the angle between line a and line b is the difference of their angles with y.

Line b is at 74° to y (on the right side), line a is at 39° to y (on the left side), so the angle between them is 74° + 39° = 113°? Let's think.

Actually, at the intersection of a and b, the vertical angles are equal, and we can use the fact that the sum of angles in a triangle or use corresponding angles.

Another way: consider the triangle formed by the two transversals and the parallel line, but perhaps simpler to use the angles we have.

Let's look at the angles around the intersection of a and b.

We know that at line b × line y, the angle below line y on the right is ∠3 = 74°.

This angle is between line b and line y.

Similarly, at line a × line y, the angle below line y on the left is ∠6 = 39°, between line a and line y.

Now, the angle between line a and line b at the point where they cross can be found by considering the triangle formed by the two lines and the parallel line, but since they cross above, we can use the fact that the alternate interior angles or something.

Perhaps use the concept that the angle between the two transversals is constant.

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

From line b × line y: the angle between line b and line y is 74° on the bottom-right, so the acute angle is 74°, but since it's obtuse or acute, 74° is acute, so line b is at 74° to the horizontal.

Similarly, line a is at 39° to the horizontal, since ∠6 = 39° is the angle between line a and line y on the bottom-left, so if we measure from the horizontal, line a is at 39° below horizontal on the left, but since it's a straight line, the angle with the horizontal is 39°.

Actually, the angle that a line makes with the horizontal can be taken as the acute angle, but for direction, let's define.

Suppose line y is horizontal.

At line b × line y, the angle in the bottom-right is 74°, which means that line b is coming down to the right at an angle of 74° from the horizontal? No.

If the angle between line b and line y is 74°, and it's in the bottom-right, then if line y is horizontal, line b is at an angle of 74° below the horizontal on the right side.

Similarly, for line a, at line a × line y, the angle in the bottom-left is 39°, so line a is at 39° below the horizontal on the left side.

But since line a is going up to the right, from the bottom-left intersection, it is rising, so at the top, it is above.

Perhaps it's better to think of the slope.

The angle that line b makes with the positive x-axis (right) is -74° or 286°, but for simplicity, the acute angle with the horizontal is 74° for line b, and for line a, since it's going up to the right, and at the top intersection with y, the angle on the bottom-left is 39°, which means that the angle between line a and the horizontal is 39°, and since it's on the left, but for the line itself, the angle with the horizontal is 39°.

Actually, for line a, when it crosses line y, the angle on the bottom-left is 39°, which is the angle between line a and line y, so the acute angle is 39°, and since line a is rising to the right, the angle with the horizontal is 39°.

For line b, at line b × line y, the angle on the bottom-right is 74°, so the acute angle with the horizontal is 74°, and since it's falling to the right, the angle with the horizontal is 74° below, so the magnitude is 74°.

Now, when two lines intersect, the angle between them is the difference of their angles with the horizontal.

So for line a at 39° above horizontal (since it's rising), and line b at 74° below horizontal, so the angle between them is 39° + 74° = 113°.

Then, at the intersection, the vertical angles are 113°, and the adjacent angles are 180° - 113° = 67°.

So for the intersection of a and b:

- The angles are 113° and 67° alternately.

Now, which is which?

In the diagram, ∠9 is likely the angle at the top, between the two lines.

Typically, ∠9 is the angle in the top part of the intersection.

From the diagram description, ∠9 is probably the angle between the two lines in the upper region.

Since line a is rising to the right, line b is falling to the right, so at their intersection, the angle above might be the smaller one or larger.

Let's calculate.

The angle between the two lines is |m1 - m2|, but in terms of direction.

The slope of line a: tan(39°) upwards

Slope of line b: tan(-74°) or tan(180°-74°) = tan(106°), but the angle between them is the absolute difference of their inclinations.

Inclination of line a: 39° (from positive x-axis)

Inclination of line b: 180° - 74° = 106° (since it's falling to the right, so from positive x-axis, it's 180° - 74° = 106°)

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

Oh! So the acute angle is 67°, and the obtuse is 113°.

So at the intersection, the angles are 67° and 113°.

Now, which is ∠9?

