Proportional Parts in Triangles and Parallel Lines worksheet from Kuta Software.

Let’s solve each problem one by one. These problems use the Triangle Proportionality Theorem (also called the Side-Splitter Theorem), which says:

> If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

We’ll apply this idea to each diagram.

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Problem 1:

We have a big triangle with a line inside parallel to the base. The left side is split into 15 and ? (top part). The bottom is split into 14 total, with 4 on the right — so the left segment of the base is 14 - 4 = 10.

So we set up the proportion:

Top left / Bottom left = Top right / Bottom right
But actually, since the line is parallel, the segments are proportional:

The whole left side is divided into 15 (bottom) and x (top).
The base is divided into 10 (left) and 4 (right).

Wait — let me reorient.

Actually, looking at the diagram: the inner line is parallel to the base. So the small top triangle is similar to the big triangle.

The left side of the big triangle is 15 + x? Wait — no.

Looking again: the label “15” is on the lower part of the left side, and “?” is on the upper part. The base is 14 total, with 4 on the right — so the left part of the base is 10.

Since the line is parallel, the ratios of corresponding parts are equal.

So:
Upper left / Lower left = Upper base / Lower base? No.

Better: The entire left side is split into two parts: top = ?, bottom = 15.
The base is split into left = 10, right = 4.

Because the line is parallel, the ratio of the top segment to the bottom segment on the left should equal the ratio of the top base segment to the bottom base segment? Not exactly.

Actually, the correct way: The small triangle on top is similar to the whole triangle.

So the height (or side) ratio equals the base ratio.

Let’s call the unknown top part “x”. Then the full left side is x + 15.

The base of the small top triangle corresponds to the 10-unit segment? Wait — no.

Actually, the inner line cuts the left side into x (top) and 15 (bottom), and cuts the base into 10 (left) and 4 (right)? That doesn’t make sense because the base is horizontal.

I think I misread. Let me reinterpret.

In problem 1: There’s a large triangle. A line parallel to the base cuts the left side and the right side. On the left side, from top to bottom: first segment is “?”, second is 15. On the base, from left to right: first segment is 14? Wait, no — the base is labeled 14 total, and there’s a mark showing 4 on the right end.

Actually, looking at standard diagrams: when a line is drawn parallel to the base, it creates a smaller similar triangle on top.

So if the left side has total length = ? + 15, and the base of the small triangle is proportional.

But the base of the small triangle isn't given directly. Instead, the base of the big triangle is 14, and the part under the small triangle is... wait.

Perhaps the 14 is the entire base, and the 4 is the part from the cut point to the right vertex. So the left part of the base (under the small triangle) is 14 - 4 = 10.

Then, since the triangles are similar:

Small triangle base / Big triangle base = Small triangle side / Big triangle side

So: 10 / 14 = ? / (? + 15)

Let x = ?

Then: 10/14 = x/(x+15)

Cross-multiply: 10(x + 15) = 14x
10x + 150 = 14x
150 = 4x
x = 37.5

That seems high. Let me check.

Alternatively, maybe the 15 is the entire left side? But the diagram shows “?” on top and 15 below, so likely x and 15 are segments.

Another approach: The line parallel to the base divides the two sides proportionally.

So: (top left) / (bottom left) = (top right) / (bottom right)

But we don’t have right side lengths.

Wait — in problem 1, only left side and base are labeled. Perhaps the 14 is the entire base, and the 4 is the segment from the foot of the parallel line to the right vertex. So the segment from left vertex to the foot is 10.

Then, the ratio of the segments on the left side should equal the ratio of the segments on the base.

That is: ? / 15 = 10 / 4 ? That would be if the line was dividing the sides in the same ratio, but that's not how it works.

Standard theorem: If a line parallel to one side intersects the other two sides, then it divides them proportionally.

So for the left side: divided into segments A and B. For the right side: divided into C and D. Then A/B = C/D.

But here, we have left side: top = ?, bottom = 15. Base: but the base is not one of the sides being intersected; the two sides being intersected are the left and right legs.

I think I need to assume that the 14 is the entire base, and the 4 is part of it, but that might not help.

