Triangle proportionality worksheet

Draw a line across a triangle parallel to one side and it cuts the other two sides in the same ratio. That gives you a proportion with one unknown, and cross-multiplying finishes it.

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Sigma Prep · sigmaprep.io/worksheets Name Triangle Proportionality Date Period Answer each question. Find $x$ when a piece is an expression In $△ Z G Q$ , $\overline{T K} ∥ \overline{G Q}$ , with $T$ on $\overline{Z G}$ and $K$ on $\overline{Z Q}$ . $Z T = 5$ , $T G = 2 x - 14$ , $Z K = 7$ and $K Q = 14$ . Find $x$ . In $△ C B T$ , $\overline{K A} ∥ \overline{B T}$ . Find $x$ . In $△ E R M$ , $\overline{N L} ∥ \overline{R M}$ , with $N$ on $\overline{E R}$ and $L$ on $\overline{E M}$ . $E N : N R = 4 : 5$ , $E L = 8$ and $L M = 3 x - 5$ . Find $x$ . In $△ G T N$ , $\overline{C H} ∥ \overline{T N}$ and $C$ is the midpoint of $\overline{G T}$ . Find $x$ . In $△ A P W$ , $\overline{L N} ∥ \overline{P W}$ . Find $x$ . In $△ V Z R$ , $\overline{Q X} ∥ \overline{Z R}$ , with $Q$ on $\overline{V Z}$ and $X$ on $\overline{V R}$ . $V Q = 6$ , $Q Z = 9$ , $V X = 2 x - 6$ and $X R = 18$ . Find $x$ . In $△ K Y G$ , $\overline{Q E} ∥ \overline{Y G}$ , with $Q$ on $\overline{K Y}$ and $E$ on $\overline{K G}$ . $K Q : Q Y = 1 : 2$ , $E G = 14$ and $K E = 3 x - 2$ . Find $x$ . Use three parallel lines $\overset{\leftrightarrow}{C A} ∥ \overset{\leftrightarrow}{E P} ∥ \overset{\leftrightarrow}{S L}$ . Find $x$ . Two straight braces cross the front edges of three parallel shelves. The shelves cut one brace into pieces of $y$ cm and 6 cm, and the other into pieces of 15 cm and 10 cm, in the same order from the top. Find $y$ . Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $x$ m long and the other is $42$ m long. Between the second and third rows, the first path is $12$ m long and the other is $18$ m long. Find $x$ . $\overline{S H} ∥ \overline{U C} ∥ \overline{D W}$ . Find $x$ . Lines $ℓ$ , $m$ and $n$ are parallel. One transversal crosses them at $H$ , $Y$ and $A$ , and another at $R$ , $N$ and $S$ . If $H Y = 35$ , $R N = 21$ and $N S = 9$ , find $Y A$ . Three parallel streets cross two straight avenues. Along the first avenue the two blocks between the streets are 6 m and $x$ m long. Along the second avenue the matching blocks are 4 m and 2 m long. Find $x$ . Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $15$ m long and the other is $x$ m long. Between the second and third rows, the first path is $10$ m long and the other is $6$ m long. Find $x$ . Decide whether the segment is parallel to the third side A ladder-style frame has two sides meeting at the top and a base along the floor. A crossbar joins the two sides. On one side the crossbar is 20 in from the top and 15 in from the floor; on the other side it is 16 in from the top and 11 in from the floor, measured along the sides. Is the crossbar parallel to the floor? In the figure, $J R = 29$ . Is $\overline{M L} ∥ \overline{Z R}$ ? In $△ J T Y$ , $S$ is on $\overline{J T}$ and $B$ is on $\overline{J Y}$ . $J S = 15$ , $J T = 27$ , $J B = 20$ and $J Y = 36$ . Is $\overline{S B} ∥ \overline{T Y}$ ? Use the lengths marked. Is $\overline{L Z} ∥ \overline{G E}$ ? In $△ U M C$ , $L$ is on $\overline{U M}$ and $Q$ is on $\overline{U C}$ . $U L = 18$ , $L M = 12$ , $U Q = 15$ and $Q C = 10$ . Is $\overline{L Q} ∥ \overline{M C}$ ? In $△ V J L$ , $U$ is on $\overline{V J}$ and $P$ is on $\overline{V L}$ . $V U = 28$ , $U J = 12$ , $V P = 35$ and $V L = 50$ . Is $\overline{U P} ∥ \overline{J L}$ ? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Triangle Proportionality

Date Period

Answer each question.