In the diagram, ∠9 is likely the angle that is vertically opposite to the angle we can find from other parts.

We can use the fact that at line b × line y, ∠3 = 74°, and this is the angle between line b and line y.

At line a × line y, ∠6 = 39°, angle between line a and line y.

Now, the angle between line a and line b at the top can be found by considering the triangle formed by the two lines and the parallel line, but since they intersect above, the angle at the intersection is equal to the difference of the angles they make with the parallel line.

Specifically, the angle between the two transversals is |74° - 39°| = 35°? No, that's not right.

Let's think of the directions.

From the point where line b crosses line y, line b is going down to the right at 74° to the horizontal.

From the point where line a crosses line y, line a is going down to the left at 39° to the horizontal? No, line a is going up to the right, so from its crossing with y, it is going down to the left at 39° to the horizontal.

So at the top, the angle between the two lines is the sum of the angles they make with the horizontal, because they are on opposite sides.

Line b is at 74° below horizontal on the right, line a is at 39° below horizontal on the left, so when they meet, the angle between them is 74° + 39° = 113°.

Yes, as I had earlier.

So the angle at the intersection of a and b is 113° for the vertical angles that are "outside", and 67° for the "inside".

In the diagram, ∠9 is probably the angle that is between the two lines in the upper part, which would be the 113° angle, since the lines are diverging.

Let's assume that.

To confirm, we can use the fact that the sum of angles around the point is 360°, and we can find other angles.

Another way: use the parallel lines and corresponding angles.

For example, at line b × line z, we can find angles, then use that.

Let's do intersection 4: line b and line z.

Since y || z, and line b is a transversal, then corresponding angles are equal.

At line b × line y, we have ∠3 = 74° (bottom-right)

At line b × line z, the corresponding angle would be the bottom-right angle, which is ∠18.

Because both are on the same side of the transversal and both below the parallel lines.

So ∠18 = ∠3 = 74° (corresponding angles)

Similarly, the top-right angle at line b × line y is ∠1 = 106°, so corresponding angle at line b × line z is the top-right angle, which is ∠16.

So ∠16 = ∠1 = 106° (corresponding)

Then, at line b × line z:

- ∠16 = 106°
- ∠18 = 74°
- Then ∠15 and ∠17 can be found.

Vertical angles: ∠15 and ∠18 are vertical? Let's see.

At this intersection:

- ∠15: top-left
- ∠16: top-right
- ∠17: bottom-left
- ∠18: bottom-right

Then:
- ∠15 and ∠18 are vertical angles? No, ∠15 and ∠18 are not opposite; typically, ∠15 and ∠18 are adjacent if it's a standard cross.

Vertical angles:
- ∠15 and ∠18 are not vertical; rather, ∠15 and the angle opposite to it.

If ∠15 is top-left, then vertical is bottom-right, which is ∠18.

Yes, so ∠15 = ∠18 = 74° (vertical)

Similarly, ∠16 and ∠17 are vertical, so ∠17 = ∠16 = 106°

Also, check: ∠15 + ∠16 = 74° + 106° = 180°, which is good for adjacent angles on the straight line z.

So for intersection 4 (line b × line z):
- ∠15 = 74°
- ∠16 = 106°
- ∠17 = 106°
- ∠18 = 74°

Now, back to intersection 3: line a and line b cross.

We have angles from other parts.

For example, at line a × line z, we have ∠13 = 39° (top-right)

At line b × line z, we have ∠15 = 74° (top-left)

Now, at the intersection of a and b, the angle can be found from the triangle formed by the two lines and line z, but since they intersect above, we can use the fact that the angle at the intersection is equal to the difference of the angles they make with line z.

At line z, line a makes an angle of 39° with z (since ∠13 = 39°, which is the angle between line a and line z on the top-right)

Line b makes an angle of 74° with line z on the top-left (∠15 = 74°)

Since both are on the same side of the transversals, and line z is straight, the angle between line a and line b at their intersection is |74° - 39°| = 35°? No.

Let's think of the directions.

From line z, line a is going up to the right at 39° to the horizontal (since ∠13 = 39° is the angle between line a and line z on the top-right, so if line z is horizontal, line a is at 39° above horizontal on the right side).

