Similar Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. These problems use the Triangle Proportionality Theorem (also called the Side-Splitter Theorem) and properties of similar triangles.

The main rule to remember: If a line is parallel to one side of a triangle, it splits the other two sides proportionally. This means you can set up ratios like $\frac{\text{top part}}{\text{bottom part}} = \frac{\text{top part}}{\text{bottom part}}$ or $\frac{\text{small triangle side}}{\text{big triangle side}} = \frac{\text{small triangle side}}{\text{big triangle side}}$.

Problem 1

Setup: We have a large right triangle with a parallel line inside.
* Left side segments: $12$ (top) and $x$ (bottom). Total left side = $12 + x$.
* Bottom side segments: $6$ (left) and $8$ (right). Total bottom side = $6 + 8 = 14$.
* Wait, looking at the diagram, the parallel line creates a smaller triangle on top? No, the parallel line connects the hypotenuse and the vertical leg? Let's look closer.
* The vertical leg has segments $12$ and $x$.
* The horizontal leg has segments $6$ and $8$.
* The line segment connecting them is parallel to the hypotenuse? No, usually in these diagrams, the internal line is parallel to the base or one of the legs.
* Let's re-examine diagram 1. It looks like a large right triangle. There is a line segment inside. The vertical side is split into $12$ and $x$. The horizontal side is split into $6$ and $8$. The internal line connects the point between $12/x$ and $6/8$.
* Actually, usually, the "parallel" assumption applies to the line *inside* the triangle being parallel to the *third* side. Here, the internal line seems parallel to the hypotenuse? That would be unusual for standard orientation.
* Let's look at the standard interpretation: The line segment labeled with no length is parallel to the side opposite the vertex where the sides $12+x$ and $6+8$ meet? No.
* Let's assume the standard "Side-Splitter" configuration. The line inside is parallel to the base (horizontal)? No, it's slanted. Is it parallel to the vertical side? No.
* Let's look at the vertices. The big triangle has a vertical leg and a horizontal leg. The inner line connects the vertical leg and the horizontal leg. For the Triangle Proportionality Theorem to apply directly as $\frac{a}{b} = \frac{c}{d}$, the inner line must be parallel to the third side (the hypotenuse).
* If the inner line is parallel to the hypotenuse, then the small triangle at the corner (with legs $x$ and $6$) is similar to the large triangle (with legs $12+x$ and $14$)? Or is the top triangle similar?
* Let's look at the position of $x$. $x$ is the bottom part of the vertical leg. $12$ is the top part. $6$ is the left part of the horizontal leg. $8$ is the right part.
* If the inner line is parallel to the hypotenuse, then the triangle formed by the corner (legs $x$ and $6$) is similar to the large triangle? No, that would mean the angle at the corner is shared. Yes.
* So, $\frac{\text{vertical leg of small}}{\text{horizontal leg of small}} = \frac{\text{vertical leg of large}}{\text{horizontal leg of large}}$.
* Small triangle legs: $x$ and $6$.
* Large triangle legs: $x + 12$ and $6 + 8 = 14$.
* Ratio: $\frac{x}{6} = \frac{x + 12}{14}$.
* Cross multiply: $14x = 6(x + 12)$.
* $14x = 6x + 72$.
* $8x = 72$.
* $x = 9$.

*Alternative Interpretation:* What if the inner line is parallel to the vertical leg? Then we'd use corresponding segments on the transversal lines. But the diagram shows the inner line connecting the two legs. The only way "parallel" makes sense here is if that inner segment is parallel to the hypotenuse. Let's double check with the other common setup: Maybe the triangle is rotated?
Let's try the ratio of parts: $\frac{\text{top}}{\text{bottom}} = \frac{\text{left}}{\text{right}}$? This applies if the parallel line cuts two sides. Here the parallel line cuts the two legs. So it is parallel to the hypotenuse.
The proportionality theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
So, $\frac{\text{segment 1 on leg A}}{\text{segment 2 on leg A}} = \frac{\text{segment 1 on leg B}}{\text{segment 2 on leg B}}$.
Which segments correspond? The ones adjacent to the same vertex.
The vertex is the bottom-left corner (the right angle). The segments touching this vertex are $x$ and $6$. The segments further away are $12$ and $8$.
So, $\frac{x}{12} = \frac{6}{8}$.
Let's solve this: $8x = 72 \rightarrow x = 9$.
Both methods yield $x = 9$. This confirms the logic.

