Similarity ( ~ )
Using Ratios to Determine Similar Triangles
The Side Splitter Theorem
When to Use Side-Splitter vs. Similar Triangles
You’re in the big leagues now. At this point, all your geometric building blocks are in place, and you’re stating to work with more than one figure at a time to solve problems. Anytime you are learning geometric similarity (or translations), it’s helpful to have some tracing paper and scissors handy so that you can see if figures, or parts of figures can be superimposed. You know by now that seeing drawings and illustrations is an important part of learning new concepts, but you may need a little more hands on for help for this one in addition to downloading the study guide. If superimposing shapes is boggling your brain, reach out for a consultation.
Even though some of the ways to determine similarity and congruence might look the same, they’re actually very different. You understand that congruence means that all the measurements for two shapes (or lines, or angles) are exactly the same. Similarity is different. It means that the shape is the same, but the size is different. Imagine building a model airplane that is an exact replica of the original, just a miniature version. The shape and all proportions are the same, but the sizes are very different. The airplanes are just like each other, but not the same. The scale is different, and the scale factor is the ratio of corresponding measures of two similar figures.
This first method uses some theorems that look a lot like the congruency theorems you learned earlier, but be careful not to mix them up! They are different! The AA ~ postulate says that if two angles on two separate triangles are the same measure, the triangles are similar … even if they look different because of very different sizes. The SSS~ and SAS~ theorems are where things might get a little confusing for some because there are also SSS congruency and SAS congruency theorems. The difference is that in the similarity (~) theorems, the corresponding parts in separate triangles must be proportional and not the same. The SSS similarity theorem says that if all the corresponding sides of two triangles are proportional, the triangles are similar. The SAS similarity theorem requires that one corresponding angle on two separate triangles must be congruent. If the two sides of the first triangle that include that angle are proportional to the two sides that include the angle on the second triangle, the triangles are similar.
Tune in because this is a big one. This theorem states that if there is a line that is parallel to one side of a triangle, then that line divides the other two sides that it intersects proportionally. This means that the original triangle whose side was split is similar to the smaller triangle created when you split it. Once you split a side in this way you have options for the proportions you can set up and compare. Not only can you compare the length of the original side to the length of the corresponding side on the new smaller triangle, but there is also proportionality between the two segments created from the original side. In simpler terms, you can either compare part to whole, or part to part. Fun fact, the converse of this theorem is also true. If two sides of a triangle are split proportionally, then the line segment that splits them is parallel to the third side.
If you are provided with the lengths of the parallel sides, you need to use the similar triangles strategy and cannot use side splitter. If the problem gives you the whole side length of the split side, you can use either, although a similar triangle strategy may be easier. If you’re given parts or segments of the split side, you must use side splitter because it is the only strategy that allows you to look at the proportions of part to part.