Solved Name: Unit 6: Similar Triangles Homework 5: Parallel ... - Free Printable

Let’s solve each problem one by one. These problems use the Triangle Proportionality Theorem (also called the Side-Splitter Theorem):
> If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

That means: if you have a triangle with a line inside it parallel to one side, the segments on the left side are in the same ratio as the segments on the right side.

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Problem 1:

We have a big triangle split by a line parallel to the base. The left side is split into parts: top = x, bottom = 32 → total left side = x + 32? Wait — actually, looking at the diagram description:

Actually, from the image description:
Left side has segment labeled “x” on top and “32” below → so total left side = x + 32? But wait — no, the arrows show that the inner line splits the left side into two parts: top part = x, bottom part = 32? Or is 32 the whole left side?

Wait — let me re-read the labels based on standard diagrams.

In problem 1:
The large triangle has a line drawn inside parallel to the base.
On the left side: from top vertex down to the parallel line is labeled x, and from there down to the bottom vertex is 32. So total left side = x + 32?
But actually, looking again — maybe 32 is the entire left side? No, because there's an arrow pointing to the lower part labeled 32, and upper part labeled x. Similarly, on the right side: upper part is not labeled, but lower part is 9, and the base is split into 24 and 9? Wait — no.

Actually, standard setup:
The parallel line cuts the two sides. On the left side, the segment above the parallel line is x, and below is 32. On the right side, the segment above is unknown, but below is 9. And the base is divided into 24 (left part) and 9 (right part)? That doesn’t make sense.

Wait — perhaps the numbers 24 and 9 are the lengths along the base? Actually, no — in similar triangles or proportional parts, when a line is parallel to the base, it creates proportional segments on the two sides.

Looking at typical textbook problems like this:

Problem 1 likely shows:
- Left side: from top to parallel line = x; from parallel line to bottom = 32 → so total left = x + 32
- Right side: from top to parallel line = ? ; from parallel line to bottom = 9
- But also, the base is split? Or maybe 24 and 9 are the segments on the base? That wouldn't be used directly.

Wait — I think I misread. Let me reinterpret based on common diagrams.

Actually, in many such worksheets, for problem 1:

It’s a triangle with a line parallel to the base cutting the two legs.
On the left leg: the segment from the top vertex to the parallel line is labeled x, and from the parallel line to the bottom vertex is 32.
On the right leg: the segment from the top vertex to the parallel line is unlabeled, but from the parallel line to the bottom vertex is 9.
And the base is divided into two parts: left part = 24, right part = 9? That can’t be — because the parallel line should create proportional segments on the *sides*, not necessarily on the base unless specified.

Wait — another possibility: the numbers 24 and 9 are the lengths of the segments on the *base* created by the foot of the parallel line? But that would only matter if we’re using coordinate geometry or something else.

I think I need to assume the standard proportionality:

If a line parallel to the base cuts the two sides, then:

(top segment on left) / (bottom segment on left) = (top segment on right) / (bottom segment on right)

But in problem 1, we don’t have the top segment on the right. Unless...

Wait — look at the diagram description again:
“Solve for x.”
Labels: left side has x (top), 32 (bottom); right side has ? (top), 9 (bottom); and the base has 24 and 9? That seems inconsistent.

Perhaps 24 and 9 are the lengths of the two parts of the base? But then how does that relate?

Another thought: maybe the 24 and 9 are the lengths of the segments on the *two sides*? No.

Let me try a different approach. In many textbooks, for such a diagram:

The triangle has vertices A (top), B (bottom left), C (bottom right).
A line DE is drawn parallel to BC, with D on AB, E on AC.
Then AD/DB = AE/EC.

In problem 1:
Suppose AD = x, DB = 32 → so AB = x + 32
AE = ?, EC = 9 → so AC = ? + 9
But we don’t know AE.

Unless the 24 is AE? That might be it!

Looking back: the user wrote:
"1. [diagram] x, 32, 24, 9"

Probably:
On left side: top = x, bottom = 32
On right side: top = 24, bottom = 9
And the line connecting them is parallel to the base.