Problem 2

Setup: A triangle with a line parallel to the base.
* Left side segments: $15$ (top) and $3$ (bottom).
* Right side segments: $x$ (top) and $5$ (bottom).
* Since the lines are parallel, the ratios of the corresponding segments are equal.
* $\frac{\text{Top Left}}{\text{Bottom Left}} = \frac{\text{Top Right}}{\text{Bottom Right}}$
* $\frac{15}{3} = \frac{x}{5}$
* Simplify the left side: $5 = \frac{x}{5}$
* Multiply both sides by 5: $x = 25$.

Problem 3

Setup: A triangle with a line parallel to the base.
* Left side segments: The top part is not labeled, but the whole left side isn't given either. Wait, let's look at the labels.
* Top small triangle side: $8$.
* Right side segments: Top part is unlabeled? No, the label $35$ is for the bottom trapezoid part? Or the whole side?
* Looking closely at crop 3: The label $8$ is on the top left side of the small triangle. The label $x$ is the base of the small triangle. The label $20$ is the base of the large triangle. The label $35$ is on the right side. It looks like it indicates the entire right side of the large triangle.
* If $35$ is the whole right side, we don't know how it's split.
* Let's re-read the diagram. Usually, numbers placed along a segment indicate the length of that specific segment.
* Label $8$: Top-left side of the small triangle.
* Label $x$: Base of the small triangle.
* Label $20$: Base of the large triangle.
* Label $35$: Right side of the large triangle? Or the bottom part? The number $35$ is centered on the lower part of the right side. It likely represents the segment from the parallel line to the bottom vertex.
* If $35$ is the bottom-right segment, we still need the top-right segment to use side-splitter.
* However, we can use Similar Triangles. The small triangle is similar to the large triangle.
* We need another pair of corresponding sides. We have the bases ($x$ and $20$). We have the left side of the small triangle ($8$). We do NOT have the left side of the large triangle. We have the right side information ($35$), but we don't know the top part.
* Let's look really closely at image 3 again. Is $35$ the whole side? The bracket or line for $35$ seems to cover the whole right side. If $35$ is the whole right side, we still can't solve it unless we assume the triangle is isosceles or something, which we can't.
* Let's reconsider the placement. Maybe $8$ is the top part of the left side, and there is no label for the bottom part?
* Let's look at Problem 4 for context. In Prob 4, $3$ and $5$ are clearly segments. $9$ is the top side. $x$ is the parallel chord.
* Back to Prob 3. Maybe I am misinterpreting the labels.
* Left side: Top segment is $8$.
* Right side: Bottom segment is $35$? If so, we are stuck.
* What if $35$ is the *top* right segment? Then $\frac{8}{\text{bottom left}} = \frac{35}{\text{bottom right}}$. Still stuck.
* What if the triangle is isosceles? No indication.
* Let's look at the similar triangles ratio: $\frac{\text{Small Left}}{\text{Large Left}} = \frac{\text{Small Base}}{\text{Large Base}}$.
* We don't have Large Left.
* Is it possible that $8$ and $35$ are the *whole* sides?
* If Left Whole = $8$ and Right Whole = $35$? No, $8$ is clearly just the top part.
* Let's look at the label $35$ again. It is next to the right side. The line for the dimension might span the whole side. If the whole right side is $35$, and the whole left side is... unknown.
* Wait, is it possible that the left side segments are $8$ and something else, and the right side segments are something and $35$?
* Let's try another interpretation. Maybe the label $8$ refers to the *entire* left side? No, it's placed next to the top segment.
* Let's look at similar problems online. Often, if one side is missing info, you can't solve it.
* Is there a typo in my reading?
* Left side: Top part $8$.
* Right side: Bottom part $35$?
* Base small: $x$.
* Base large: $20$.
* This problem seems unsolvable with standard interpretations unless I'm missing a label.
* Let's look at the image again very carefully.
* Ah, look at the left side. There is a tick mark? No.
* Look at the right side. The number $35$ is positioned similarly to the $8$.
* Could the left side be $8$ (top) and the right side be $35$ (top)? If top-left=$8$ and top-right=$35$, we still don't know the bottom parts.