Find $x$ when a piece is an expression
In $△ Z G Q$ , $\overline{T K} ∥ \overline{G Q}$ , with $T$ on $\overline{Z G}$ and $K$ on $\overline{Z Q}$ . $Z T = 5$ , $T G = 2 x - 14$ , $Z K = 7$ and $K Q = 14$ . Find $x$ .
In $△ C B T$ , $\overline{K A} ∥ \overline{B T}$ . Find $x$ .
In $△ E R M$ , $\overline{N L} ∥ \overline{R M}$ , with $N$ on $\overline{E R}$ and $L$ on $\overline{E M}$ . $E N : N R = 4 : 5$ , $E L = 8$ and $L M = 3 x - 5$ . Find $x$ .
In $△ G T N$ , $\overline{C H} ∥ \overline{T N}$ and $C$ is the midpoint of $\overline{G T}$ . Find $x$ .
In $△ A P W$ , $\overline{L N} ∥ \overline{P W}$ . Find $x$ .
In $△ V Z R$ , $\overline{Q X} ∥ \overline{Z R}$ , with $Q$ on $\overline{V Z}$ and $X$ on $\overline{V R}$ . $V Q = 6$ , $Q Z = 9$ , $V X = 2 x - 6$ and $X R = 18$ . Find $x$ .
In $△ K Y G$ , $\overline{Q E} ∥ \overline{Y G}$ , with $Q$ on $\overline{K Y}$ and $E$ on $\overline{K G}$ . $K Q : Q Y = 1 : 2$ , $E G = 14$ and $K E = 3 x - 2$ . Find $x$ .

Use three parallel lines
$\overset{\leftrightarrow}{C A} ∥ \overset{\leftrightarrow}{E P} ∥ \overset{\leftrightarrow}{S L}$ . Find $x$ .
Two straight braces cross the front edges of three parallel shelves. The shelves cut one brace into pieces of $y$ cm and 6 cm, and the other into pieces of 15 cm and 10 cm, in the same order from the top. Find $y$ .
Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $x$ m long and the other is $42$ m long. Between the second and third rows, the first path is $12$ m long and the other is $18$ m long. Find $x$ .
$\overline{S H} ∥ \overline{U C} ∥ \overline{D W}$ . Find $x$ .
Lines $ℓ$ , $m$ and $n$ are parallel. One transversal crosses them at $H$ , $Y$ and $A$ , and another at $R$ , $N$ and $S$ . If $H Y = 35$ , $R N = 21$ and $N S = 9$ , find $Y A$ .
Three parallel streets cross two straight avenues. Along the first avenue the two blocks between the streets are 6 m and $x$ m long. Along the second avenue the matching blocks are 4 m and 2 m long. Find $x$ .
Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $15$ m long and the other is $x$ m long. Between the second and third rows, the first path is $10$ m long and the other is $6$ m long. Find $x$ .

Decide whether the segment is parallel to the third side
A ladder-style frame has two sides meeting at the top and a base along the floor. A crossbar joins the two sides. On one side the crossbar is 20 in from the top and 15 in from the floor; on the other side it is 16 in from the top and 11 in from the floor, measured along the sides. Is the crossbar parallel to the floor?
In the figure, $J R = 29$ . Is $\overline{M L} ∥ \overline{Z R}$ ?
In $△ J T Y$ , $S$ is on $\overline{J T}$ and $B$ is on $\overline{J Y}$ . $J S = 15$ , $J T = 27$ , $J B = 20$ and $J Y = 36$ . Is $\overline{S B} ∥ \overline{T Y}$ ?
Use the lengths marked. Is $\overline{L Z} ∥ \overline{G E}$ ?
In $△ U M C$ , $L$ is on $\overline{U M}$ and $Q$ is on $\overline{U C}$ . $U L = 18$ , $L M = 12$ , $U Q = 15$ and $Q C = 10$ . Is $\overline{L Q} ∥ \overline{M C}$ ?
In $△ V J L$ , $U$ is on $\overline{V J}$ and $P$ is on $\overline{V L}$ . $V U = 28$ , $U J = 12$ , $V P = 35$ and $V L = 50$ . Is $\overline{U P} ∥ \overline{J L}$ ?