Perhaps the 14 is the length from left vertex to the point where the parallel line meets the base? No.

Let me look at problem 2 for clue.

Problem 2: Left side has 15 and 24? Wait, labels: on the left, from bottom to top: 15 and then 24? No, the diagram shows on the left side: bottom segment 15, top segment 24? But that can't be because 24 is larger.

Actually, in problem 2: the big triangle has height or side labeled 24 on the right? Let's read.

Problem 2: There's a triangle. A line parallel to the base. On the left side, from bottom to top: 15 and then the top part is not labeled, but on the right side, from bottom to top: ? and 24. And the base is 24 total, with 15 on the left? This is confusing.

Perhaps I should use the property that the ratios of the corresponding sides are equal.

For problem 1: Let’s denote the unknown as x.

The small triangle on top has base = let's say b, and the big triangle has base 14.

The left side of small triangle is x, left side of big triangle is x + 15.

Since similar, x / (x+15) = b / 14.

But we don't know b.

From the base, the part from left to the cut is 14 - 4 = 10, which is the base of the trapezoid or something.

Actually, the segment of the base from the left vertex to the point where the parallel line meets the base is 10, and from there to the right vertex is 4.

Then, the small triangle on top has base 10? No, that doesn't make sense because the parallel line is above, so the small triangle's base is the parallel line itself, which is not on the base.

I think I have it backward.

In standard configuration: the parallel line is closer to the apex, so it creates a small triangle on top similar to the big triangle.

The base of the small triangle is the parallel line, and the base of the big triangle is the bottom.

The distance from apex to parallel line is proportional.

On the left side, from apex to parallel line is x, from parallel line to base is 15, so total left side is x+15.

On the base, the entire base is 14, but the part corresponding to the small triangle is not directly given.

However, the segment from the left vertex to the point where the parallel line meets the left side is x, and from there to the base is 15.

Similarly, on the base, from left vertex to the projection or something.

Perhaps the 4 is the length from the right vertex to the point where the parallel line meets the right side? But in problem 1, only left side and base are labeled.

Let's look at the diagram description: "1) ? on top left, 15 on bottom left, base 14 with 4 on the right end."

So, likely, the base is divided into two parts: from left to the foot of the perpendicular or something, but it's not specified.

Another idea: perhaps the 14 is the length of the base, and the 4 is the length of the segment on the base from the right vertex to the point where the parallel line meets the base. So the left part is 10.

Then, by the basic proportionality theorem, the line parallel to the base divides the two sides proportionally.

So for the left side: the ratio of the top segment to the bottom segment is x/15.

For the right side, we don't have information, but for the base, the ratio of the left segment to the right segment is 10/4 = 5/2.

But that's not directly related.

I recall that in such cases, the ratio is between the segments of the sides, not the base.

Let's think differently. Suppose the parallel line intersects the left side at P and the right side at Q. Then AP/PB = AQ/QC, where A is apex, B and C are base vertices.

In problem 1, A is top, B is bottom left, C is bottom right.

P is on AB, Q on AC.

AP = ?, PB = 15, so AB = ? + 15.

On the base BC, but the base is not divided by the parallel line; the parallel line is above, so it doesn't intersect the base.

I think I have a fundamental mistake.

In the diagram, the parallel line is inside the triangle, parallel to the base, so it intersects the two legs, not the base.

So for problem 1: the left leg is divided into two parts: top = ?, bottom = 15.

The right leg is divided into two parts: top = let's say y, bottom = z, but not given.

The base is 14, but that's the whole base, not divided.

The 4 is probably the length from the right vertex to the point where the parallel line meets the right leg? But that would be on the leg, not on the base.

Perhaps the 4 is on the base, indicating that from the right vertex to the foot of the parallel line's projection or something.

This is ambiguous. Let me try to assume that the 4 is the length of the segment on the base from the right vertex to the point directly below where the parallel line meets the right leg, but that's complicated.

Perhaps in some diagrams, the number on the base indicates the segment corresponding.

Let's look at problem 3 for clarity.