Yes! That makes sense. So the parallel line cuts the left side into x and 32, and the right side into 24 and 9.

So by Triangle Proportionality Theorem:

x / 32 = 24 / 9

Because the ratios of the segments on each side should be equal.

Is that correct? Actually, yes — if the line is parallel to the base, then the ratio of the upper segment to the lower segment on one side equals that on the other side.

So:

x / 32 = 24 / 9

Simplify 24/9 = 8/3

So x / 32 = 8/3

Multiply both sides by 32:

x = (8/3) * 32 = 256 / 3 ≈ 85.333...

But that seems messy. Maybe I have the ratio backwards.

Alternative interpretation: sometimes it's set up as the whole side over part, but let's think.

Standard theorem: if DE || BC, then AD/AB = AE/AC, or AD/DB = AE/EC.

AD/DB = AE/EC is the side-splitter form.

So if AD = x, DB = 32, AE = 24, EC = 9, then:

x / 32 = 24 / 9

Same as before.

But 24/9 reduces to 8/3, so x = 32 * 8 / 3 = 256/3.

But perhaps the 24 and 9 are switched? Or maybe 24 is the whole right side?

Another possibility: the 24 is the length from top to parallel line on the right, and 9 is from parallel line to bottom, same as left.

But then why is the answer not integer? Maybe it's correct.

Let me check problem 2 to see the pattern.

Problem 2:
Left side: top = 20, bottom = 14
Right side: top = ?, bottom = x
Base has 33? Probably the hypotenuse or something.

Diagram: triangle with a line parallel to the base. Left side split into 20 (top) and 14 (bottom). Right side split into ? (top) and x (bottom). And the base is labeled 33? But 33 might be the whole right side or something.

This is confusing without seeing the image.

Perhaps in problem 2, the 33 is the length of the side from top to bottom on the right, so if bottom part is x, top part is 33 - x.

And since the line is parallel, the ratios are equal:

20 / 14 = (33 - x) / x

That could be.

Similarly, for problem 1, if the right side has top = 24, bottom = 9, then total right side = 33, but that's not given.

I recall that in some diagrams, the numbers on the base are the segments, but for proportionality, we use the sides.

Let me look for a standard solution approach.

Perhaps for problem 1, the 24 and 9 are the lengths of the segments on the base, but that doesn't help directly.

Another idea: maybe the triangle is divided such that the parallel line creates two similar triangles, and the bases are 24 and 9, but that would mean the small triangle has base 24, large has base 24+9=33, so ratio 24/33, and then the sides are proportional.

For example, if the small triangle on top has height corresponding to x, and the large triangle has height x+32, then x/(x+32) = 24/33.

Let me try that.

Assume that the parallel line creates a small triangle on top similar to the large triangle.

Then, the ratio of similarity is the ratio of corresponding sides.

If the base of the small triangle is 24, and the base of the large triangle is 24 + 9 = 33, then ratio k = 24/33 = 8/11.

Then, the left side of the small triangle is x, and the left side of the large triangle is x + 32, so:

x / (x + 32) = 24/33 = 8/11

Then cross-multiply:

11x = 8(x + 32)

11x = 8x + 256

3x = 256

x = 256/3 ≈ 85.333

Same as before.

But perhaps in the diagram, the 24 and 9 are not on the base, but on the sides.

Let's look at problem 3 for clue.

Problem 3:
Left side: top = ? , bottom = 20, total left = 36? Labels: "x+5", "30", "36", "20"

From description: "x+5" on the right side top, "30" on right side bottom, "36" on left side total, "20" on left side bottom.

So left side: from top to parallel line = 36 - 20 = 16? Or is 36 the whole left side, and 20 is the bottom part, so top part = 16.

Right side: top = x+5, bottom = 30.

Since the line is parallel, the ratios should be equal:

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

So 16 / 20 = (x+5) / 30

Simplify 16/20 = 4/5

So 4/5 = (x+5)/30

Multiply both sides by 30:

24 = x + 5

x = 19

That makes sense, integer answer.

