Geometry worksheet on parallel lines cut by a transversal, requiring students to calculate missing angles and identify angle relationships.

Let’s solve this step by step.

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

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

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

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Step 1: Use vertical angles and linear pairs at each intersection

#### Top-left intersection (lines y and b):
Given: angle between line y (left) and transversal b (up) = 74° → this is the angle above line y, left of transversal b.

So:

- ∠2 is vertically opposite to the 74°? Wait — let’s label carefully.

Actually, looking at standard labeling:

At intersection of line y and transversal b:

The 74° is shown in the upper-left quadrant. So:

→ The angle directly across (vertical angle) is ∠3 → so ∠3 = 74°

→ Adjacent angles on a straight line add to 180°, so:

∠1 + 74° = 180° → ∠1 = 106°
∠2 + 74° = 180° → ∠2 = 106°? Wait — no.

Wait — if 74° is in the top-left, then:

Top-right = ∠1
Bottom-left = ∠2
Bottom-right = ∠3

Standard position:

If we have two lines crossing, four angles:

Label them clockwise from top-left:

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

Then:

Vertical angles:
74° and ∠3 are vertical → ∠3 = 74°
∠1 and 2 are vertical → so ∠1 = 2

Linear pair:
74° + ∠1 = 180° → ∠1 = 106° → so ∠2 = 106°

✔ So:
∠1 = 106°
∠2 = 106°
∠3 = 74°

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#### Bottom-left intersection (lines z and transversal a):

Given: angle below line z, left of transversal a = 39° → that’s ∠14? Let’s see.

Looking at diagram:

At line z and transversal a:

Angles labeled:

Left side: ∠12 (top-left), ∠14 (bottom-left)
Right side: ∠13 (top-right), ? (bottom-right not labeled yet)

But 39° is shown in the bottom-left → that’s ∠14 = 39°

Then:

Vertical angle to ∠14 is ∠13 → so ∠13 = 39°

Adjacent angles:

∠12 + ∠14 = 180° → ∠12 = 141°
∠13 + ∠14 = 180°? No — ∠13 and ∠14 are adjacent? Actually, ∠13 and ∠14 are on same side of transversal but different sides of line z.

Better: at intersection of z and a:

Four angles:

Top-left: ∠12
Top-right: ∠13
Bottom-right: ? (not labeled)
Bottom-left: ∠14 = 39°

Vertical angles:

∠12 and bottom-right are vertical → but bottom-right isn’t labeled.
∠13 and ∠14 are NOT vertical — actually, ∠13 and the angle opposite it (which would be bottom-right) are vertical.

Wait — better approach:

Since ∠14 = 39°, its vertical angle is ∠13? No — vertical angle to ∠14 is the angle diagonally opposite — which is ∠13 only if they are across from each other.

In standard cross:

If you have:

A | B
C | D

Then A and D are vertical, B and C are vertical.

So if ∠14 is bottom-left (C), then vertical angle is top-right (B) → which is ∠13.

Yes! So ∠13 = ∠14 = 39°? No — wait, vertical angles are equal, but ∠14 and ∠13 are adjacent if they share a side.

I think I’m confusing myself.

Let me define clearly:

At intersection of line z (horizontal) and transversal a (diagonal going up-right):

Assume:

- Above line z, left of transversal a: ∠12
- Above line z, right of transversal a: ∠13
- Below line z, left of transversal a: ∠14 = 39°
- Below line z, right of transversal a: let's call it X (not labeled)

Then:

Vertical angles:

∠12 and X are vertical → so ∠12 = X
∠13 and ∠14 are vertical → so ∠13 = ∠14 = 39°

Linear pairs:

∠12 + ∠13 = 180° → ∠12 = 180 - 39 = 141°
∠14 + X = 180° → X = 141° → matches ∠12

Also, ∠13 + X = 180°? 39 + 141 = 180 → yes.

So:

∠12 = 141°
∠13 = 39°
∠14 = 39°
(and the unlabeled bottom-right is 141°)

But in the diagram, ∠15 and ∠17 are also there — those are on line z and transversal b.

Hold on — we have two transversals: a and b.

Transversal a cuts both y and z.
Transversal b cuts both y and z.

And they intersect each other somewhere in the middle, forming angles ∠8, ∠9, ∠10, ∠11.

So let’s handle intersections separately.