* Could the left side be $8$ (total) and right side $35$ (total)? If total left=$8$ and total right=$35$, and base small=$x$, base large=$20$. We still need the ratio of similarity. We don't have the top-left segment length specifically to compare to the total.
* Wait, let's look at Problem 1 again. $12$ and $x$ were parts.
* Let's look at Problem 3 again. Is it possible that the left side is split into $8$ and... wait, is the "8" actually the *whole* left side? And the "35" is the *whole* right side?
* If Small Triangle is similar to Large Triangle:
* We need the ratio of similarity $k = \frac{\text{Small Side}}{\text{Large Side}}$.
* If we assume the triangle is isosceles (Left=Right), then Left Total = Right Total. But $8 \neq 35$.
* Let's try one more look. Is it possible that the label on the left is not $8$? It looks like an $8$.
* Is it possible the label on the right is not $35$? It looks like $35$.
* Is it possible the base is not $20$? It looks like $20$.
* Let's assume the standard case where we compare the small triangle to the large triangle using the sides we *do* have.
* We have Base Small ($x$) and Base Large ($20$).
* We have Side Left Top ($8$).
* We have Side Right Bottom ($35$??).
* This feels wrong. Let's look at the spacing. The $35$ is centered on the lower segment. The $8$ is centered on the upper segment.
* Is there any other number? No.
* Could the "8" be the *bottom* left segment? No, it's near the top vertex.
* Could the "35" be the *top* right segment? It's near the middle/bottom.
* Let's consider that I might be misidentifying the parallel lines. "Assume all segments that appear to be parallel are parallel." The horizontal line $x$ is parallel to base $20$.
* Okay, what if the triangle is a right triangle or has some other property? No marks.
* Let's guess the most likely intended structure based on typical worksheets. Usually, you are given enough info to form a proportion.
* Possibility A: The left side total is given? No.
* Possibility B: The right side top part is given? No.
* Possibility C: The labels refer to the whole sides?
* If Left Side Total = $8$? Then Top Part < $8$.
* If Right Side Total = $35$?
* This doesn't help find $x$ without knowing where the cut is.
* Possibility D: I am misreading the number $8$. Could it be a variable? No.
* Possibility E: I am misreading the number $35$. Could it be $3$ and $5$? No, it's clearly $35$.
* Possibility F: The label $8$ is the top segment, and the label $35$ is the whole right side? And maybe the triangle is isosceles? If it's isosceles, Left Total = Right Total = $35$. Then Top Left = $8$. Bottom Left = $27$.
* Then we can use Side Splitter: $\frac{\text{Top Left}}{\text{Bottom Left}} = \frac{\text{Top Right}}{\text{Bottom Right}}$.
* We don't know the split on the right.
* But we can use Similar Triangles: $\frac{\text{Small Left}}{\text{Large Left}} = \frac{\text{Small Base}}{\text{Large Base}}$.
* $\frac{8}{35} = \frac{x}{20}$.
* $35x = 160$.
* $x = \frac{160}{35} = \frac{32}{7} \approx 4.57$.
* Possibility G: The label $35$ is the bottom segment, and the label $8$ is the top segment, and the triangle is isosceles?
* If Isosceles, Top Right = Top Left = $8$.
* Then we have Top Left=$8$, Bottom Left=$?$, Top Right=$8$, Bottom Right=$35$.
* By Side Splitter: $\frac{\text{Top Left}}{\text{Bottom Left}} = \frac{\text{Top Right}}{\text{Bottom Right}}$.
* $\frac{8}{\text{Bottom Left}} = \frac{8}{35}$. This implies Bottom Left = $35$.
* So Large Left Side = $8 + 35 = 43$.
* Now use Similar Triangles: $\frac{\text{Small Left}}{\text{Large Left}} = \frac{\text{Small Base}}{\text{Large Base}}$.
* $\frac{8}{43} = \frac{x}{20}$.
* $43x = 160$.
* $x = \frac{160}{43} \approx 3.72$.
* Possibility H: There is a typo in the problem or my understanding. Let's look at Problem 4.
* Left side: $3$ (top), $5$ (bottom).
* Top side: $9$.
* Parallel chord: $x$.
* This is a different orientation. The parallel line cuts the two sides meeting at the bottom vertex? No, the parallel line $x$ is "horizontal", cutting the left and right sides. The top side is $9$.
* Wait, in Problem 4, the parallel line $x$ is parallel to the top side $9$?
* If $x$ is parallel to the side labeled $9$, then the small triangle (bottom) is similar to the large triangle.
* Left side of large triangle = $3 + 5 = 8$.
* Left side of small triangle = $5$.
* Ratio of similarity = $\frac{\text{Small}}{\text{Large}} = \frac{5}{8}$.
* Therefore, $\frac{x}{9} = \frac{5}{8}$.
* $x = \frac{45}{8} = 5.625$.
* This makes perfect sense.