Problem 3: Triangle with a line parallel to the base. Left side: top 8, bottom 20. Base: left part ?, right part 18.

So here, the base is divided into two parts: left = ?, right = 18, and the whole base is ? + 18.

By proportionality, the ratio of the segments on the left side should equal the ratio on the right side, but we don't have right side.

The theorem says that the line divides the two sides proportionally, so for the left side: 8/20 = for the right side: top/bottom.

But we don't have right side lengths.

However, for the base, the segments are proportional to the sides.

Actually, the correct application is: the ratio of the top segment to the bottom segment on one side equals the ratio on the other side.

But in problem 3, we have left side: 8 and 20, so ratio 8:20 = 2:5.

Then on the right side, if we had top and bottom, it would be the same ratio.

But we have the base divided into ? and 18.

The base is not one of the sides; the sides are the legs.

I think I need to realize that the parallel line creates similar triangles, so the ratio of corresponding sides is constant.

For problem 3: the small triangle on top has left side 8, big triangle has left side 8+20=28.

So ratio of similarity is 8/28 = 2/7.

Then the base of the small triangle is (2/7) times the base of the big triangle.

But the base of the big triangle is ? + 18, and the base of the small triangle is the part above, which is not given.

The segment on the base from left to the cut is ?, which is the base of the trapezoid, not the small triangle.

Perhaps the ? is the length from the left vertex to the point where the parallel line meets the base, but that doesn't make sense because the parallel line is not on the base.

I recall that in some textbooks, when a line is drawn parallel to the base, and it intersects the two legs, then the segments are proportional, and also, the base is divided in the same ratio if you consider the projections, but it's messy.

Let's search for a standard approach.

For problem 1: let's assume that the 14 is the entire base, and the 4 is the length of the segment on the base from the right vertex to the point where the parallel line's extension meets, but that's not it.

Perhaps the 4 is the length of the right part of the base, and 14 is the left part, but the diagram says "14" with an arrow covering the whole base, and "4" on the right end, so likely 14 is total, 4 is right segment, so left segment is 10.

Then, by the triangle proportionality theorem, the line parallel to the base divides the two legs proportionally, and also, the ratio of the segments on the legs is equal to the ratio of the segments on the base? No.

Actually, there is a corollary: if a line is parallel to one side of a triangle, then it divides the other two sides proportionally, and moreover, the ratio is the same as the ratio of the distances from the apex.

But for the base, the length from the left vertex to the foot of the parallel line's intersection with the base is not defined.

I think I found the issue: in these diagrams, the numbers on the base indicate the lengths of the segments created by the feet of the perpendiculars or something, but in reality, for the proportionality, we only need the legs.

Let's look at problem 2.

Problem 2: Triangle with a line parallel to the base. On the left side, from bottom to top: 15 and then the top part is not labeled, but on the right side, from bottom to top: ? and 24. And the base is 24 total, with 15 on the left? The diagram shows "15" on the left part of the base, "24" on the whole base? No, it says "15" and "24" on the base, with 15 on left, 24 on right? That can't be because 15+24=39, but the base is labeled as 24? Confusing.

In problem 2: "15" is on the left part of the base, "24" is on the whole base? Or "24" is on the right side.

Let's read the user's description: "2) ? on the right side top, 24 on the right side bottom? No.

From the text: "2) ? on the right side, 24 on the right side? Let's see: "2) ? on the right side top, and 24 on the right side bottom? But it says "24" and "?" on the right side, with "15" and "24" on the base.

Perhaps: on the base, left segment 15, right segment 24, but that would make base 39, but it's labeled as 24? No.

Another possibility: in problem 2, the "24" is the length of the right side from bottom to top, and "?" is the top part, but that doesn't make sense.

Let's assume that for problem 2: the big triangle has right side of length 24 + ? , but the diagram shows "24" on the lower part of the right side, and "?" on the upper part, and on the base, "15" on the left, "24" on the right? But 15+24=39, and the base is not labeled as 39.

Perhaps the "24" on the base is the whole base, and "15" is the left segment, so right segment is 9.

Then, for the right side, lower part is 24, upper part is ?.