Sigma Prep · sigmaprep.io/worksheets Name Triangle ProportionalityAnswer keyVersion 1 Date Period Answer each question. Find $x$ when a piece is an expression In $△ Z G Q$ , $\overline{T K} ∥ \overline{G Q}$ , with $T$ on $\overline{Z G}$ and $K$ on $\overline{Z Q}$ . $Z T = 5$ , $T G = 2 x - 14$ , $Z K = 7$ and $K Q = 14$ . Find $x$ . $12$ In $△ C B T$ , $\overline{K A} ∥ \overline{B T}$ . Find $x$ . $7$ In $△ E R M$ , $\overline{N L} ∥ \overline{R M}$ , with $N$ on $\overline{E R}$ and $L$ on $\overline{E M}$ . $E N : N R = 4 : 5$ , $E L = 8$ and $L M = 3 x - 5$ . Find $x$ . $5$ In $△ G T N$ , $\overline{C H} ∥ \overline{T N}$ and $C$ is the midpoint of $\overline{G T}$ . Find $x$ . $2$ In $△ A P W$ , $\overline{L N} ∥ \overline{P W}$ . Find $x$ . $10$ In $△ V Z R$ , $\overline{Q X} ∥ \overline{Z R}$ , with $Q$ on $\overline{V Z}$ and $X$ on $\overline{V R}$ . $V Q = 6$ , $Q Z = 9$ , $V X = 2 x - 6$ and $X R = 18$ . Find $x$ . $9$ In $△ K Y G$ , $\overline{Q E} ∥ \overline{Y G}$ , with $Q$ on $\overline{K Y}$ and $E$ on $\overline{K G}$ . $K Q : Q Y = 1 : 2$ , $E G = 14$ and $K E = 3 x - 2$ . Find $x$ . $3$ Use three parallel lines $\overset{\leftrightarrow}{C A} ∥ \overset{\leftrightarrow}{E P} ∥ \overset{\leftrightarrow}{S L}$ . Find $x$ . $20$ Two straight braces cross the front edges of three parallel shelves. The shelves cut one brace into pieces of $y$ cm and 6 cm, and the other into pieces of 15 cm and 10 cm, in the same order from the top. Find $y$ . $9$ cm Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $x$ m long and the other is $42$ m long. Between the second and third rows, the first path is $12$ m long and the other is $18$ m long. Find $x$ . $28$ m $\overline{S H} ∥ \overline{U C} ∥ \overline{D W}$ . Find $x$ . $8$ Lines $ℓ$ , $m$ and $n$ are parallel. One transversal crosses them at $H$ , $Y$ and $A$ , and another at $R$ , $N$ and $S$ . If $H Y = 35$ , $R N = 21$ and $N S = 9$ , find $Y A$ . $15$ Three parallel streets cross two straight avenues. Along the first avenue the two blocks between the streets are 6 m and $x$ m long. Along the second avenue the matching blocks are 4 m and 2 m long. Find $x$ . $3$ m Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $15$ m long and the other is $x$ m long. Between the second and third rows, the first path is $10$ m long and the other is $6$ m long. Find $x$ . $9$ m Decide whether the segment is parallel to the third side A ladder-style frame has two sides meeting at the top and a base along the floor. A crossbar joins the two sides. On one side the crossbar is 20 in from the top and 15 in from the floor; on the other side it is 16 in from the top and 11 in from the floor, measured along the sides. Is the crossbar parallel to the floor?No In the figure, $J R = 29$ . Is $\overline{M L} ∥ \overline{Z R}$ ?No In $△ J T Y$ , $S$ is on $\overline{J T}$ and $B$ is on $\overline{J Y}$ . $J S = 15$ , $J T = 27$ , $J B = 20$ and $J Y = 36$ . Is $\overline{S B} ∥ \overline{T Y}$ ?Yes Use the lengths marked. Is $\overline{L Z} ∥ \overline{G E}$ ?No In $△ U M C$ , $L$ is on $\overline{U M}$ and $Q$ is on $\overline{U C}$ . $U L = 18$ , $L M = 12$ , $U Q = 15$ and $Q C = 10$ . Is $\overline{L Q} ∥ \overline{M C}$ ?Yes In $△ V J L$ , $U$ is on $\overline{V J}$ and $P$ is on $\overline{V L}$ . $V U = 28$ , $U J = 12$ , $V P = 35$ and $V L = 50$ . Is $\overline{U P} ∥ \overline{J L}$ ?Yes Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Triangle ProportionalityAnswer keyVersion 1