Similarly, for problem 1, if we assume that the 24 and 9 are the segments on the right side, with 24 being the top part, 9 the bottom part, and on the left side, x is top, 32 is bottom, then:

x / 32 = 24 / 9

x = 32 * 24 / 9 = 32 * 8 / 3 = 256/3, which is not nice.

But in problem 3, the 36 is the whole left side, 20 is the bottom part, so top part is 16.

In problem 1, perhaps the 32 is the whole left side, and x is the top part, so bottom part is 32 - x? But the label says "x" and "32" with arrows, suggesting x is top, 32 is bottom.

Perhaps for problem 1, the 24 is the whole right side, and 9 is the bottom part, so top part is 24 - 9 = 15.

Then x / 32 = 15 / 9 = 5/3

x = 32 * 5 / 3 = 160/3, still not good.

Another possibility: the 24 and 9 are the lengths of the bases of the small and large triangles, but that would be unusual.

Let's consider that in some diagrams, the number on the base is the segment, but for proportionality, we use the sides.

Perhaps for problem 1, the parallel line divides the left side into x and 32, and the right side into y and 9, and the base is divided into 24 and 9, but that doesn't help.

I recall that in the Triangle Proportionality Theorem, if a line parallel to one side intersects the other two sides, then it divides those sides proportionally, meaning the ratio of the segments on one side equals the ratio on the other side.

So for problem 1, if left side is divided into segments of length a and b, right side into c and d, then a/b = c/d.

In the diagram, likely a = x, b = 32, c = 24, d = 9, so x/32 = 24/9.

But 24/9 = 8/3, so x = 32 * 8 / 3 = 256/3.

Perhaps it's b/a = d/c, but that would be 32/x = 9/24 = 3/8, so x = 32 * 8 / 3 = same thing.

Or perhaps the 24 is the whole right side, so if bottom is 9, top is 15, then x/32 = 15/9 = 5/3, x = 160/3.

Still not integer.

Let's look at problem 2.

Problem 2: left side: top = 20, bottom = 14, so total left = 34.

Right side: bottom = x, and the whole right side is 33? Or the base is 33.

The label "33" is on the hypotenuse or the right side.

Probably, the right side has length 33, and it's divided into top part and bottom part x.

So if the line is parallel, then the ratio of top to bottom on left equals top to bottom on right.

So 20/14 = (33 - x)/x

Because top right = 33 - x, bottom right = x.

So 20/14 = (33 - x)/x

Simplify 20/14 = 10/7

So 10/7 = (33 - x)/x

Cross-multiply: 10x = 7(33 - x)

10x = 231 - 7x

17x = 231

x = 231/17 = 13.588... not nice.

231 ÷ 17 = 13.588, not integer.

Perhaps 33 is the base, not the side.

Another idea: in problem 2, the 33 is the length of the side from top to bottom on the right, so if the bottom part is x, top part is 33 - x, and left side top 20, bottom 14, so 20/14 = (33 - x)/x, same as above.

But 231/17 is approximately 13.588, not good for homework.

Perhaps the ratio is of the whole side to the part.

Let's try for problem 2: if the small triangle on top is similar to the large triangle, then the ratio of corresponding sides.

If the left side of small triangle is 20, left side of large triangle is 20+14=34, so ratio 20/34 = 10/17.

Then the right side of small triangle is say y, right side of large triangle is 33, so y/33 = 10/17, y = 330/17, then x = 33 - y = 33 - 330/17 = (561 - 330)/17 = 231/17, same as before.

So perhaps it's correct, but let's see problem 4.

Problem 4: left side: top = x+8, bottom = 20? Labels: "x+8", "2x-5", "20", "22.5"

From description: "x+8" on left top, "2x-5" on right top, "20" on left bottom, "22.5" on right bottom.

So left side: top = x+8, bottom = 20

Right side: top = 2x-5, bottom = 22.5

Since parallel, (x+8)/20 = (2x-5)/22.5

Let me solve that.

First, 22.5 = 45/2, so:

(x+8)/20 = (2x-5)/(45/2) = (2x-5) * 2 / 45 = (4x-10)/45

So (x+8)/20 = (4x-10)/45

Cross-multiply:

45(x+8) = 20(4x-10)

45x + 360 = 80x - 200

360 + 200 = 80x - 45x

560 = 35x

x = 560 / 35 = 16

Nice! Integer.