Then by proportionality, the ratio of the segments on the left side should equal the ratio on the right side.

But we don't have left side segments.

On the left side, we have no labels, only on the base and right side.

This is frustrating.

Let's try a different strategy. In many such problems, the number on the base indicates the length from the vertex to the point where the parallel line's projection falls, but for simplicity, in problem 1, if we assume that the line parallel to the base divides the left side into x and 15, and the base into 10 and 4 (since 14-4=10), then by the theorem, x/15 = 10/4, because the ratios are equal.

Is that correct? Let me verify with a known example.

Suppose a triangle with base 10, and a line parallel to base at half height, then it should divide the legs in 1:1 ratio, and the base of the small triangle is 5, so the segment on the base from left to the cut is 5, from cut to right is 5, so ratio 1:1, and on the leg, top:bottom = 1:1, so yes, x/15 = 10/4 would give x = 37.5, which is large, but mathematically possible.

In that case, for problem 1: x/15 = 10/4 = 5/2, so x = 15 * 5/2 = 75/2 = 37.5

Then for problem 3: left side top 8, bottom 20, so ratio 8/20 = 2/5.

Base: left segment ?, right segment 18, so ?/18 = 2/5, so ? = 36/5 = 7.2

But let's check if that makes sense.

In problem 3, if the ratio is 2:5, then the base should be divided in 2:5, so left: right = 2:5, so if right is 18, left is (2/5)*18 = 7.2, yes.

And for the whole base, 7.2 + 18 = 25.2, and the small triangle base would be proportional, but we don't need it.

So this seems consistent.

For problem 2: on the base, left segment 15, right segment ? but the diagram shows "15" and "24" on the base, with "24" likely the whole base or the right segment.

In problem 2: "15" on the left part of the base, "24" on the whole base? Or "24" on the right part.

From the text: "2) ? on the right side top, 24 on the right side bottom? Let's see the user's input: "2) ? on the right side, and 24 on the right side? It says "24" and "?" on the right side, with "15" and "24" on the base.

Perhaps "15" is the left segment of the base, "24" is the right segment of the base, so base is 15+24=39.

Then on the right side, lower part is 24, upper part is ?.

Then by proportionality, the ratio of the segments on the left side should equal the ratio on the right side.

But we don't have left side segments.

On the left side, no labels, so perhaps the ratio is based on the base segments.

The theorem is that the line divides the two sides proportionally, so for the left side, let's say top A, bottom B, for right side top C, bottom D, then A/B = C/D.

Also, the base is divided into segments proportional to the sides, but in this case, the base segments are 15 and 24, so ratio 15:24 = 5:8.

Then for the right side, if bottom is 24, top is ?, then ?/24 = 5/8? Why 5/8?

Actually, the ratio of the segments on the base is equal to the ratio of the segments on the legs only if the triangle is isosceles or something, but in general, it's not.

I think my initial assumption is wrong.

Let's go back to the similar triangles approach.

For problem 2: the small triangle on top is similar to the big triangle.

On the right side, the big triangle has side length = ? + 24 (if ? is top, 24 is bottom).

The base of the big triangle is 15 + 24 = 39? But the diagram may have "24" as the whole base.

In the user's input for problem 2: "2) ? on the right side top, 24 on the right side bottom? And "15" on the left part of the base, "24" on the whole base? It's ambiguous.

Perhaps "24" is the length of the right side from bottom to top, and "?" is the top part, but that doesn't make sense.

Another idea: in problem 2, the "24" is the length of the altitude or something, but unlikely.

Let's look at the numbers.

Perhaps for problem 2: the base is 24, with 15 on the left, so right segment is 9.

Then on the right side, the lower part is 24, upper part is ?.

Then by proportionality, the ratio of the segments on the left side should equal the ratio on the right side.

But we don't have left side.

Unless the left side is not labeled, but the ratio can be found from the base.

I recall that in some configurations, the ratio is the same as the ratio of the bases of the similar triangles.

For problem 2: the small triangle on top has base equal to the length of the parallel line, which is not given, but the big triangle has base 24 (assume), and the segment from left to the cut is 15, so the base of the small triangle is 15? No.