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Intersection 1: Line y and transversal b → already did:

∠1 = 106°
∠2 = 106°
∠3 = 74°

(∠4, ∠5, ∠6, ∠7 are on line y and transversal a — we’ll do next)

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Intersection 2: Line y and transversal a

This is the top-right intersection.

No given angles here, but since lines y and z are parallel, and transversal a cuts them, we can use corresponding angles or alternate interior once we know something on z.

From earlier, at line z and transversal a, we have ∠13 = 39° (above line z, right of transversal a)

Since y || z, and transversal a, then:

Corresponding angle to ∠13 is ∠5 (above line y, right of transversal a) → so ∠5 = ∠13 = 39°

Similarly, alternate interior: ∠13 and ∠6 are alternate interior? Let’s see.

Actually, ∠13 is above z, right of a → corresponding is above y, right of a → ∠5

So ∠5 = 39°

Then at intersection y and a:

Angles:

Top-left: ∠4
Top-right: ∠5 = 39°
Bottom-right: ∠7
Bottom-left: ∠6

Vertical angles:

∠4 and ∠7 are vertical → ∠4 = ∠7
∠5 and 6 are vertical → ∠6 = ∠5 = 39°

Linear pairs:

∠4 + ∠5 = 180° → ∠4 = 180 - 39 = 141° → so ∠7 = 141°

∠6 + ∠7 = 180° → 39 + 141 = 180 → good.

So:

∠4 = 141°
∠5 = 39°
∠6 = 39°
∠7 = 141°

---

Now, Intersection 3: Line z and transversal b

This is bottom-right intersection.

We don’t have direct given, but we can use parallel lines with what we know from top.

From top-left, at line y and transversal b, we had ∠3 = 74° (below line y, right of transversal b)

Since y || z, and transversal b, then:

Alternate interior angle to ∠3 is ∠15 (above line z, left of transversal b) → so ∠15 = ∠3 = 74°

Corresponding angle to ∠3 is ∠17 (below line z, right of transversal b) → so ∠17 = ∠3 = 74°? Wait, corresponding should be same relative position.

∠3 is below y, right of b → corresponding is below z, right of b → which is ∠17 → so ∠17 = 74°

Now at intersection z and b:

Angles:

Top-left: ∠15
Top-right: ∠16
Bottom-right: ∠18
Bottom-left: ∠17

We have ∠15 = 74° (from alternate interior)
∠17 = 74° (corresponding to ∠3)

But ∠15 and ∠17 are not vertical — let’s assign properly.

Actually, if ∠15 is top-left, ∠17 is bottom-left — they are on same side of transversal, different sides of line z.

Better: use vertical and linear.

We know ∠15 = 74° (as alternate interior to ∠3)

Then vertical angle to ∠15 is ∠18 → so ∠18 = 74°

Adjacent: ∠15 + ∠16 = 180° → ∠16 = 106°

∠17 + ∠18 = 180° → ∠17 = 180 - 74 = 106°? But earlier I said ∠17 = 74° — contradiction.

Mistake!

Let’s clarify positions.

At line z and transversal b:

Assume:

- Above line z, left of transversal b: ∠15
- Above line z, right of transversal b: ∠16
- Below line z, left of transversal b: ∠17
- Below line z, right of transversal b: ∠18

From parallel lines:

∠3 is at line y, below y, right of b → so that’s like “south-east” at top intersection.

Corresponding angle at bottom intersection should be same relative position: below z, right of b → which is ∠18.

So ∠18 = ∠3 = 74°

Alternate interior: ∠3 and ∠15 are alternate interior? ∠3 is below y, right of b; ∠15 is above z, left of b — yes, they are alternate interior → so ∠15 = ∠3 = 74°

Now, at intersection z and b:

∠15 = 74° (top-left)
∠18 = 74° (bottom-right) — vertical to ∠15? Yes! Because top-left and bottom-right are vertical → so ∠15 and ∠18 are vertical → both 74° — consistent.

Then:

∠16 (top-right) and ∠17 (bottom-left) are vertical → so ∠16 = ∠17

Linear pair: ∠15 + ∠16 = 180° → 74 + ∠16 = 180 → ∠16 = 106° → so ∠17 = 106°

Check: ∠17 + ∠18 = 106 + 74 = 180 → good.

So:

∠15 = 74°
∠16 = 106°
∠17 = 106°
∠18 = 74°

Earlier I thought ∠17 was corresponding to ∠3, but no — corresponding to ∠3 (below y, right of b) is ∠18 (below z, right of b) — yes, that’s correct.