Date Period

Answer each question.

Find $x$ when a piece is an expression
In $△ Z G Q$ , $\overline{T K} ∥ \overline{G Q}$ , with $T$ on $\overline{Z G}$ and $K$ on $\overline{Z Q}$ . $Z T = 5$ , $T G = 2 x - 14$ , $Z K = 7$ and $K Q = 14$ . Find $x$ . $12$
In $△ C B T$ , $\overline{K A} ∥ \overline{B T}$ . Find $x$ . $7$
In $△ E R M$ , $\overline{N L} ∥ \overline{R M}$ , with $N$ on $\overline{E R}$ and $L$ on $\overline{E M}$ . $E N : N R = 4 : 5$ , $E L = 8$ and $L M = 3 x - 5$ . Find $x$ . $5$
In $△ G T N$ , $\overline{C H} ∥ \overline{T N}$ and $C$ is the midpoint of $\overline{G T}$ . Find $x$ . $2$
In $△ A P W$ , $\overline{L N} ∥ \overline{P W}$ . Find $x$ . $10$
In $△ V Z R$ , $\overline{Q X} ∥ \overline{Z R}$ , with $Q$ on $\overline{V Z}$ and $X$ on $\overline{V R}$ . $V Q = 6$ , $Q Z = 9$ , $V X = 2 x - 6$ and $X R = 18$ . Find $x$ . $9$
In $△ K Y G$ , $\overline{Q E} ∥ \overline{Y G}$ , with $Q$ on $\overline{K Y}$ and $E$ on $\overline{K G}$ . $K Q : Q Y = 1 : 2$ , $E G = 14$ and $K E = 3 x - 2$ . Find $x$ . $3$

Use three parallel lines
$\overset{\leftrightarrow}{C A} ∥ \overset{\leftrightarrow}{E P} ∥ \overset{\leftrightarrow}{S L}$ . Find $x$ . $20$
Two straight braces cross the front edges of three parallel shelves. The shelves cut one brace into pieces of $y$ cm and 6 cm, and the other into pieces of 15 cm and 10 cm, in the same order from the top. Find $y$ . $9$ cm
Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $x$ m long and the other is $42$ m long. Between the second and third rows, the first path is $12$ m long and the other is $18$ m long. Find $x$ . $28$ m
$\overline{S H} ∥ \overline{U C} ∥ \overline{D W}$ . Find $x$ . $8$
Lines $ℓ$ , $m$ and $n$ are parallel. One transversal crosses them at $H$ , $Y$ and $A$ , and another at $R$ , $N$ and $S$ . If $H Y = 35$ , $R N = 21$ and $N S = 9$ , find $Y A$ . $15$
Three parallel streets cross two straight avenues. Along the first avenue the two blocks between the streets are 6 m and $x$ m long. Along the second avenue the matching blocks are 4 m and 2 m long. Find $x$ . $3$ m
Two straight paths cross three parallel rows of a vineyard. Between the first and second rows, one path is $15$ m long and the other is $x$ m long. Between the second and third rows, the first path is $10$ m long and the other is $6$ m long. Find $x$ . $9$ m