If the parallel line is close to the apex, the small triangle's base is small.

Perhaps the 15 is the length from the left vertex to the point where the parallel line meets the left side, but that's on the leg.

I think I need to assume that for all problems, the number on the base indicates the length of the segment from the vertex to the point where the parallel line's foot is, but for simplicity, in problem 1, with left side segments x and 15, base segments 10 and 4, then x/15 = 10/4, as I had.

And for problem 3, 8/20 = ?/18, so ? = (8/20)*18 = (2/5)*18 = 36/5 = 7.2

For problem 2: let's assume that on the base, left segment 15, right segment 24, so ratio 15:24 = 5:8.

Then on the right side, if the lower part is 24, upper part is ?, then ?/24 = 5/8, so ? = 15.

But why would the ratio be the same?

Perhaps the ratio of the top to bottom on each side is equal to the ratio of the corresponding base segments.

In problem 2, if the base is divided into 15 and 24, then for the right side, the top segment / bottom segment = 15/24 = 5/8, so if bottom is 24, top is (5/8)*24 = 15.

So ? = 15.

Similarly, for problem 1: base segments 10 and 4, ratio 10:4 = 5:2, so on left side, top/bottom = 5/2, so x/15 = 5/2, x = 37.5

For problem 3: base segments ? and 18, and on left side, top/bottom = 8/20 = 2/5, so ?/18 = 2/5, ? = 7.2

Now for problem 4: triangle with a line parallel to the base. Labels: on the left side, top 7, bottom 15? The diagram shows "7" on the top left, "15" on the bottom left, and on the right side, "2" on the top, "12" on the bottom? And the base is not labeled, but we have to find nothing, wait, the task is to find missing length, but in problem 4, all are given? No, in problem 4, it's "find the missing length", but in the diagram, all seem labeled: 7, 15 on left, 2, 12 on right, and no question mark. Perhaps I misread.

In the user's input: "4) 7 on top left, 15 on bottom left, 2 on top right, 12 on bottom right" — but then what is missing? Perhaps it's to verify or something, but the instruction is "find the missing length", so maybe there is a question mark.

In the text: "4) 7 2 15 12" — perhaps the 2 is on the top right, and we need to find something else, but it's not specified.

Perhaps for problem 4, the missing length is not indicated, but in the diagram, there might be a question mark on one of the segments.

To save time, let's proceed with the pattern.

For problem 5: "Solve for x." Diagram: left side 45, with 5x on the top part? "45" on the whole left side, "5x" on the top part, "20" on the bottom part of the base, "36" on the whole base.

So, the line parallel to the base divides the left side into 5x (top) and 45-5x (bottom)? But 45 is the whole left side, so if 5x is top, then bottom is 45-5x.

On the base, whole base 36, bottom part 20, so the left part of the base is 36-20=16? Or 20 is the left segment.

Assume that on the base, the segment from left to the cut is 20, from cut to right is 16, since 36-20=16.

Then by proportionality, the ratio of top to bottom on left side equals ratio of left to right on base.

So 5x / (45-5x) = 20 / 16 = 5/4

So 5x / (45-5x) = 5/4

Divide both sides by 5: x / (9 - x) = 1/4 (since 45/5=9)

So 4x = 9 - x

5x = 9

x = 9/5 = 1.8

Then 5x = 9, and 45-5x = 36, and 9/36 = 1/4, and 20/16 = 5/4, not equal.

Mistake.

If 5x is the top segment, and 45 is the whole left side, then bottom segment is 45 - 5x.

Ratio top/bottom = 5x / (45-5x)

On the base, if 20 is the left segment (corresponding to the top triangle), and 16 is the right segment, then for the similar triangles, the ratio should be top/bottom = left_base / right_base? No.

In similar triangles, the ratio of corresponding sides is constant.

The small triangle on top has left side 5x, big triangle has left side 45, so ratio 5x/45 = x/9.

The base of the small triangle is the parallel line, which is not given, but the base of the big triangle is 36, so if the small triangle's base is b, then b/36 = x/9.

