High School Geometry: Special Triangles Explained

by Jana Sosnowski
The study of triangles in geometry includes properties for special triangles.

The study of triangles in geometry includes properties for special triangles.

Triangles are the primary polygons studied in high school geometry, and they lay the foundation for math studies throughout high school. Once students learn basic properties of triangles (such as all angles add up to equal 180 degrees or all triangles contain three sides and three angles), they begin to learn that there are special triangles with standard properties.

Isosceles Triangles

Two sides of an isosceles triangle are equal in length. The two sides with identical dimensions are called legs, and the third side is called the base. Two angles formed at the ends of the base are called base angles, and they are also equivalent. Given one of the base angles, it is possible to find the measures of all three angles. With identical base angles, add the two angles together and subtract from 180 degrees to find the value of the third angle opposite the base. Drawing a perpendicular segment from the base to the opposite angle creates two right triangles, which can also be used to determine the length of a missing side length.

Special Right Triangles

Special right triangles will always include a 90-degree angle. The special right triangles also have standard angle measures -- the 45-45-90 triangle and the 30-60-90 triangle. You can use the unique properties of special right triangles to find the lengths of all three sides given one side length.

45-45-90 Triangles

This type of right triangle has two 45-degree angles. With congruent angles, the 45-45-90 triangle is also an isosceles triangle, which means you can also use the properties of an isosceles triangle to solve an equation, and find the missing measurements. The two legs across from the 45-degree angles will be congruent, and the length of the hypotenuse can be determined using the Pythagorean Theorem.

30-60-90 Triangles

In addition to the 90-degree angle, this right triangle contains 30-degree and 60-degree angles. The legs in this triangle are referred to as the short leg and the long leg. The short leg is found directly across from the 30-degree angle, and the long leg is directly across from the 60-degree angle. Given the shortest leg of the triangle, multiply by two to find the hypotenuse and by √3 to find the longer leg. You can find the length of the shorter leg by dividing the hypotenuse by two. You can also find the shorter leg by dividing the longer leg by √3.

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