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AA similarity simply stands for "angle-angle" similarity. It's a useful concept for us to understand - especially as we get further into advanced math concepts. But what exactly *is *angle-angle similarity? What can it teach us about triangles and geometry? Let's find out:

AA similarity, also known as Angle-Angle Similarity, states that if two triangles have two pairs of corresponding angles that are congruent, then the triangles are similar. In the context of mathematics, "congruent" means "equal in measure." It's important to note that similarity is different from congruence. Similar triangles have the same shape but not necessarily the same size, while congruent triangles have both the same shape and size. The symbol for congruence is "≅," while the symbol for similarity is "~".

Consider the following image:

We can see that $\angle A\cong \angle D$. We can also see that $\angle E\cong \angle B$. Because of the principles of AA similarity, we now know that these two triangles are similar. Note that this rule applies even if we're not dealing with right triangles. Many other rules of similarity only apply to right triangles.

Aside from helping us learn new math skills, why is it important to know whether two triangles are similar?

One of the most obvious applications is the measurement of objects that are too tall or difficult to measure by hand. For example, we might look at the shadow created by an object and use the shadow to determine the length or height of the object. Because we can examine *angles* and proportions rather than distances, we don't necessarily need to measure everything by hand if we apply principles like AA similarity.

Eratosthenes used similar principles to measure shadows using vertical objects at different locations, allowing him to calculate the Earth's circumference with relatively high levels of accuracy.

Triangle Proportionality Theorem

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