And ∠15 is alternate interior to ∠3 — also correct.

Good.

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Now, the middle intersection: where transversals a and b cross each other.

This forms angles ∠8, ∠9, ∠10, ∠11.

We need to find these.

Note that this intersection is between the two transversals, not involving the parallel lines directly, but we can use triangle or just vertical/linear.

Actually, we can consider the triangle formed or use the fact that around a point, angles sum to 360°, and vertical angles are equal.

But perhaps easier: look at the angles around this point.

We can find some of these angles using the triangles or by noting that they are related to the other angles via straight lines.

For example, consider the line from top-left to bottom-right: that’s transversal b.

Along transversal b, from top to bottom, we have:

At top: angles ∠1, ∠2, ∠3 — but specifically, the angle between line y and transversal b on the lower side is ∠3 = 74°.

As we go down transversal b, when it crosses transversal a, it forms angles.

Similarly, transversal a has at top: ∠5 = 39°, etc.

Perhaps consider the triangle formed by the two transversals and one of the parallel lines? Not necessary.

Better: at the intersection of a and b, the angles are determined by the directions of the lines.

We can find the angle between the two transversals.

For example, at the top, on line y, the angle between transversal b and transversal a.

On line y, from left to right:

First transversal b comes in, making ∠3 = 74° with line y (below).

Then transversal a comes in, making ∠6 = 39° with line y (below).

The angle between the two transversals on the lower side of line y would be |74° - 39°| = 35°? Not exactly, because they are on the same side.

Actually, the angle between transversal b and transversal a at their intersection can be found by considering the triangle or using the fact that the sum of angles around a point is 360°.

Let’s denote the intersection point of a and b as P.

At P, four angles: ∠8, ∠9, ∠10, ∠11.

Typically, ∠8 and ∠10 are vertical, ∠9 and ∠11 are vertical.

To find them, we can use the fact that along transversal b, the angles on a straight line must sum appropriately.

Consider the path along transversal b from top to bottom.

At the top intersection (y and b), the angle between line y and transversal b on the side towards the middle is ∠3 = 74°.

As we move down transversal b, when we reach the intersection with a, the angle that transversal b makes with the horizontal might change, but since it's a straight line, the direction is constant.

Actually, the key is that the angle between the two transversals is fixed.

We can calculate the angle between transversal a and transversal b by looking at the angles they make with the parallel lines.

For example, transversal b makes an angle of 74° with line y (since ∠3 = 74°, and it's the acute angle? Or obtuse?).

∠3 = 74° is the angle below line y, right of transversal b — so if line y is horizontal, transversal b is going down to the right, making 74° with the horizontal? Or with the line.

Actually, in geometry, when we say the angle between a transversal and a line, it's the smaller angle, but here it's specified.

Perhaps use the triangle formed by the two transversals and the segment between the parallel lines.

Notice that between line y and line z, the two transversals form a sort of "X", and we can consider the quadrilateral or the triangles.

Another way: the sum of angles in the triangle formed by the two transversals and one of the parallel lines.

For example, consider the triangle formed by points: intersection of y and b, intersection of y and a, and intersection of a and b.

That might be complicated.

Let's think differently.

At the intersection of a and b, the angles can be found by noting that they are supplementary to the angles on the straight lines.

For instance, along transversal a, from top to bottom:

At top (y and a), the angle below line y, left of transversal a is ∠6 = 39°.

As we go down transversal a, when we reach the intersection with b, the angle that transversal a makes with the horizontal is still the same, but now we have another line.

The angle between transversal a and transversal b at their intersection can be calculated as the difference of the angles they make with the parallel lines, but since the lines are parallel, the alternate interior angles are equal, so the angle between the transversals should be the same regardless.

Let's calculate the angle between the two transversals.

Suppose we consider the direction.

Assume line y and z are horizontal.

Transversal b: at line y, it makes an angle of 74° with line y on the lower side. Since it's cutting down to the right, the angle with the positive x-axis might be 180° - 74° = 106° if measured from positive x, but let's think in terms of slope.

The angle that transversal b makes with the horizontal line y is 74° below the horizontal, so its direction is 74° south of east, or -74° from positive x-axis.

Similarly, transversal a: at line y, it makes an angle of ∠6 = 39° with line y on the lower side. From the diagram, transversal a is going down to the left? No, in the diagram, transversal a is going from top-right to bottom-left, I think.

Let's look back at the diagram description.

