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Consider the triangles $△ABC$ and $△DEF,$ where
$∠A≅∠Dand∠B≅∠E.$
If one of these triangles can be mapped to the other using only similarity transformations, they are similar. As the only condition for segments to be congruent is to have the same length, it's possible to dilate $△DEF$ by some factor so that $AB$ and the image of $DE$ are congruent.

By the ASA congruence theorem, the triangles are congruent. Consequently, there exists a rigid motion that maps $△D_{′}E_{′}F_{′}$ onto $△ABC.$ Thus, it is possible to combine a dilation with some rigid motion, both being similarity transformations, to map $△DEF$ onto $△ABC$. This means that the original triangles indeed are similar.

Statement | Reason |

$∠A≅∠D$ and $∠B≅∠E$ | Given |

$AB≅D_{′}E_{′}$ | By construction |

$∠A≅∠D_{′}$ and $∠B≅∠E_{′}$ | Dilations preserve angles |

$△ABC≅△D_{′}E_{′}F_{′}$ | ASA congruence theorem |

Exists rigid motion from $△D_{′}E_{′}F_{′}$ to $△ABC$ | Congruence definition |

Exists similarity transformation from $△DEF$ to $△ABC$ | Dilation and rigid motion are similarity transformations |

$△ABC∼△DEF$ | Similarity definition |

Show that $△ABC$ and $△ADE$ are similar.

Show 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 $△ABC$ are given. Using that the sum of a triangle's interior angles is $180_{∘}$, we can find the unknown angle measure.

$m∠A+m∠B+m∠C=180_{∘}$

$59_{∘}+m∠B+73_{∘}=180_{∘}$

AddTermsAdd terms

$m∠B+132_{∘}=180_{∘}$

SubEqn$LHS−132_{∘}=RHS−132_{∘}$

$m∠B=48_{∘}$

The measure of angle $B$ is $48_{∘}.$

We can now see that $∠B≅∠ADE.$ Thus, both $△ABC$ and $△ADE$ have the interior angles $59_{∘}$ and $48_{∘}.$ By the AA Similarity Theorem, the triangles are similar, which we were to show.

The Grim Reaper, who is $5$ feet tall, is standing $16$ feet away from a street lamp at night. The Grim Reaper's shadow cast by the street lamp light is $8$ feet long. How tall is the street lamp?

Show 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 $x,$ 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 $24$ feet.

Since the triangles are similar, the ratios between corresponding side lengths are the same. This gives us the equation $5x =824 ,$ which we can solve for $x.$

The street lamp is $15$ feet high.

The Pythagorean Theorem states that for every right triangle with side lengths $a,$ $b,$ and $c,$ where $c$ is the length of the hypotenuse, $a_{2}+b_{2}=c_{2}.$
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 $x$ and $y$ is $c.$ 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, $a_{2}$ and $b_{2}$ 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 $a_{2}$ can be found. $xa =ac ⇒a_{2}=cx$ The hypotenuse and bottom leg of the medium and largest triangle gives a similar expression, but for $b_{2}$ instead. $yb =bc ⇒b_{2}=cy$ These can now be substituted in the left-hand side of the Pythagorean Theorem, to show that the sum of $a_{2}$ and $b_{2}$ indeed is $c_{2}.$ Remember that the sum of $x$ and $y$ is $c.$ $a_{2}+b_{2}a_{2}+b_{2}a_{2}+b_{2}a_{2}+b_{2} =cx+cy=c(x+y)=c⋅c=c_{2} $

Thus, the Pythagorean Theorem is true for all right triangles. {{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

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