Worksheets · Geometry

Similar triangles worksheet

Two triangles are similar when two angles match (AA), when all three side ratios match (SSS), or when two side ratios match and the angle between them does (SAS). In nested triangles a line parallel to the base cuts off a smaller similar triangle, and its sides are in the same ratio as the whole ones.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

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Name

Similar Triangles

Date Period

Answer each question.

  1. Decide if the triangles are similar, and name the criterion

    Are they similar? Give the criterion if they are.6711121422
  2. Are the triangles similar? If so, by which criterion?ABCDE1061511
  3. Similar or not? If similar, name the criterion.62°48°62°57°
  4. Decide whether the triangles are similar. If they are, name the criterion.ABCDE10121518
  5. Which criterion, if any, shows the triangles similar?141848°212748°

  1. Find xx in overlapping similar triangles

    MN‾∥KL‾, so △JMN∼△JKL. Find x.8x2451JMNKL
  2. DE‾∥BC‾. Use the similar triangles to find x.122324xADEBC
  3. DE‾∥BC‾. Find x.5x1518ADEBC

  1. Find xx or a marked side in similar triangles

    The triangles are similar. Find x.x − 418x + 430
  2. The two triangles are similar. Solve for x.42x − 14126
  3. The triangles are similar, small to large in the ratio 2:5. Find x.2x − 2430

  1. Write the similarity statement and ratio

    The triangles are similar. Complete the statement △KLJ∼△   and give the ratio of corresponding sides, △KLJ to the other.JKL6NOM8
  2. AB‾∥CD‾. Complete the statement △EAB∼△   and give the ratio of corresponding sides, △EAB to the other.ABCDE68912
  3. The triangles are similar. Complete the statement △DFE∼△   and give the ratio of corresponding sides, △DFE to the other.DEF212427ABC283236

  1. Use shadows to find a height or length

    A 6 ft student stands 4 ft from a pole with a spotlight on top and casts a 2 ft shadow away from it. How tall is the pole?
  2. A 3 ft child casts a 2 ft shadow. A 30 ft statue nearby casts a shadow at the same time. How long is the statue's shadow?
  3. A 15 m tree casts a 30 m shadow. At the same time the garden shed beside it casts a 4 m shadow. How tall is the garden shed?

  1. Find xx in triangles that meet tip to tip

    Find x.ABCDE812x36
  2. Find x.ABCDE121024x + 8
  3. Find the value of x.ABCDEx + 186x − 5712

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Similar TrianglesAnswer keyVersion 1

Date Period

Answer each question.

  1. Decide if the triangles are similar, and name the criterion

    Are they similar? Give the criterion if they are.6711121422Yes, SSS
  2. Are the triangles similar? If so, by which criterion?ABCDE1061511Not similar
  3. Similar or not? If similar, name the criterion.62°48°62°57°Not similar
  4. Decide whether the triangles are similar. If they are, name the criterion.ABCDE10121518Yes, SAS
  5. Which criterion, if any, shows the triangles similar?141848°212748°Yes, SAS

  1. Find xx in overlapping similar triangles

    MN‾∥KL‾, so △JMN∼△JKL. Find x.8x2451JMNKL9
  2. DE‾∥BC‾. Use the similar triangles to find x.122324xADEBC46
  3. DE‾∥BC‾. Find x.5x1518ADEBC6

  1. Find xx or a marked side in similar triangles

    The triangles are similar. Find x.x − 418x + 43016
  2. The two triangles are similar. Solve for x.42x − 141268
  3. The triangles are similar, small to large in the ratio 2:5. Find x.2x − 243018

  1. Write the similarity statement and ratio

    The triangles are similar. Complete the statement △KLJ∼△   and give the ratio of corresponding sides, △KLJ to the other.JKL6NOM8△OMN, 3:4
  2. AB‾∥CD‾. Complete the statement △EAB∼△   and give the ratio of corresponding sides, △EAB to the other.ABCDE68912△EDC, 2:3
  3. The triangles are similar. Complete the statement △DFE∼△   and give the ratio of corresponding sides, △DFE to the other.DEF212427ABC283236△ACB, 3:4

  1. Use shadows to find a height or length

    A 6 ft student stands 4 ft from a pole with a spotlight on top and casts a 2 ft shadow away from it. How tall is the pole?18 ft
  2. A 3 ft child casts a 2 ft shadow. A 30 ft statue nearby casts a shadow at the same time. How long is the statue's shadow?20 ft
  3. A 15 m tree casts a 30 m shadow. At the same time the garden shed beside it casts a 4 m shadow. How tall is the garden shed?2 m

  1. Find xx in triangles that meet tip to tip

    Find x.ABCDE812x3624
  2. Find x.ABCDE121024x + 812
  3. Find the value of x.ABCDEx + 186x − 571213

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

The triangles are similar. Find x

  1. The small triangle has side 4 where the big one has 4 + 6 = 10.
  2. So the bases are in the ratio 4 to 10.
  3. x = 15 times 4 over 10.

The answer is 6.

Where students go wrong

Using the lower piece instead of the whole side. The ratio compares the small triangle to the big one, and the big one runs all the way from the top vertex.

Also called AA similarity, SSS and SAS similarity, nested similar triangles or proportional sides.

What you can put on this worksheet

AA, SSS or SAS similarity, or not similar
Which criterion, if any, shows the triangles similar? — two triangles with all three sides marked, labeled 3, 4, 5, 6, 8, 10 → Yes, SSS
A missing side in nested similar triangles
MN‾∥KL‾. Use the similar triangles to find x. — a triangle with a smaller similar triangle inside it, sharing the top vertex → 12
A side written in x
The triangles are similar. Find x. — two similar triangles, matching angles marked alike, labeled 5, 15 → 10
The similarity statement and ratio
The triangles are similar. Complete the statement △DFE∼△   and give the ratio of corresponding sides, △DFE to the other. → △JKL, 4:5
Shadows
A 6 ft person casts a 3 ft shadow. A 38 ft building nearby casts a shadow at the same time. How long is the building's shadow? → 19 ft
Bowtie triangles: a missing side
Find the value of x. — segments a d and b c crossing at e, making two triangles → 5
Bowtie triangles: similar or not
Are the triangles similar? If so, by which criterion? — segments a d and b c crossing at e, making two triangles → Yes, SAS
One triangle inside another: similar or not
Is △JMN∼△JKL? If so, by which criterion: AA, SAS or SSS? — a triangle with a segment across it making a smaller triangle at the top, labeled J, K, L, M, N → Yes, AA
One triangle inside another: a piece in x
DE‾∥BC‾. Find x. — a triangle with a segment across it making a smaller triangle at the top → 8
One triangle inside another: the leftover piece
DE‾∥BC‾. Find x. — a triangle with a segment across it making a smaller triangle at the top → 7

Questions about these worksheets

Yes. Every similar triangles 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 similar triangles 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 10: aa, sss or sas similarity, or not similar, a missing side in nested similar triangles, a side written in x, the similarity statement and ratio, shadows, bowtie triangles: a missing side, bowtie triangles: similar or not, one triangle inside another: similar or not, one triangle inside another: a piece in x and one triangle inside another: the leftover piece. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the lower piece instead of the whole side. The ratio compares the small triangle to the big one, and the big one runs all the way from the top vertex.