* Now back to Problem 3 with this "Similar Triangles" mindset.
* We established that simply having one segment on the left ($8$) and one on the right ($35$) is insufficient unless we assume symmetry or misread the labels.
* Let's look at the label $8$ again. Is it possible it says $S$? No.
* Is it possible the label on the right is not $35$ but $3$ and $5$ separated? No.
* Is it possible the label $8$ is the whole left side?
* If Whole Left = $8$.
* And the label $35$ is the whole right side?
* Then we still don't know the ratio.
* Let's look at the visual proportions. The top triangle looks roughly half the height of the total?
* Let's reconsider the "Isosceles" assumption. In many geometry problems, if a triangle looks isosceles and lacks specific angle markers, it might not be. BUT, if information is missing, it's a strong hint.
* However, there is another possibility. What if the label $35$ is actually $3.5$? Or $5$?
* Let's look at the font. The $3$ and $5$ in problem 2 are distinct. The $35$ in problem 3 looks like a two-digit number.
* Let's try one more interpretation: Thales Theorem / Intercept Theorem.
* Maybe the lines aren't parallel to the base? "Assume all segments that appear to be parallel are parallel." The line $x$ appears parallel to the base $20$.
* Okay, what if the label $8$ is the bottom left segment?
* Visually, $8$ is next to the top segment.
* What if the label $35$ is the top right segment?
* Visually, $35$ is next to the bottom segment.
* Let's assume the question is valid and I'm missing a standard convention.
* Could the "8" and "35" be the lengths of the parallel sides?
* No, $x$ and $20$ are the parallel sides.
* Could the "8" be the left side and "35" be the right side of the small triangle?
* If Small Left = $8$ and Small Right = $35$?
* Then we don't know the large triangle's sides to get the ratio.
* Could the "8" be the left side and "35" be the right side of the large triangle?
* If Large Left = $8$ and Large Right = $35$?
* Then we don't know the small triangle's sides.
* Crucial Insight: Look at the position of the numbers in Problem 1, 2, 5, 6, 7, 8. They always denote segments.
* In Problem 3, is it possible that the left side is split into $8$ and $8$? (i.e. midpoint?) No indication.
* Is it possible that the right side is split into $35$ and $35$?
* Let's look at the solution key for similar worksheets online (Lindsay Bowden Geometry).
* Common problem type: Given two sides of the small triangle and one side of the large? No.
* Given segments on one side and corresponding segments on the other?
* Let's assume there is a typo in the book/worksheet and the left side was meant to have two numbers, or the right side.
* HOWEVER, look at the number $35$ again. Is it possible it is $3$ and $5$? Like, top is $3$, bottom is $5$?
* If Top Right = $3$ and Bottom Right = $5$.
* And Top Left = $8$.
* Then $\frac{\text{Top Left}}{\text{Bottom Left}} = \frac{\text{Top Right}}{\text{Bottom Right}} \Rightarrow \frac{8}{\text{Bottom Left}} = \frac{3}{5}$.
* Bottom Left = $\frac{40}{3}$.
* Total Left = $8 + \frac{40}{3} = \frac{64}{3}$.
* Similarity Ratio $\frac{\text{Small Left}}{\text{Total Left}} = \frac{8}{64/3} = \frac{24}{64} = \frac{3}{8}$.
* $\frac{x}{20} = \frac{3}{8} \Rightarrow x = \frac{60}{8} = 7.5$.
* This yields a clean decimal answer. This is a very plausible scenario if the "35" is actually a poorly spaced "3" and "5". But visually it really looks like "35".
* Let's try another combination. What if Left Side is $8$ (top) and $?$ (bottom), and Right Side is $?$ (top) and $35$ (bottom)?
* What if the triangle is Isosceles?
* If Isosceles, Left Side = Right Side.
* If the labels indicate the whole sides: Left=$8$, Right=$35$. Contradiction.
* If the labels indicate segments: Top-Left=$8$. Bottom-Right=$35$.
* If Isosceles, Top-Right=$8$ and Bottom-Left=$35$.
* Then Total Side = $8+35=43$.
* Ratio $\frac{\text{Small}}{\text{Large}} = \frac{8}{43}$.
* $x = 20 \cdot \frac{8}{43} = \frac{160}{43} \approx 3.72$. Not a clean number.
* What if Top-Left=$8$ and Top-Right=$8$ (Isosceles top part), and we are missing the bottom?
* Let's look at Problem 3 again. Is the "8" actually an $S$ for side? No.
* Is the "35" actually $3.5$?
* If Top-Left=$8$, Bottom-Right=$3.5$. Assume Isosceles $\rightarrow$ Top-Right=$8$, Bottom-Left=$3.5$.
* Total Side = $11.5$.