So for problem 4, x=16.

Now for problem 1, perhaps the 24 and 9 are not both on the right side.

In problem 1, maybe the 24 is the length of the base of the small triangle, and 9 is the additional base, but that doesn't fit.

Another possibility: in problem 1, the parallel line creates a trapezoid, and the 24 and 9 are the lengths of the two bases, but then we need heights or something.

Perhaps for problem 1, the 32 and 9 are the segments on the sides, and 24 is the base of the small triangle, but that's complicated.

Let's assume that in problem 1, the left side is divided into x (top) and 32 (bottom), and the right side is divided into 24 (top) and 9 (bottom), and the line is parallel, so x/32 = 24/9.

But 24/9 = 8/3, so x = 32 * 8 / 3 = 256/3.

Perhaps it's 32/x = 9/24, then 32/x = 3/8, so x = 32 * 8 / 3 = same.

Or perhaps the 24 is the whole right side, so if bottom is 9, top is 15, then x/32 = 15/9 = 5/3, x = 160/3.

Still not good.

Let's look at problem 5.

Problem 5: left side: top = 4, bottom = 2x+4? Labels: "4", "2x+4", "3", "x+7"

From description: "4" on left top, "2x+4" on left bottom, "3" on right top, "x+7" on right bottom.

So left side: top = 4, bottom = 2x+4

Right side: top = 3, bottom = x+7

Parallel line, so 4 / (2x+4) = 3 / (x+7)

Cross-multiply:

4(x+7) = 3(2x+4)

4x + 28 = 6x + 12

28 - 12 = 6x - 4x

16 = 2x

x = 8

Good.

Problem 6: left side: top = 28, bottom = 82? Labels: "28", "82", "2x+8", "5x-4"

From description: "28" on left top, "82" on left bottom? But 28 and 82 are both on left? That can't be.

Probably "28" is the top part of left side, "82" is the bottom part of left side, so total left = 28+82=110.

Right side: top = 2x+8, bottom = 5x-4.

So 28/82 = (2x+8)/(5x-4)

Simplify 28/82 = 14/41

So 14/41 = (2x+8)/(5x-4)

Cross-multiply:

14(5x-4) = 41(2x+8)

70x - 56 = 82x + 328

-56 - 328 = 82x - 70x

-384 = 12x

x = -32

Negative? That can't be for length.

Perhaps 28 and 82 are not both on left. Maybe 28 is the whole left side, and 82 is something else.

The label "82" might be the whole left side, and "28" is the top part, so bottom part = 82 - 28 = 54.

Then right side: top = 2x+8, bottom = 5x-4.

So 28/54 = (2x+8)/(5x-4)

Simplify 28/54 = 14/27

So 14/27 = (2x+8)/(5x-4)

Cross-multiply:

14(5x-4) = 27(2x+8)

70x - 56 = 54x + 216

70x - 54x = 216 + 56

16x = 272

x = 17

Good.

So for problem 6, x=17.

Now back to problem 1. Perhaps in problem 1, the 32 is the whole left side, and x is the top part, so bottom part = 32 - x.

And on the right side, 24 is the whole right side, 9 is the bottom part, so top part = 24 - 9 = 15.

Then, since parallel, the ratio of top to bottom should be equal, but which ratio?

If we use the side-splitter: (top left) / (bottom left) = (top right) / (bottom right)

So x / (32 - x) = 15 / 9 = 5/3

Then 3x = 5(32 - x)

3x = 160 - 5x

8x = 160

x = 20

Oh! Nice integer.

And it makes sense.

In the diagram, likely the 32 is the entire left side, with x being the upper segment, so lower segment is 32 - x.

Similarly, on the right side, 24 is the entire right side, with 9 being the lower segment, so upper segment is 24 - 9 = 15.

Then by proportionality: x / (32 - x) = 15 / 9

As above, x=20.

We can verify: if x=20, left upper=20, lower=12, ratio 20/12=5/3

Right upper=15, lower=9, ratio 15/9=5/3, same.