In the original problem, transversal a is labeled with arrowheads: at top, it's going up-right, at bottom, down-left, so it's sloping down to the left.

Transversal b is going down to the right.

So at line y, for transversal a: the angle below line y, left of transversal a is ∠6 = 39°.

Since transversal a is going down to the left, the angle it makes with the horizontal line y on the lower side is 39°, but on the left side.

So if line y is horizontal, transversal a is making 39° with it on the lower-left side, so its direction is 180° + 39° = 219° from positive x-axis, or -141°.

Transversal b is making 74° with line y on the lower-right side, so its direction is -74° or 286°.

Then the angle between the two transversals is |286° - 219°| = 67°, or the smaller angle between them is min(67, 360-67) = 67°.

But this might be overcomplicating.

Perhaps use the fact that in the region between the parallel lines, the consecutive interior angles or something.

Another idea: consider the triangle formed by the two transversals and the line connecting the two points on the parallel lines, but that's not helpful.

Let's use the straight line property.

For example, along transversal b, from the top intersection to the bottom intersection, it's a straight line, so the angles on one side should add up.

At the top, on the lower side of line y, the angle between line y and transversal b is ∠3 = 74°.

At the bottom, on the upper side of line z, the angle between line z and transversal b is ∠15 = 74° (as we found).

Since the lines are parallel, and transversal b is straight, the angle it makes with the parallel lines is the same, which is consistent.

Now, at the intersection with transversal a, the angle that transversal b makes with transversal a can be found by considering the polygon or the sum.

Perhaps the easiest way is to realize that the angles at the intersection of a and b are determined by the other angles via vertical angles and the fact that they are part of the figure.

Let's list what we have so far:

From top-left (y and b):
∠1 = 106°, ∠2 = 106°, ∠3 = 74°

From top-right (y and a):
∠4 = 141°, ∠5 = 39°, ∠6 = 39°, ∠7 = 141°

From bottom-left (z and a):
∠12 = 141°, ∠13 = 39°, ∠14 = 39° (and the unlabeled is 141°)

From bottom-right (z and b):
∠15 = 74°, ∠16 = 106°, ∠17 = 106°, ∠18 = 74°

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

Let me denote the angles as per diagram:

Typically, ∠8, 9, ∠10, ∠11 are around the center.

Assume that ∠8 is the angle between the lower part of b and the lower part of a, etc.

To find them, we can use the fact that on a straight line, angles sum to 180°.

For example, consider the line from top-left to bottom-right: that's transversal b.

Along this line, at the top, the angle between line y and transversal b on the side towards the center is ∠3 = 74°.

As we move down, when we reach the intersection with a, the angle between transversal b and the line to the left or right.

Notice that the angle between transversal b and transversal a at their intersection is the same as the angle between them at any point, and we can find it from the triangle formed with the parallel lines.

Consider the quadrilateral formed by the two parallel lines and the two transversals. It's a trapezoid with non-parallel sides being the transversals.

In this trapezoid, the sum of consecutive interior angles is 180°, but we already used that.

The key is that the two transversals intersect, and the vertical angles are equal, and adjacent angles are supplementary.

Moreover, we can find one of the angles by considering the triangle formed by, say, the point on y between the two transversals, the intersection point, and the point on z.

For example, take the segment on line y between the two transversals. The length doesn't matter, but the angles at the ends.

At the left end (intersection with b), the angle inside the trapezoid is ∠3 = 74°.

At the right end (intersection with a), the angle inside the trapezoid is ∠6 = 39°.

Then, the two transversals meet at the bottom, forming a triangle with the segment on y.

In that triangle, the two base angles are 74° and 39°, so the vertex angle at the intersection of a and b is 180° - 74° - 39° = 67°.

Is that correct?

Let's see: the triangle is formed by:

- Point A: intersection of y and b
- Point B: intersection of y and a
- Point C: intersection of a and b

Then, at point A, the angle of the triangle is the angle between line y and transversal b, which is ∠3 = 74°, but is that the angle inside the triangle?

If we consider the triangle ABC, with A on y-b, B on y-a, C on a-b.

Then, at A, the angle between AB (which is along y) and AC (which is along b) is indeed ∠3 = 74°, assuming we take the angle inside the triangle.

Similarly, at B, the angle between BA (along y) and BC (along a) is ∠6 = 39°.

Then, at C, the angle is 180° - 74° - 39° = 67°.

This angle at C is the angle between the two transversals, which is one of the angles at the intersection, say ∠9 or ∠11.