Decide whether the segment is parallel to the third side
A ladder-style frame has two sides meeting at the top and a base along the floor. A crossbar joins the two sides. On one side the crossbar is 20 in from the top and 15 in from the floor; on the other side it is 16 in from the top and 11 in from the floor, measured along the sides. Is the crossbar parallel to the floor?No
In the figure, $J R = 29$ . Is $\overline{M L} ∥ \overline{Z R}$ ?No
In $△ J T Y$ , $S$ is on $\overline{J T}$ and $B$ is on $\overline{J Y}$ . $J S = 15$ , $J T = 27$ , $J B = 20$ and $J Y = 36$ . Is $\overline{S B} ∥ \overline{T Y}$ ?Yes
Use the lengths marked. Is $\overline{L Z} ∥ \overline{G E}$ ?No
In $△ U M C$ , $L$ is on $\overline{U M}$ and $Q$ is on $\overline{U C}$ . $U L = 18$ , $L M = 12$ , $U Q = 15$ and $Q C = 10$ . Is $\overline{L Q} ∥ \overline{M C}$ ?Yes
In $△ V J L$ , $U$ is on $\overline{V J}$ and $P$ is on $\overline{V L}$ . $V U = 28$ , $U J = 12$ , $V P = 35$ and $V L = 50$ . Is $\overline{U P} ∥ \overline{J L}$ ?Yes

How to do these

In triangle $A B C$ , $D E$ is parallel to $B C$ . $A D = 4$ , $D B = 6$ , $A E = 8$ . Find $E C$

AD over DB equals AE over EC.
4 over 6 equals 8 over EC.
Cross-multiply: 4 times EC is 48.

The answer is $12$ .

Where students go wrong

Mixing a piece with a whole. AD over DB equals AE over EC, piece to piece, and AD over AB equals AE over AC, piece to whole; but AD over AB against AE over EC gives the wrong answer.

Also called side-splitter theorem, triangle proportionality theorem or parallel line in a triangle.

What you can put on this worksheet

Find x in a triangle: In $△ B N Q$ , $\overline{G Z} ∥ \overline{N Q}$ and $G$ is the midpoint of $\overline{B N}$ . Find $x$ . — triangle bnq cut by gz, labeled B, N, Q, G, Z, 13 → $13$
Named segments, three parallel lines, whole sides: In $△ Z H V$ , $\overline{N Y} ∥ \overline{H V}$ , with $N$ on $\overline{Z H}$ and $Y$ on $\overline{Z V}$ . If $Z N : N H = 3 : 5$ and $Z Y = 9$ , find $Y V$ . → $15$
Word problems and the converse: Two straight roads start at a crossroads. Main Street crosses both of them, and Oak Street, parallel to Main Street, crosses both too, nearer the crossroads. Along the first road, Oak Street is 12 mi from the crossroads and 20 mi from Main Street. Along the second road, Oak Street is 9 mi from the crossroads. How far is Oak Street from Main Street along the second road? → $15$ mi

Questions about these worksheets

Yes. Every triangle proportionality sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Geometry is usually taken around 9th or 10th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the triangle proportionality questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: find x in a triangle, named segments, three parallel lines, whole sides and word problems and the converse. Tick as many as you want and set how many of each, or let it spread them evenly.

Mixing a piece with a whole. AD over DB equals AE over EC, piece to piece, and AD over AB equals AE over AC, piece to whole; but AD over AB against AE over EC gives the wrong answer.

Triangle proportionality worksheet

Select your difficulty

Find xxx when a piece is an expression

Use three parallel lines

Decide whether the segment is parallel to the third side

Find xxx when a piece is an expression

Use three parallel lines

Decide whether the segment is parallel to the third side

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these triangle proportionality worksheets free?+

Do the triangle proportionality worksheets come with an answer key?+

Are these triangle proportionality worksheets printable?+

What grade are these triangle proportionality worksheets for?+

Can I get a different version of the same triangle proportionality worksheet?+

Is this a free alternative to Kuta Software for triangle proportionality?+

Can I choose how hard the triangle proportionality questions are?+

What kinds of question are on a triangle proportionality worksheet?+

What do students get wrong with triangle proportionality?+

Find $x$ when a piece is an expression

Find $x$ when a piece is an expression