Perfect.

So for problem 1, x=20.

Now problem 2.

Problem 2: left side: top=20, bottom=14, so total left=34.

Right side: the whole side is 33? Or the base is 33.

From description: "20", "14", "33", "x"

Probably, the right side has length 33, and x is the bottom part, so top part = 33 - x.

Then, since parallel, top/left bottom = top/right bottom? No.

Ratio: top left / bottom left = top right / bottom right

So 20 / 14 = (33 - x) / x

As before, 10/7 = (33 - x)/x

10x = 7(33 - x)

10x = 231 - 7x

17x = 231

x = 231/17 = 13.588... not integer.

But 231 ÷ 17 = 13.588, not nice.

Perhaps 33 is the base, not the side.

Another possibility: in problem 2, the 33 is the length of the side from top to bottom on the right, but x is the top part, and bottom is not labeled, but the diagram shows x on the bottom.

The user said: "2. [diagram] 20, 14, 33, x"

And in the text: "20" on left top, "14" on left bottom, "33" on the hypotenuse or right side, "x" on the bottom of right side.

Perhaps the 33 is the whole right side, and x is the bottom part, so top is 33 - x.

But then x=231/17.

Maybe the ratio is of the whole side to the part.

Or perhaps for problem 2, the line is parallel, so the small triangle on top is similar to the large triangle.

Small triangle left side = 20, large triangle left side = 20+14=34, so ratio 20/34=10/17.

Then small triangle right side = y, large triangle right side = 33, so y/33 = 10/17, y=330/17, then x = 33 - y = 33 - 330/17 = (561-330)/17=231/17, same.

But 231 and 17, 17*13=221, 231-221=10, so 13 and 10/17, not integer.

Perhaps 33 is not the right side, but the base.

Let's assume that the base is divided into two parts, but that doesn't help for proportionality of sides.

Another idea: in some diagrams, the number on the base is the segment, but for the theorem, we use the sides.

Perhaps for problem 2, the 33 is the length of the side from the top to the point where the parallel line meets, but that doesn't make sense.

Let's look at the answer choices or think differently.

Perhaps in problem 2, the x is on the top, not bottom.

The user said: "x" is on the bottom of the right side.

But let's calculate with the values.

20/14 = 10/7 ≈ 1.4286

If x is bottom right, top right = 33 - x, so (33 - x)/x = 10/7

Then 7(33 - x) = 10x

231 - 7x = 10x

231 = 17x

x = 231/17 = 13.588...

But perhaps it's acceptable, or maybe I have the ratio wrong.

Another possibility: the ratio is bottom/top or something.

Or perhaps the 33 is the whole right side, and x is the top part, so bottom is 33 - x.

Then 20/14 = x/(33 - x)

Then 10/7 = x/(33 - x)

10(33 - x) = 7x

330 - 10x = 7x

330 = 17x

x = 330/17 ≈ 19.411, still not good.

Perhaps the 33 is not related to the right side length, but to the base.

Let's skip and come back.

Problem 3 we did earlier: left side whole = 36, bottom part = 20, so top part = 16.

Right side: top = x+5, bottom = 30.

So 16/20 = (x+5)/30

4/5 = (x+5)/30

x+5 = 24

x = 19

Good.

Problem 4: x=16, as calculated.

Problem 5: x=8.

Problem 6: x=17.

Problem 7: find CE.

Diagram: points A,B,C,D,E.

A to B = 6, B to C = x-5, C to D = 14, D to E = 2x+3.

And lines: from A to E, and from B to D, and BD is parallel to AE? The arrows suggest that BD is parallel to AE.

The diagram has arrows on BD and on AE, indicating they are parallel.

Also, points are colinear: A-B-C on one line, C-D-E on another line? No.

Typically, it's a triangle or quadrilateral.

From description: "A B x-5 C" on one line, then "C D 14" , "D E 2x+3", and "A to E" with arrow, "B to D" with arrow, and BD || AE.

Also, "find CE".

CE is from C to E, which is CD + DE = 14 + (2x+3) = 2x+17, but we need to find x first.

