ACT and SAT Math Tips: Similar Triangles, the Mini-Me of the Math World

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Similar triangles occur frequently on both the ACT and the SAT. If you think of them as the Mini-Me of the standardized testing math world, you'll never fail to recognize them when they appear on your test. Let's review their properties and then look how they may try to sneak onto the ACT and the SAT. Similar triangles are look-alike fathers and sons. A chip off the ol' block, the smaller son is a spitting image of his larger dad:

Triangles1.pngTriangles that have the exact same shape but different area are called similar triangles. The corresponding angle measurements of similar triangles are equal, and the corresponding side lengths are proportional:


They are fairly easy to identify when they are sitting next to each other, but similar triangles can be pretty sneaky on the ACT and on the SAT, with one hiding inside the other or one balancing on the other between parallel tightropes.  There are three situations that may make similar triangles difficult to identify:

1. In the first set of disguised similar triangles, line RS is parallel to line YZ and thus the two triangles are similar:


2. When triangles have parallel bases and share the opposite vertex angles, they are also similar:


3. Finally, the best-disguised set of similar triangles occur when one right triangle is inside of another right triangle and they share a complete side:


Let's look at how recognizing the similar triangle relationship can save you valuable time on your test:


Many students will abandon this question because they do not think enough information has been provided to solve it. But start with triangle ABC and DIAGRAM it using your knowledge of the 30:60:90 triangle:

triangles7.pngTriangles ABC and ACX are similar; we know this because they are both right triangles and share angle A. Redraw triangle ACX next to triangle ABC so they look like similar triangles. Then DIAGRAM triangle ACX. We know AC = 8, and since ACX is also a 30:60:90, we can complete the side lengths. Since the sides of ACX are proportional to the sides of ABC, we know we have correctly determined all side lengths:

triangles8.pngNow find the area of ACX:


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Image: This is where I normally give credit to the people who graciously post fantastic photos on the internet under the Creative Commons license. But not today! Oh, no! That very basic--yet completely awesome--graphic of the father and son triangle was created by yours truly. Never had one lesson!