In the diagram, likely ∠9 is the angle at the top of the intersection, but let's assume that the angle we calculated, 67°, is the acute angle between the transversals.

Then, at the intersection, the four angles are: two of 67° and two of 180° - 67° = 113°, since vertical angles are equal, and adjacent are supplementary.

So, typically, ∠8 and ∠10 are vertical, ∠9 and ∠11 are vertical.

If the angle in the triangle is 67°, and it's the angle at C between the two transversals, and if we assume that this is the angle that is "inside" the trapezoid, then it might be ∠11 or ∠9.

Looking at the diagram description, ∠9 is probably the angle at the top of the intersection, between the upper parts of the transversals.

In our triangle, the angle at C is between the segments from C to A and C to B, which are parts of the transversals going upwards, so it should be the angle at the top, which is ∠9.

So ∠9 = 67°

Then, since vertical angles, ∠11 = ∠9 = 67°

Then, adjacent angles: ∠8 + ∠9 = 180° → ∠8 = 113°
∠10 + ∠9 = 180° → ∠10 = 113°

And ∠8 and ∠10 are vertical, so both 113° — good.

So:

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

Now, let's verify with another method.

For example, along transversal a, from top to bottom.

At top (y and a), the angle below line y, left of transversal a is ∠6 = 39°.

As we go down transversal a, when we reach the intersection with b, the angle between transversal a and the line to the left or right.

The angle that transversal a makes with the horizontal is constant.

At the bottom, at z and a, the angle above line z, right of transversal a is ∠13 = 39°.

Since the lines are parallel, and transversal a is straight, the angle it makes with the parallel lines is the same, which is consistent.

Now, at the intersection with b, the angle between transversal a and transversal b should be the same as before.

We can consider the angle on the other side.

For example, at the top, on the upper side of line y, for transversal a, the angle is ∠5 = 39°, but that's on the other side.

I think the triangle method is solid.

So we'll go with that.

Thus:

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

Now, let's compile all answers.

List of all 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°
∠15 = 74°
∠16 = 106°
∠17 = 106°
∠18 = 74°

Now, for the relationships:

a. ∠9 & ∠11

From above, ∠9 = 67°, ∠11 = 67°, and they are vertical angles? In the diagram, if ∠9 and ∠11 are opposite each other at the intersection, then they are vertical angles.

In standard labeling, if ∠9 is top, ∠11 is bottom, then yes, vertical angles.

So relationship: vertical angles

b. ∠15 & ∠3

∠15 = 74°, ∠3 = 74°

∠3 is at top, below y, right of b
∠15 is at bottom, above z, left of b

Since y || z, and transversal b, these are alternate interior angles.

Yes, because they are on opposite sides of the transversal and between the parallel lines.

So: alternate interior angles

c. ∠14 & ∠5

14 = 39°, ∠5 = 39°

∠14 is at bottom, below z, left of a
∠5 is at top, above y, right of a

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

∠5 is above y, right of a
∠14 is below z, left of a — not the same position.

Alternate exterior? ∠5 is above y, right of a — so exterior if we consider the region outside, but usually for parallel lines, we classify based on position.

Note that ∠5 and ∠13 are corresponding (both above the parallel lines, right of transversal a), and ∠13 and ∠14 are vertical angles? Earlier we had ∠13 = 39°, ∠14 = 39°, and they are vertical angles at the bottom intersection.

∠13 and ∠14 are vertical angles, so equal.

∠5 and ∠13 are corresponding angles (since y||z, transversal a), so ∠5 = ∠13

Thus ∠5 = ∠14

But what is the direct relationship between ∠5 and ∠14?

They are not adjacent, not vertical, not corresponding directly.

Actually, ∠5 and ∠14 are alternate exterior angles? Let's see.

Exterior angles are outside the parallel lines.

∠5 is above y, so if y is the top line, above it is exterior.

∠14 is below z, so below the bottom line is exterior.

And they are on opposite sides of the transversal: ∠5 is right of a, ∠14 is left of a.

So yes, they are alternate exterior angles.

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

So relationship: alternate exterior angles

Some might say they are just equal due to combination, but the specific pair relationship is alternate exterior.

Confirm:

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

Yes, ∠5 is outside (above y), right of transversal a.

∠14 is outside (below z), left of transversal a.

Opposite sides of transversal, and both exterior → alternate exterior angles.

Perfect.

So summary:

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

Now, final answer.

──────────────────────────────────────

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