Since BD || AE, and they are cut by transversals AC and CE or something.

Points: likely, A, B, C are on one straight line, with AB=6, BC=x-5, so AC = AB + BC = 6 + x - 5 = x+1.

Then from C, there is a line to E, with D on it, CD=14, DE=2x+3, so CE = CD + DE = 14 + 2x + 3 = 2x+17.

And BD is a line from B to D, and AE is from A to E, and BD || AE.

So, we have two lines: line ACE and line ADE or something.

Actually, it's like triangle ACE, with B on AC, D on CE, and BD || AE.

Yes, that makes sense.

So in triangle ACE, B on AC, D on CE, BD || AE.

Then by Basic Proportionality Theorem (Thales' theorem), AB/BC = ED/DC? Let's see.

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

Here, BD || AE, and AE is a side, but BD is intersecting AC and CE.

Actually, in triangle ACE, side AE is one side, but BD is parallel to AE, and B is on AC, D is on CE.

So, the line BD is parallel to side AE, and it intersects the other two sides: AC and CE.

Side AC and side CE are the two sides from C.

Triangle ACE has vertices A, C, E.

Side AE is opposite to C.

Line BD is parallel to AE, with B on AC, D on CE.

Then, by the converse or direct theorem, it should divide the sides proportionally.

Specifically, AB/BC = ED/DC? Let's think.

The theorem says that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Here, the line BD is parallel to side AE, and it intersects sides AC and CE.

Side AC is from A to C, side CE is from C to E.

So, on side AC, it is divided at B, so segments AB and BC.

On side CE, it is divided at D, so segments CD and DE.

The proportion should be AB/BC = CD/DE? Or AB/AC = CD/CE, etc.

Standard formula: in triangle ACE, with BD || AE, B on AC, D on CE, then AB/BC = ED/DC? I think it's AB/BC = AD/DE, but AD is not defined.

Recall: the ratio is of the segments from the common vertex.

From vertex C, the line BD parallel to AE.

Then, CB/BA = CD/DE? Let's derive.

Since BD || AE, triangles CBD and CAE are similar? Not necessarily.

Actually, angle at C is common, and since BD || AE, angle CBD = angle CAE (corresponding angles), so triangle CBD ~ triangle CAE by AA similarity.

Yes!

So triangle CBD ~ triangle CAE.

Then, corresponding sides proportional.

CB / CA = CD / CE = BD / AE

Now, CB = BC = x-5

CA = AC = AB + BC = 6 + (x-5) = x+1

CD = 14

CE = CD + DE = 14 + (2x+3) = 2x+17

So, CB / CA = CD / CE

So (x-5) / (x+1) = 14 / (2x+17)

Now solve for x.

Cross-multiply:

(x-5)(2x+17) = 14(x+1)

Expand both sides:

Left: x*2x + x*17 -5*2x -5*17 = 2x^2 + 17x -10x -85 = 2x^2 +7x -85

Right: 14x + 14

So:

2x^2 +7x -85 = 14x + 14

Bring all to left:

2x^2 +7x -85 -14x -14 = 0

2x^2 -7x -99 = 0

Solve quadratic: discriminant d = b^2 -4ac = 49 + 792 = 841 = 29^2

So x = [7 ± 29]/4

x = (7+29)/4 = 36/4 = 9

x = (7-29)/4 = -22/4 = -5.5 (discard, since length)

So x=9

Then CE = 2x+17 = 2*9 +17 = 18+17=35

Now back to problem 2.

With x=9 for problem 7, but for problem 2, let's assume that the 33 is the whole right side, and x is the bottom part, and we have to accept x=231/17, but that's unlikely.

Perhaps in problem 2, the 33 is the length of the base, and the triangle is right-angled or something, but the diagram shows a triangle with a line parallel to the base.

Another idea: in problem 2, the 33 is the length of the side from the top to the bottom on the right, but the x is not on that side; perhaps x is on the base.

The user said: "x" is on the bottom of the right side, but maybe it's on the base.

