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DFN: we call a triangle equilateral if all sides of the triangle are the same length

DFN:we call two triangles similar if corresponding angles are equal, and corresponding sides are proportional.

First show that all corresponding sides are proportional:

Consider a equilateral triangle with side lengths 1, all other equal lateral triangles sides can be expressed as S*(1), where S is some scalar. Hence all equilateral triangles sides are proportional to each other.

Next, show that all corresponding angles are equal:

The angle between two sides of a triangle is related to the length of the sides. These relationships are called sin, cos, and tan. Knowing that the cos(x), where x is one of the angles in the triangle, is the adjacent divided by the hypotenuse we see that cos(x)=(1/2)c/a, since a = c (because its equal lateral) we are left with cos(x)=(1/2) which means x = 60 degrees. this can be applied to all three angles, which shows that all three angles are 60 degrees.

/ \

/ | \

a / | \ b

/__ |__\

c

We have now shown that all equal lateral triangles are similar because they all have proportional sides, and they all have equal angles.

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16y ago
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Q: Are all equilateral triangles are similar?
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