Line b is going up to the left at 74° to the horizontal, because at line b × line z, ∠15 = 74° is top-left, so line b is at 74° above horizontal on the left side.

So when they intersect, the angle between them is 39° + 74° = 113°, same as before.

So the angle at the intersection is 113° for the vertical angles that are "between" the lines in the upper region.

In the diagram, ∠9 is likely this angle.

Typically, ∠9 is the angle at the top of the intersection, which would be the angle between the two lines above, so 113°.

Then the vertical angle to it is also 113°, which might be ∠10 or ∠11.

Let's define the angles at the intersection of a and b.

Assume:
- ∠8: top-left
- ∠9: top-right
- ∠10: bottom-right
- ∠11: bottom-left

Then, the angle between the two lines in the top-right region is ∠9.

Since line a is going up to the right, line b is going up to the left, so in the top-right, the angle between them is the angle from line b to line a, which is 180° - (angle of b + angle of a) = 180° - (74° + 39°) = 180° - 113° = 67°? I'm confusing myself.

Let's calculate the actual angle.

The direction of line a: from the intersection, it goes down to the left at 39° to the horizontal, or up to the right at 39°.

From the intersection point, line a has a slope corresponding to 39° above horizontal to the right.

Line b has a slope corresponding to 74° above horizontal to the left, which is 180° - 74° = 106° from positive x-axis.

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

So the acute angle is 67°, obtuse is 113°.

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

We can use the fact that at line a × line y, we have ∠5 = 39°, which is the angle between line a and line y on the top-right.

At line b × line y, we have ∠1 = 106°, which is the angle between line b and line y on the top-right.

Then, at the intersection of a and b, the angle ∠9 can be found as the difference or sum.

Consider the triangle formed by the two lines and line y, but they don't form a triangle with line y because they intersect above.

The angle at the intersection of a and b is equal to the difference of the angles they make with the parallel line.

Specifically, the angle between the two transversals is |m1 - m2|, but in this case, since they are on the same side, it's the difference.

From line y, line a makes an angle of 39° with y (at ∠5), line b makes an angle of 106° with y (at ∠1), but 106° is obtuse, so the acute angle is 74°, but for the direction, the angle from the horizontal.

Perhaps it's easier to use the following: the sum of the angles in the quadrilateral or use parallel lines properties.

Let's use the fact that the alternate interior angles or corresponding.

Another approach: the angle ∠9 is vertically opposite to the angle that is between the two lines at the bottom, but we can calculate it from the angles at the bottom.

At line a × line z, ∠13 = 39° (top-right)

At line b × line z, ∠15 = 74° (top-left)

Now, these two angles are on the same side of line z, and the lines a and b are crossing above, so the angle between a and b at their intersection is equal to the difference of these two angles if they were on the same side, but they are on opposite sides.

Specifically, the angle between line a and line b is |74° - 39°| = 35°? That can't be right because earlier calculation gave 67° or 113°.

Let's calculate the actual value.

Suppose we have line z horizontal.

At line a × line z, the angle on the top-right is 39°, so the angle between line a and line z is 39°, so the slope of line a is tan(39°) .

At line b × line z, the angle on the top-left is 74°, so the angle between line b and line z is 74°, so the slope of line b is tan(74°) but in the other direction, so the angle with the positive x-axis is 180° - 74° = 106°.

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

So the acute angle is 67°, obtuse is 113°.

In the diagram, for the intersection of a and b, the angle ∠9 is likely the angle that is in the region towards line y, which would be the larger angle, 113°, because the lines are spreading out.

We can verify with the sum.

At the intersection, the four angles sum to 360°.

Also, we can find one angle from the parallel lines.

For example, the angle ∠9 and the angle at line a × line y on the bottom-left, which is ∠6 = 39°, are related.

Specifically, ∠9 and ∠6 are alternate interior angles or something, but not directly.

Notice that line a is a straight line, so the angle at line a × line b and line a × line y are on the same line.

On line a, the angles at the two intersections with the parallel lines and with line b.

From line a × line y to line a × line b, the angle along line a.

At line a × line y, the angle between line a and line y is 39° (at ∠6).

