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Similarity

Similar Triangles

Rule

Angle-Angle Similarity Theorem

If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.

Similar Triangles

If and then

Proof

The Angle-Angle Similarity Theorem can be proven by using transformations. Consider the triangles and with and congruent, and and also congruent.

Similar Triangles
If one of these triangles can be mapped onto the other by using only similarity transformations, it can be concluded that they are similar. First, will be translated so that one of its vertices coincides with its corresponding vertex in
Similar Triangles
Next, the image of the translation will be rotated about the common vertex.
Similar Triangles
Finally, the image of the rotation will be dilated about the same vertex by a certain scale factor.
Similar Triangles
Since could be mapped onto by a combination of similarity transformations, it can be concluded that the triangles are similar.


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Exercise

Show that and are similar.

Show Solution
Solution

If we show that the two triangles have two angles in common then, by the AA Similarity Theorem, they must be similar. Therefore, we need to find more angle measures. Two of the three angle measures of are given. Using that the sum of a triangle's interior angles is , we can find the unknown angle measure.

The measure of angle is

We can now see that Thus, both and have the interior angles and By the AA Similarity Theorem, the triangles are similar, which we were to show.

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Exercise

The Grim Reaper, who is feet tall, is standing feet away from a street lamp at night. The Grim Reaper's shadow cast by the street lamp light is feet long. How tall is the street lamp?

Skills Find the height of the street light using similarity.svg
Show Solution
Solution

First, we have to make sense of the information, preferably by adding it to the sketch of the situation. Under the assumption that the lamp post and the Grim Reaper make right angles against the ground, two right triangles can be drawn. We are interested in the length corresponding to the height of the lamp post.

As these triangles both have a right angle, and share the angle on the right-hand side, they are similar by the AA Similarity Theorem. Notice that the base of the larger triangle measures feet.

Since the triangles are similar, the ratios between corresponding side lengths are the same. This gives us the equation which we can solve for

The street lamp is feet high.


Proof

Pythagorean Theorem with Similarity

The Pythagorean Theorem states that for every right triangle with side lengths and where is the length of the hypotenuse, This can be proven using similar triangles.
Consider a right triangle with the given side lengths.

Now, draw a segment from the right angle to the hypotenuse so that the segment is perpendicular to the hypotenuse. Name all the new lengths that were created.

Notice that the sum of and is The two new, smaller triangles both have a right angle and share another angle with the larger triangle. Thus, by the AA Similarity Theorem, they are both similar to the larger triangle and, by extension, each other.

As they are similar, the ratios between sides in the triangles are the same for each triangle. Using this, and can both be expressed in an alternate fashion. Looking at the hypotenuse and left-most leg of the smallest and largest triangle, the following expression for can be found. The hypotenuse and bottom leg of the medium and largest triangle gives a similar expression, but for instead. These can now be substituted in the left-hand side of the Pythagorean Theorem, to show that the sum of and indeed is Remember that the sum of and is Thus, the Pythagorean Theorem is true for all right triangles.


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