Also, just as two 30-60-90 triangles form an equilateral triangles, two 45-45-90 triangles form a square. We explain the colossal importance of this fact when we cover polygons a little later in this chapter.
Similar Triangles
Similar triangles have the same shape but not necessarily the same size. Or, if you prefer more math-geek jargon, two triangles are “similar” if the ratio of the lengths of their corresponding sides is constant (which you now know means that their corresponding angles must be congruent). Take a look at a few similar triangles:
As you may have assumed from the figure above, the symbol for “is similar to” is ~. So, if triangle ABC is similar to triangle DEF, we write ABC ~ DEF.
There are two crucial facts about similar triangles.
- Corresponding angles of similar triangles are identical.
- Corresponding sides of similar triangles are proportional.
For ABC ~ DEF, the corresponding angles are 
The corresponding sides are AB/DE = BC/EF = CA/FD.
The SAT usually tests similarity by presenting you with a single triangle that contains a line segment parallel to one base. This line segment creates a second, smaller, similar triangle. In the figure below, for example, line segment DE is parallel to CB, and triangle ABC is similar to triangle AE.
After presenting you with a diagram like the one above, the SAT will ask a question like this:
Notice that this question doesn’t tell you outright that DE and CB are parallel. But it does tell you that both lines form the same angle, xº, when they intersect with BA, so you should be able to figure out that they’re parallel. And once you see that they’re parallel, you should immediately recognize that ABC ~ AED and that the corresponding sides of the two triangles are in constant proportion. The question tells you what this proportion is when it tells you that AD = 2 /3AC. To solve for DE, plug it into the proportion along with CB:
Congruent Triangles
Congruent triangles are identical. Some SAT questions may state directly that two triangles are congruent. Others may include congruent triangles without explicit mention, however.
Two triangles are congruent if they meet any of the following criteria:
- All the corresponding sides of the two triangles are equal. This is known as the Side-Side-Side (SSS) method of determining congruency.
- The corresponding sides of each triangle are equal, and the mutual angles between those corresponding sides are also equal. This is known as the Side-Angle-Side (SAS) method of determining congruency
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- The two triangles share two equal corresponding angles and also share any pair of corresponding sides. This is known as the Angle-Side-Angle (ASA) method of determining congruency
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