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.
The small triangle has side 4 where the big one has 4 + 6 = 10.
So the bases are in the ratio 4 to 10.
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
Are the triangles similar? If so, by which postulate?→ Yes, SSS
A missing side in nested similar triangles
The triangles are similar. Find x→ 12
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 2: aa, sss or sas similarity, or not similar and a missing side in nested similar triangles. 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.