At line a × line b, the angle between line a and line b is θ.

Then, since line y and line b are not parallel, it's hard.

Perhaps use the fact that the triangle formed by the two transversals and the parallel line has angles that sum to 180°.

Consider the triangle formed by line a, line b, and line y.

But line a and line b intersect above line y, so they form a triangle with line y.

Yes! The two transversals and the parallel line form a triangle.

Specifically, line a and line b intersect at a point above line y, and they intersect line y at two points, so they form a triangle with vertices at: P = intersection of a and b, Q = intersection of a and y, R = intersection of b and y.

So triangle PQR, with P on top, Q on left, R on right.

Then, at Q (line a × line y), the angle is the angle between line a and line y, which is ∠6 = 39° (since it's the angle in the triangle at Q).

At R (line b × line y), the angle is the angle between line b and line y, which is ∠3 = 74° (angle in the triangle at R).

Then, at P (intersection of a and b), the angle is 180° - 39° - 74° = 67°.

Oh! So in the triangle, the angle at P is 67°.

This angle at P is the angle between line a and line b inside the triangle, which is the angle facing line y.

In the diagram, this angle is likely ∠11 or ∠8, but typically, for the intersection, the angle in the triangle is the one between the two lines on the side towards the parallel line.

In this case, since the triangle is below the intersection point P, the angle at P in the triangle is the angle between the two lines below, which would be ∠11 or ∠10.

In standard labeling, if P is the intersection, and the triangle is below, then the angle at P in the triangle is the bottom angle, which might be ∠11 or ∠10.

In the diagram, ∠11 is likely the bottom-left angle at the intersection.

So if the angle in the triangle is 67°, then ∠11 = 67°.

Then, since vertical angles are equal, the angle opposite to it is also 67°, which would be ∠9 or ∠10.

If ∠11 is bottom-left, then vertical is top-right, which is ∠9.

So ∠9 = 67°.

Then the other two angles are 180° - 67° = 113° each, so ∠8 = 113°, ∠10 = 113°.

Let me confirm.

In triangle PQR:
- At Q: angle between line a and line y is 39° — this is the angle inside the triangle, so yes, ∠PQR = 39°
- At R: angle between line b and line y is 74° — ∠PRQ = 74°
- At P: ∠QPR = 180° - 39° - 74° = 67°

This angle at P is the angle between the two lines as seen from the triangle, which is the angle between the rays going down to Q and R.

In the intersection, this is the angle between the two lines in the downward direction, which is the angle at the bottom of the intersection.

In the diagram, for the intersection of a and b, the bottom angle is likely ∠11 or ∠10.

Typically, ∠11 is labeled as the bottom-left angle, which would be the angle between line b and line a in the bottom-left region.

Since line b is on the left, line a on the right, in the bottom-left, it would be the angle between the downward part of line b and the downward part of line a, but in the triangle, it's the angle between the rays to Q and R, which are both downward, so yes, ∠11 = 67°.

Then, vertical angle to ∠11 is ∠9 (top-right), so ∠9 = 67°.

Then, the adjacent angles are 180° - 67° = 113°, so ∠8 = 113° (top-left), ∠10 = 113° (bottom-right).

Perfect.

So for intersection 3 (line a × line b):
- ∠8 = 113°
- ∠9 = 67°
- ∠10 = 113°
- ∠11 = 67°

Now, we have all angles.

Let's summarize:

From intersection 1 (b × y):
- ∠1 = 106°
- ∠2 = 106°
- ∠3 = 74°

From intersection 2 (a × y):
- ∠4 = 141°
- ∠5 = 39°
- ∠6 = 39°
- ∠7 = 141°

From intersection 3 (a × b):
- ∠8 = 113°
- ∠9 = 67°
- ∠10 = 113°
- ∠11 = 67°

From intersection 4 (a × z):
- ∠12 = 141° (top-left)
- ∠13 = 39° (top-right)
- ∠14 = 39° (bottom-left, given)
- and the bottom-right angle is not labeled, but in the diagram, it might be part of the next, but we have ∠15 etc. at the other intersection.