Let's read the user's description: "2. [diagram] 20, 14, 33, x"

And in the text: "20" on left top, "14" on left bottom, "33" on the hypotenuse, "x" on the bottom of the right side.

Perhaps the 33 is the length of the right side, and x is the bottom part, so top is 33 - x, and the ratio is 20/14 = (33 - x)/x, as before.

But 231/17 is approximately 13.588, and 17*13=221, 231-221=10, so 13 10/17, not nice.

Perhaps the 33 is the whole left side, but 20+14=34, not 33.

20+14=34, so not.

Another possibility: the 33 is the length of the base, and the triangle is divided, but for proportionality, we need the sides.

Perhaps for problem 2, the line is parallel to the left side or something, but the arrows suggest it's parallel to the base.

Let's calculate the ratio.

20/14 = 10/7

If the right side is divided into top and bottom, with bottom = x, top = y, and y + x = 33, and y/x = 10/7, then y = (10/7)x, so (10/7)x + x = 33, (17/7)x = 33, x = 33 * 7 / 17 = 231/17, same as before.

So perhaps it's correct, and we leave it as fraction.

But for homework, usually integer answers.

Perhaps in problem 2, the 33 is not the right side, but the distance or something else.

Let's look at problem 1 again; we have x=20.

For problem 2, perhaps the x is the top part.

Assume that on the right side, x is the top part, and bottom is 33 - x, but 33 is the whole, so if x is top, bottom is 33 - x, then 20/14 = x/(33 - x)

Then 10/7 = x/(33 - x)

10(33 - x) = 7x

330 - 10x = 7x

330 = 17x

x = 330/17 ≈ 19.411, still not good.

Perhaps the 33 is the base, and the triangle is isosceles or something, but unlikely.

Another idea: in some diagrams, the number on the base is the segment, and for the proportionality, if the line is parallel to the base, then the ratio of the segments on the sides is equal to the ratio of the bases, but that's for similar triangles.

For example, if the small triangle on top has base b1, large has base b2, then ratio k = b1/b2, and sides are proportional.

In problem 2, if the base of the small triangle is say p, large is q, but not given.

Perhaps the 33 is the base of the large triangle, and the base of the small triangle is not given.

I think for now, I'll assume that for problem 2, x = 231/17, but let's see if it simplifies or if I have a mistake.

Perhaps the 14 is not the bottom, but the whole left side.

Let's try that.

Suppose in problem 2, the left side has top = 20, and the whole left side = 14, but 20 > 14, impossible.

Or top = 20, bottom = 14, whole = 34.

Perhaps the 33 is the length from the top to the parallel line on the right side.

So if the right side from top to parallel line is 33, and from parallel line to bottom is x, then 20/14 = 33/x

Then 10/7 = 33/x

10x = 231

x = 23.1, not good.

20/14 = x/33, then x = 20*33/14 = 660/14 = 330/7 ≈ 47.14, not good.

I think I need to accept that for problem 2, x = 231/17, but let's calculate numerical value.

231 ÷ 17 = 13.5882352941

But perhaps in the diagram, the 33 is the whole right side, and x is the bottom, and we box it as fraction.

But for consistency, let's do all problems.

Problem 3: x=19

Problem 4: x=16

Problem 5: x=8

Problem 6: x=17

Problem 7: CE=35

For problem 2, perhaps the 33 is the base, and the triangle is right-angled at the bottom left, but the diagram shows a general triangle.

Another thought: in problem 2, the "33" might be the length of the side from the top to the bottom on the right, but the x is not on that side; perhaps x is on the base.

The user said: "x" is on the bottom of the right side, but maybe it's a typo, and it's on the base.

Assume that the base is divided into two parts, but not specified.

Perhaps the 33 is the length of the right side, and the x is the length of the base or something.

I recall that in some problems, the number on the base is used for the proportion if the line is parallel, but only if it's the corresponding side.

Let's give up and use the calculation.

So for problem 2: 20/14 = (33 - x)/x

10/7 = (33 - x)/x

10x = 7(33 - x)

10x = 231 - 7x

17x = 231

x = 231/17

Simplify: 231 and 17, gcd is 1, since 17 prime, 231÷17=13*17=221, remainder 10, so 231/17 = 13 10/17 or 231/17.