But we have the segment on the base from left to the cut is 20, which is not the base of the small triangle.

Perhaps the 20 is the length of the base of the small triangle.

In many diagrams, the number on the base under the small triangle is the base of the small triangle.

In problem 5, "20" is on the left part of the base, and "36" on the whole base, so likely 20 is the base of the small triangle, 36 is the base of the big triangle.

Then, since similar, ratio = 20/36 = 5/9.

Then on the left side, small triangle side / big triangle side = 5/9.

Small triangle side is 5x, big triangle side is 45, so 5x / 45 = 5/9

So 5x / 45 = 5/9

Simplify: x/9 = 5/9, so x = 5.

Then 5x = 25, and 25/45 = 5/9, and 20/36 = 5/9, yes.

So for problem 5, x = 5.

Similarly, for problem 1: if we assume that the 10 is the base of the small triangle, 14 is the base of the big triangle, then ratio 10/14 = 5/7.

Then on left side, small triangle side / big triangle side = 5/7.

Small triangle side is ?, big triangle side is ? + 15, so ? / (? + 15) = 5/7

7? = 5(? + 15)

7? = 5? + 75

2? = 75

? = 37.5

Same as before.

For problem 3: small triangle base is ? , big triangle base is ? + 18, ratio of sides 8/28 = 2/7, so ? / (? + 18) = 2/7

7? = 2(? + 18)

7? = 2? + 36

5? = 36

? = 7.2

Good.

For problem 2: let's assume that on the base, the left segment 15 is the base of the small triangle, and the whole base is 24, so ratio 15/24 = 5/8.

Then on the right side, small triangle side / big triangle side = 5/8.

Small triangle side is ? , big triangle side is ? + 24, so ? / (? + 24) = 5/8

8? = 5(? + 24)

8? = 5? + 120

3? = 120

? = 40

But earlier I thought 15, but that was wrong.

In problem 2, if the whole base is 24, and left segment is 15, then right segment is 9, but for the small triangle, its base is 15, big triangle base is 24, ratio 15/24 = 5/8.

On the right side, if the small triangle side is ? , big triangle side is ? + 24, then ? / (? + 24) = 5/8, so ? = 40.

But is the 24 on the right side the bottom part? In the diagram, "24" is on the lower part of the right side, so if ? is top, 24 is bottom, then big triangle side is ? + 24, yes.

So ? = 40.

For problem 4: labels: left side top 7, bottom 15, so big left side 22.

Right side top 2, bottom 12, so big right side 14.

Check if proportional: 7/15 vs 2/12 = 1/6, 7/15 ≈ 0.466, 1/6≈0.166, not equal, so perhaps not parallel, but the problem assumes it is parallel, so maybe the missing length is not here, or perhaps we need to find something else.

In problem 4, perhaps the "2" is not given, or there is a question mark.

In the user's input: "4) 7 2 15 12" — perhaps the 2 is the missing length, but it's written as 2, so likely all are given, but the task is to find missing, so maybe for problem 4, it's to verify or something, but let's skip and come back.

For problem 6: "Solve for x." Diagram: top side 28, with 8 on the right part? "28" on the whole top, "8" on the right segment, so left segment 20.

Left side: 3x-5 on the top part? "3x-5" on the left side top, "10" on the bottom part of the left side? And the base is not labeled, but we have to find x.

Assume that the line parallel to the base divides the top side into 20 and 8 (since 28-8=20), and the left side into 3x-5 (top) and 10 (bottom).

Then by proportionality, (3x-5)/10 = 20/8 = 5/2

So 3x-5 = 10 * 5/2 = 25

3x = 30

x = 10

Then 3x-5 = 25, and 25/10 = 2.5, 20/8 = 2.5, yes.

For problem 7: "Find the missing length." Diagram: three parallel lines cutting two transversals. On the left transversal, segments 15 and 6. On the right transversal, segments 25 and ?.

By the theorem of parallel lines cutting transversals, the segments are proportional.

So 15/6 = 25/?

Or 15/25 = 6/?

The ratio should be the same.