At line a × line z, we have:
- ∠12 = 141°
- ∠13 = 39°
- ∠14 = 39°
- and the fourth angle, bottom-right, should be 141°, and this is vertical to ∠12, so it's 141°, but in the diagram, this angle is not labeled with a number; instead, the next angles are for line b × line z.

In the user's text, after ∠14, it has "∠15 = ", which is at the intersection of line b and line z.

So for line a × line z, the angles are ∠12, ∠13, ∠14, and the fourth is not numbered, but we can leave it.

Similarly, for line b × line z:
- ∠15 = 74° (top-left)
- ∠16 = 106° (top-right)
- ∠17 = 106° (bottom-left)
- ∠18 = 74° (bottom-right)

Now, to confirm, at line a × line z, the bottom-right angle should be vertical to ∠12 = 141°, so it's 141°, and it's adjacent to ∠14 = 39°, and 141° + 39° = 180°, good.

Similarly for others.

So all angles are found.

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

a. ∠9 & ∠11

From above, ∠9 = 67°, ∠11 = 67°, and they are vertical angles? In the intersection of a and b, ∠9 is top-right, ∠11 is bottom-left, which are vertical angles, yes.

So they are vertical angles.

b. ∠15 & ∠3

∠15 = 74°, ∠3 = 74°

∠3 is at line b × line y, bottom-right

∠15 is at line b × line z, top-left

Since lines y and z are parallel, and line b is the transversal, then ∠3 and ∠15 are alternate interior angles.

∠3 is below line y, on the right side of transversal b.

∠15 is above line z, on the left side of transversal b.

Since y || z, and b is transversal, alternate interior angles are equal, and here both are 74°, so yes, they are alternate interior angles.

c. ∠14 & ∠5

∠14 = 39°, ∠5 = 39°

∠14 is at line a × line z, bottom-left

∠5 is at line a × line y, top-right

Since y || z, and a is transversal, then ∠14 and ∠5 are corresponding angles? Let's see.

∠14 is below line z, on the left side of transversal a.

∠5 is above line y, on the right side of transversal a.

Corresponding angles would be in the same relative position.

For example, the angle below line z on the left corresponds to the angle below line y on the left, which is ∠6.

∠5 is above line y on the right.

Actually, ∠14 and ∠5 are not corresponding; rather, ∠14 and ∠6 are corresponding, both below the parallel lines on the left side.

∠5 and ∠13 are corresponding, both above on the right side.

But ∠14 and ∠5: ∠14 is below z on left, ∠5 is above y on right.

Since the lines are parallel, and transversal a, the angle ∠14 and the angle at y on the same side.

Note that ∠5 and ∠6 are vertical angles, both 39°, and ∠6 corresponds to ∠14, so ∠5 = ∠6 = ∠14, so they are equal, but what is the relationship?

Actually, ∠14 and ∠5 are alternate exterior angles or something.

Let's see their positions.

Transversal a, parallel lines y and z.

∠5 is above line y, on the right side of transversal a.

∠14 is below line z, on the left side of transversal a.

So they are on opposite sides of the transversal, and both outside the parallel lines, so they are alternate exterior angles.

And since the lines are parallel, alternate exterior angles are equal, which they are (both 39°).

So the relationship is alternate exterior angles.

To confirm: alternate exterior angles are on opposite sides of the transversal and outside the parallel lines.

Yes, ∠5 is above y, so outside, on the right; ∠14 is below z, so outside, on the left; so opposite sides, both exterior, so alternate exterior angles.

Perfect.

So now, let's list all answers.

First, the missing angles:

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

And for the relationships:

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

Now, to box the final answer, but since it's multiple parts, I'll present it clearly.

The user asked to determine the measure of each missing angle, and name the relationships.

So in the final answer, I'll list all.

But the instruction is to provide only the final correct answer after all reasoning.

So I'll write it as:

Final Answer:

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

a. vertical angles
b. alternate interior angles
c. alternate exterior angles

Note: ∠14 was given as 39°, so it's not missing, but included for completeness.

In the diagram, ∠14 is given, so perhaps not to be solved, but in the list, it's there.

Similarly, the 74° and 39° are given, so the missing are the numbered ones.

But in the answer, we provide all as per the diagram.

I think this is correct.