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If you draw an altitude in an equilateral triangle, you will form two congruent 30º- 60º- 90º triangles. Starting with the sides of the equilateral triangle to be 2, the Pythagorean Theorem will allow us to establish pattern relationships between the sides of a 30º- 60º- 90º triangle. These relationships will be stated here as "short cut formulas" that will allow us to quickly arrive at answers regarding side lengths without applying trigonometric functions, or other means.

There are three pattern relationships that we can establish that apply ONLY to a 30º-60º-90º triangle.

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45--45--90. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30°, 60°, and 90°. The sides are in the ratio 1 : √3 : 2.

The proof of this fact is clear using trigonometry. The geometric proof is:

Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30--60--90 triangle with hypotenuse of length 2, and base BD of length 1.

The fact that the remaining leg AD has length √3 follows immediately from the Pythagorean theorem.

The 30-60-90 triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α+δ, α+2δ are the angles in the progression then the sum of the angles 3α+3δ = 180°. So one angle must be 60° the other 90° leaving the remaining angle to be 30°.source wiki

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