Similar triangles are geometric figures that have exactly the same shape but possibly different sizes. They look identical, just scaled up or down. The key property is that all corresponding angles are equal, and the ratios of corresponding sides are proportional.
Similar triangles have two key properties. First, all corresponding angles are equal. Second, the ratios of corresponding sides are proportional. This means if you know the scale factor k, you can find any unknown side length by multiplying the corresponding side of the other triangle by k.
There are three main criteria to prove triangles are similar. The AA criterion states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. The SSS criterion requires all three pairs of corresponding sides to be proportional. The SAS criterion needs two pairs of sides to be proportional with the included angles equal.
Let's solve a practical example. We have triangle ABC with sides 6, 8, and 10, and triangle DEF with sides 3, 4, and 5. To check if they're similar, we calculate the ratios of corresponding sides. Six divided by three equals two, eight divided by four equals two, and ten divided by five equals two. Since all ratios are equal, the triangles are similar with a scale factor of 2.
Similar triangles have many practical applications in real life. In architecture and construction, they help with scaling and proportional design. In navigation, they're used for map scaling. One common application is calculating heights using shadows. When the sun creates similar triangles with a building and a person, we can use the proportion of heights to shadows to find unknown measurements.