But perhaps write as improper fraction.

For the sake of completing, I'll use that.

But let's check problem 1 again; we have x=20.

Now for the response, I'll provide the answers.

So summary:

1. x = 20

2. x = 231/17

3. x = 19

4. x = 16

5. x = 8

6. x = 17

7. CE = 35

For problem 2, perhaps it's 13.588, but let's see if there's a mistake in interpretation.

Another idea: in problem 2, the 33 might be the length of the side from the top to the point, but the x is the bottom, and the ratio is of the segments.

Perhaps the line is parallel to the left side, but the arrows suggest otherwise.

Let's assume that for problem 2, the correct interpretation is that the right side has top = 33, bottom = x, and left side top = 20, bottom = 14, so 20/14 = 33/x

Then 10/7 = 33/x

x = 33 * 7 / 10 = 231/10 = 23.1, not good.

20/14 = x/33, x = 20*33/14 = 660/14 = 330/7 ≈ 47.14.

Not good.

Perhaps the 33 is the whole left side, but 20+14=34≠33.

Close to 34, perhaps typo, and it's 34.

If whole left side is 34, then for right side, if whole is s, but not given.

I think I have to go with x = 231/17 for problem 2.

So for the final answer, I'll list them.

But for the response, since the user asked to solve, and for problem 2, it's fractional, but ok.

Perhaps in problem 2, the x is the top part, and 33 is the whole, so bottom = 33 - x, and 20/14 = x/(33 - x), then x = 330/17, same issue.

Let's calculate 231/17 = 13.588, and 330/17 = 19.411, neither nice.

Another possibility: the 33 is the length of the base, and the triangle is divided, but for the proportionality, if the line is parallel to the base, then the ratio of the segments on the sides is equal, and the base segments are proportional to the sides, but that's for the similar triangles.

For example, if the small triangle has base b_s, large has base b_l, then b_s / b_l = ratio of sides.

In problem 2, if the base of the small triangle is p, large is q, but not given.

Perhaps the 33 is the base of the large triangle, and the base of the small triangle is not given, but we have the sides.

I think I need to move on.

So for the answer, I'll provide:

1. 20

2. \frac{231}{17}

3. 19

4. 16

5. 8

6. 17

7. 35

But for problem 2, let's double-check with the initial assumption.

Perhaps in problem 2, the 33 is the length from the top to the parallel line on the right side, and x is from parallel line to bottom, so then 20/14 = 33/x

Then x = 14 * 33 / 20 = (14/20)*33 = (7/10)*33 = 231/10 = 23.1

Still not good.

20/14 = x/33, x = 20*33/14 = 660/14 = 330/7 ≈ 47.14.

No.

Perhaps the 14 is the whole left side, but 20>14, impossible.

I think it's correct as 231/17.

So I'll proceed.

For the response, since the user may expect integer, but for accuracy, I'll use the calculation.

So final answers:

1. x = 20

2. x = \frac{231}{17}

3. x = 19

4. x = 16

5. x = 8

6. x = 17

7. CE = 35

For problem 2, perhaps it's 13.6, but let's box the fraction.

Or simplify 231/17 is already simplified.

So for the Final Answer section, I'll list them.

But the user said "solve for x" for 1-6, and "find CE" for 7.

So for each, the answer.

In the response, I can write:

After solving each problem using the Triangle Proportionality Theorem or similar triangles, the answers are:

1. 20

2. \frac{231}{17}

3. 19

4. 16

5. 8

6. 17

7. 35

For problem 2, to confirm, let's assume the diagram: left side divided into 20 (top) and 14 (bottom), so ratio top/bottom = 20/14 = 10/7.

Right side: if the whole side is 33, and x is the bottom part, then top part = 33 - x, so (33 - x)/x = 10/7, so 7(33 - x) = 10x, 231 - 7x = 10x, 231 = 17x, x = 231/17.

Yes.

So I think it's correct.

Final Answer:

1. 20
2. \frac{231}{17}
3. 19
4. 16
5. 8
6. 17
7. 35