* Ratio $\frac{8}{11.5} = \frac{x}{20}$.
* $x = \frac{160}{11.5} = 13.9$.
* Let's go with the most robust mathematical interpretation of potentially ambiguous diagrams in these specific worksheets: Often, if a single number is on a side, it represents the entire length of that side.
* If Left Side Total = $8$.
* If Right Side Total = $35$.
* This creates a contradiction for an isosceles triangle, but triangles don't have to be isosceles.
* BUT, we need the position of the parallel line. We don't have it.
* Wait! Look at the tick marks. Are there tick marks?
* In Problem 3, there are NO tick marks indicating congruence.
* Let's look at Problem 4 again. $3$ and $5$ are clearly segments.
* Let's look at Problem 3's "35" again. It is written with the same font size as the "20". The "8" is smaller? No.
* There is a possibility that Problem 3 is defective or relies on a visual estimation I can't make.
* However, let's look at the similar triangle Problem 5.
* Left side: $30$ (top), $x$ (bottom).
* Right side: $15$ (top), $20$ (bottom).
* Parallel lines? The line separating $30/x$ and $15/20$ is parallel to the base?
* Wait, the triangle in #5 is sideways. The vertex is on the left.
* Top side (in diagram orientation): Split into $30$ and $x$? No, the line goes across.
* Let's orient #5. Vertex at left. Two sides extending right. A vertical-ish line cuts them.
* Top side segments: $30$ (left part), $15$ (right part?? No, $15$ is on the top edge of the trapezoid?).
* Let's trace the lines.
* Triangle vertex at left.
* Top side has a segment labeled $30$. Then the parallel line. Then the rest of the side? No, the label $15$ is on the top side of the *trapezoid*? Or the top side of the small triangle?
* Actually, usually these are: Side 1 split into $A, B$. Side 2 split into $C, D$.
* In #5: One side is split into $30$ and $x$. The other side is split into $15$ and $20$.
* Which corresponds to which?
* $30$ is the segment from vertex to parallel line. $15$ is the segment from vertex to parallel line?
* Looking at the diagram: The label $30$ is on the top-left segment. The label $15$ is on the top-right segment? No, $15$ is on the top side of the small triangle?
* Let's assume the standard "bowtie" or "A-frame".
* In #5, it's an A-frame on its side.
* Side 1 (Top in pic): Segment from vertex to cut = $30$. Segment from cut to end = ? Wait, the label $15$ is inside the small triangle? No, it's on the boundary.
* Actually, looking at #5: The label $30$ is on the long top side. The label $15$ is on the short top side of the small triangle? No.
* Let's assume: Side A is split into $30$ (vertex to cut) and $x$ (cut to end)? No, $x$ is the bottom side.
* Let's restart #5.
* Vertex at left.
* Top side: Label $30$ is for the segment from vertex to the parallel line? Or the whole side?
* Bottom side: Label $x$ is for the segment from vertex to end? Or cut to end?
* Right side (vertical): Label $15$ and $20$.
* It looks like the vertical side is split into $15$ (top) and $20$ (bottom).
* The top side is split into $30$ (left) and ... wait, where is the other part?
* Ah, the label $15$ is on the top side of the small triangle? No, it's near the vertical line.
* Okay, let's look at the labels relative to the parallel line.
* The parallel line is the vertical one? No, the text says "segments that appear parallel". The vertical line and the rightmost vertical line? No, it's a triangle.
* In #5, the parallel lines are the two vertical-ish lines?
* If the two vertical lines are parallel, then we have proportional intercepts on the transversals (the top and bottom slanted lines).
* Transversal 1 (Top): Segments are $30$ and $15$?
* Label $30$ is on the left part. Label $15$ is on the right part?
* If so, Top Left = $30$, Top Right = $15$.
* Transversal 2 (Bottom): Segments are $x$ and $20$?
* Label $x$ is on the left part? Label $20$ is on the right part?
* Wait, the label $20$ is on the vertical line? No, it's inside the trapezoid.
* Let's look at the label $20$. It is next to the vertical segment on the right? No, it's floating.
* Actually, in #5, the label $15$ is on the top side of the small triangle? And $30$ is on the top side of the trapezoid?
* If Top Small = $15$ and Top Trap = $30$.
* And Bottom Small = $x$? No, $x$ is the whole bottom?
* And Bottom Trap = $20$?
* This is getting too speculative.

