Then you can compare any two corresponding angles for congruence.

By subtracting each triangle's measured, identified angles from 180В°, you can learn the measure of the missing angle. You may have to rotate one triangle to see if you can find two pairs of corresponding angles.ДЄnother challenge: two angles are measured and identified on one triangle, but two different angles are measured and identified on the other one.Д«ecause each triangle has only three interior angles, one each of the identified angles has to be congruent. Sometimes the triangles are not oriented in the same way when you look at them. Watch for trickery from textbooks, online challenges, and mathematics teachers. Since в€ A is congruent to в€ U, and в€ M is congruent to в€ T, we now have two pairs of congruent angles, so the AA Theorem says the two triangles are similar. Notice в€ M is congruent to в€ T because they each have two little slash marks. A single slash for interior в€ A and the same single slash for interior в€ U mean they are congruent. We have already marked two of each triangle's interior angles with the geometer's shorthand for congruence: the little slash marks. Here are two scalene triangles в–і J A M and в–і O U T. For AA, all you have to do is compare two pairs of corresponding angles. The two triangles could go on to be more than similar they could be identical. Triangle Similarity TheoremsДЄngle-Angle (AA) says that two triangles are similar if they have two pairs of corresponding angles that are congruent. If they both were equilateral triangles but side E N was twice as long as side H E, they would be similar triangles. They are the same size, so they are identical triangles. The two equilateral triangles are the same except for their letters. Side F O is congruent to side H E side O X is congruent to side E N, and в€ O and в€ E are the included, congruent angles. Both в€ O and в€ E are included angles between sides F O and O X on в–і F O X, and sides H E and E N on в–і H E N. Notice that в€ O on в–і F O X corresponds to в€ E on в–і H E N. To make your life easy, we made them both equilateral triangles. You cannot compare two sides of two triangles and then leap over to an angle that is not between those two sides. The included angle refers to the angle between two pairs of corresponding sides. If the ratios are congruent, the corresponding sides are similar to each other. You can establish ratios to compare the lengths of the two triangles' sides. Their comparative sides are proportional to one another their corresponding angles are identical. Triangles are easy to evaluate for proportional changes that keep them similar. You look at one angle of one triangle and compare it to the same-position angle of the other triangle. The three theorems for similarity in triangles depend upon corresponding parts.

Even if two triangles are oriented differently from each other, if you can rotate them to orient in the same way and see that their angles are alike, you can say those angles correspond. In geometry, correspondence means that a particular part on one polygon relates exactly to a similarly positioned part on another.

These three theorems, known as Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS), are foolproof methods for determining similarity in triangles. Similar triangles are easy to identify because you can apply three theorems specific to triangles. An equilateral triangle with sides 21 cm and a square with sides 14 cm would not be similar because they are different shapes. You could have a square with sides 21 cm and a square with sides 14 cm they would be similar. In geometry, two shapes are similar if they are the same shape but different sizes. This is an everyday use of the word 'similar,' but it not the way we use it in mathematics. Mint chocolate chip ice cream and chocolate chip ice cream are similar, but not the same. Similar triangles are triangles with the same shape but different side measurements. Similarity in mathematics does not mean the same thing that similarity in everyday life does. Similar Triangles (Definition, Proving, & Theorems)

Similarity statement