So 15/6 = 25/x, where x is the missing length.

15/6 = 5/2, so 5/2 = 25/x, so 5x = 50, x = 10.

If 15/25 = 6/x, then 3/5 = 6/x, 3x = 30, x = 10, same.

So ? = 10.

For problem 8: three parallel lines. On the left transversal, segments 77 and ?. On the right transversal, segments 30 and 15.

So 77/? = 30/15 = 2, so ? = 77/2 = 38.5

Or if the segments are corresponding, 77/x = 30/15 = 2, so x = 38.5

Now for problem 4: let's assume that the missing length is on the right side or something, but in the diagram, all are given, so perhaps it's to find if it's proportional, but the task is to find missing length, so maybe there is a question mark on one of the segments.

Perhaps in problem 4, the "2" is the missing length, but it's written as 2, so likely not.

Another possibility: in problem 4, the numbers are 7, 15 on left, and on right, 2 is given, 12 is given, but perhaps the 2 is on the top, and we need to find the bottom or something, but it's labeled 12.

Perhaps the missing length is the base or something, but not specified.

To resolve, let's calculate the ratio.

Left side: top 7, bottom 15, ratio 7:15

Right side: top 2, bottom 12, ratio 2:12 = 1:6

7/15 ≈ 0.466, 1/6≈0.166, not equal, so if the line is parallel, it should be equal, so perhaps the 2 is incorrect, or there is a question mark on the 2.

In many worksheets, for problem 4, the top right is missing, so let's assume that "2" is the missing length, but it's written as 2, so perhaps it's given, but we need to find another.

Perhaps the "12" is the whole right side, and "2" is the top, so bottom is 10, but then ratio 2/10 = 1/5, still not 7/15.

7/15 = 0.466, 1/5=0.2, not equal.

Perhaps the missing length is on the base, but not labeled.

For the sake of completing, let's assume that in problem 4, the top right is missing, and it's denoted as "?", but in the text it's "2", so perhaps it's 2, and we need to find something else.

Another idea: in problem 4, the "7" and "15" are on the left, "2" and "12" on the right, but perhaps the 2 is not the top, but the bottom or something.

Let's calculate what it should be.

If the line is parallel, then 7/15 = x/12, so x = 7/15 * 12 = 84/15 = 5.6

Or 7/x = 15/12, etc.

But since the problem asks to find missing length, and in the diagram there might be a question mark, perhaps for problem 4, the top right is missing, so ? = 7/15 * 12 = 5.6

But in the user's input, it's "2", so likely not.

Perhaps "2" is the answer, but we need to verify.

I think for consistency, in problem 4, if we assume that the ratio should be equal, and if 7/15 = 2/12, but 7/15 = 0.466, 2/12=0.166, not equal, so perhaps the missing length is the bottom right or something.

Let's look at the numbers: 7, 15, 2, 12. Perhaps the 2 is on the top right, and we need to find the bottom right, but it's given as 12.

Perhaps the "12" is the whole right side, and "2" is the top, so bottom is 10, but then 7/15 vs 2/10 = 1/5, not equal.

7/15 = x/10, x = 70/15 = 14/3 ≈4.666, not 2.

I think there might be a typo, or in some versions, the number is different.

For the sake of time, let's assume that in problem 4, the top right is missing, and it's ? , and from proportion, 7/15 = ?/12, so ? = 7*12/15 = 84/15 = 28/5 = 5.6

But since the user has "2", perhaps it's 2, and we need to find another.

Another possibility: in problem 4, the "7" is the whole left side, "15" is the bottom, so top is 7-15? Negative, impossible.

Perhaps "7" is top, "15" is bottom, "2" is top right, "12" is bottom right, and the missing length is not here, but the task is to find if it's proportional, but the instruction is "find the missing length", so likely there is a question mark on one of the segments.

Perhaps for problem 4, the missing length is the base, but not labeled.

I recall that in some problems, they ask for the length of the parallel line, but here it's not specified.

To move forward, let's list the answers we have:

1) 37.5

2) 40

3) 7.2

4) ?

5) x=5

6) x=10

7) 1