* Let's step back and solve the clear ones first, then revisit the ambiguous ones with the most likely standard conventions.

### Problem 4
* Triangle with parallel line $x$ cutting sides.
* Side 1 (Left): Split into $3$ (top) and $5$ (bottom). Total = $8$.
* Side 2 (Right): Not labeled with segments.
* Side 3 (Top): Length $9$.
* Line $x$ is parallel to Side 3 (Top).
* Therefore, Small Triangle (bottom) is similar to Large Triangle.
* Wait, the parallel line $x$ is "below" the top side?
* The diagram shows a triangle pointing down? No, pointing up.
* Top side is $9$.
* Parallel line $x$ is inside.
* Left side segments: $3$ (top, near vertex?) and $5$ (bottom).
* If the vertex is at the top, then the small triangle is at the top.
* Small Triangle Side (Left) = $3$.
* Large Triangle Side (Left) = $3 + 5 = 8$.
* Small Triangle Base = $x$.
* Large Triangle Base = $9$.
* Ratio: $\frac{\text{Small Left}}{\text{Large Left}} = \frac{\text{Small Base}}{\text{Large Base}}$.
* $\frac{3}{8} = \frac{x}{9}$.
* $8x = 27$.
* $x = \frac{27}{8} = 3.375$.

### Problem 5
* Triangle on its side. Vertex at left.
* Two transversals intersected by two parallel vertical lines.
* Top transversal segments: $30$ (between vertex and first parallel line) and $15$ (between first and second parallel line)?
* Label $30$ is on the segment from vertex to the first vertical line.
* Label $15$ is on the segment from the first vertical line to the second? No, the triangle ends at the second vertical line?
* Actually, it looks like a single triangle with one parallel line inside.
* Vertex at left.
* Top side: Segment from vertex to parallel line = $30$. Segment from parallel line to right vertex = $15$?
* Bottom side: Segment from vertex to parallel line = $x$? No, $x$ is the whole bottom? Or the part?
* Right side (vertical base): Split into $20$ and ...?
* Let's look at the labels $15$ and $20$. They are near the right side.
* $15$ is on the top part of the right vertical side?
* $20$ is on the bottom part of the right vertical side?
* If the right side is the base, and the line inside is parallel to it...
* Then the left vertex is the apex.
* Top side segments: $30$ (apex to cut) and $15$ (cut to base?? No, $15$ is labeled on the top edge of the trapezoid?).
* Let's assume the standard Side-Splitter.
* Side 1 (Top): Split into $30$ and $15$.
* Side 2 (Bottom): Split into $x$ and $20$?
* If $30$ corresponds to $x$ (both touch the vertex) and $15$ corresponds to $20$ (both touch the base).
* Then $\frac{30}{15} = \frac{x}{20}$.
* $2 = \frac{x}{20} \Rightarrow x = 40$.
* This seems very likely. Clean integer answer.
* Verification: $\frac{\text{Top Left}}{\text{Top Right}} = \frac{\text{Bottom Left}}{\text{Bottom Right}}$.
* $\frac{30}{15} = \frac{x}{20}$.
* $x = 40$.

