Similar Triangles Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion. Subjects Near Me. Download our free learning tools apps and test prep books. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape.
Thus, they always form similar triangles. The diagram below makes this much more clear. If one angle moves, the other two must move in accordance to create a triangle. So with any movement, the three angles move in concert to create a new triangle with the same shape.
Hence, any triangles with three pairs of congruent angles will be similar. Sometimes the triangles are not oriented in the same way when you look at them. You may have to rotate one triangle to see if you can find two pairs of corresponding angles. Another challenge: two angles are measured and identified on one triangle, but two different angles are measured and identified on the other one.
Because each triangle has only three interior angles, one each of the identified angles has to be congruent. Then you can compare any two corresponding angles for congruence.
The second theorem requires an exact order: a side, then the included angle, then the next side. The Side-Angle-Side SAS Theorem states if two sides of one triangle are proportional to two corresponding sides of another triangle, and their corresponding included angles are congruent, the two triangles are similar. Here are two triangles, side by side and oriented in the same way. With their included angle the same, these two triangles are similar.
This theorem states that if two triangles have proportional sides, they are similar. This might seem like a big leap that ignores their angles, but think about it: the only way to construct a triangle with sides proportional to another triangle's sides is to copy the angles.
Notice we have not identified the interior angles. They all are the same ratio when simplified. They all are 1 2. So even without knowing the interior angles, we know these two triangles are similar, because their sides are proportional to each other.
Now that you have studied this lesson, you are able to define and identify similar figures, and you can describe the requirements for triangles to be similar they must either have two congruent pairs of corresponding angles, two proportional corresponding sides with the included corresponding angle congruent, or all corresponding sides proportional.
If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar. We could also use Trigonometry to calculate the other two sides using the Law of Cosines :.
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