### Problem 6
* Triangle with vertex at top? No, vertex at bottom?
* It's a triangle with a line parallel to the left side?
* Let's identify the parallel lines. The internal line and the right side? No.
* The internal line and the left side?
* Let's look at the shape. It's a triangle. A line cuts through it.
* Labels: $6$ and $8$ on the left side?
* $6$ is top part. $8$ is bottom part.
* Labels: $2$ and $x$ on the right side?
* $2$ is top part. $x$ is bottom part.
* The line connecting the split points is parallel to the third side (the rightmost vertical-ish side? No, the base?).
* Actually, looking at the orientation, the "base" is the side on the right?
* If the internal line is parallel to the right side, then the triangle on the left is similar to the large triangle?
* No, the parallel line is the one *inside*. It is parallel to the side *opposite* the vertex where the sides $6+8$ and $2+x$ meet?
* The sides $6+8$ and $2+x$ meet at the bottom-left vertex? No.
* Let's trace the perimeter.
* Left side: Split into $6$ and $8$.
* Right side: Split into $2$ and $x$.
* The line connecting these splits is parallel to the third side (the top/right side?).
* Wait, the diagram shows the parallel line is parallel to the side labeled... nothing?
* Let's assume the standard A-frame.
* Vertex at top.
* Left leg: $6$ (top), $8$ (bottom).
* Right leg: $2$ (top), $x$ (bottom).
* Internal line parallel to base.
* Then $\frac{\text{Top Left}}{\text{Bottom Left}} = \frac{\text{Top Right}}{\text{Bottom Right}}$.
* $\frac{6}{8} = \frac{2}{x}$.
* $6x = 16$.
* $x = \frac{16}{6} = \frac{8}{3} \approx 2.67$.

### Problem 7
* Triangle with vertex at top-left?
* Side 1 (Top): Split into $15$ and $4$?
* $15$ is the long part from vertex. $4$ is the short part at the end?
* Or $15$ is the whole side?
* Label $15$ is centered on the first segment. Label $4$ is on the second segment.
* So Top Side segments: $15$ and $4$.
* Side 2 (Bottom): Split into $x$ and $10$?
* Label $x$ is on the first segment (from vertex).
* Label $10$ is on the second segment.
* Parallel line connects the splits.
* Ratio: $\frac{\text{Segment 1 Top}}{\text{Segment 2 Top}} = \frac{\text{Segment 1 Bottom}}{\text{Segment 2 Bottom}}$.
* $\frac{15}{4} = \frac{x}{10}$.
* $4x = 150$.
* $x = \frac{150}{4} = 37.5$.

### Problem 8
* Triangle with vertex at bottom-left?
* Side 1 (Left): Split into $4$ and $5$?
* $4$ is top part. $5$ is bottom part?
* Wait, the vertex is at the bottom left.
* So the segments starting from the vertex are the "first" ones.
* Label $5$ is adjacent to the vertex. Label $4$ is further away.
* So Left Side: $5$ (vertex to cut), $4$ (cut to end).
* Side 2 (Right/Top): Split into $3$ and $x$?
* Label $3$ is adjacent to the vertex? No, $3$ is on the internal parallel line?
* No, $3$ is on the side.
* Let's look at the position