The triangle is a basic two-dimensional geometric figure that every math student should be familiar with. In this article I have summarized the definitions and formulas regarding triangles that the average math student should know. I pulled this information from Saxon Algebra 2, Third Edition. This particular edition does an excellent job of integrating teaching Geometry along with teaching Algebra. Here are the math facts about triangles the math student should know:
To find the perimeter of a triangle add the length of each of the three sides.
To find the area of a triangle multiple the base times the height and divide by 2.
The sum of the three angles of a triangle is 180 degrees.
The largest angle of the three angles is always opposite the longest side and the smallest angle of the three is always opposite the shortest side.
If a triangle has an angle equal to 90 degrees, it is called a “right triangle.”
If a triangle has an angle greater than 90 degrees, it is called an “obtuse triangle.”
If all three angles of a triangle are less than 90 degrees, it is called an “acute triangle.”
If all three sides of the triangle are of different lengths, it is called a “scalene triangle.”
If the triangle has at least two sides that are of equal lengths, it is called an “isosceles triangle.”
A triangle that has three equal angles also has three equal sides. This triangle is called an “equilateral triangle.”
The angles for an equilateral triangle are all 60 degrees.
If two sides of a triangle are of equal length, then the angles opposite those sides are also equal.
With a right triangle, the side opposite the right angle (90 degrees) is called the “hypotenuse.”
The Pythagorean theorem states that a2 + b2 = c2, where c is the length of the hypotenuse and a and b are the length of the other two triangle sides.
If two triangles have the same angles, then they have the same shape and are called “similar triangles.”
The sides opposite equal angles in similar triangles are called “corresponding sides.”
The ratios of the lengths of corresponding sides in similar triangles are equal; the corresponding sides are related to each by a number called a “scale factor.”
An important theorem the math student should know for proving two triangles are similar is “AA means AAA.” This is also represented as “AA à AAA.” This theorem states that if you can prove two angles in one triangle are the same as two angles in a second triangle, can prove that all three angles are the same, and ergo the two triangles are similar.
“Congruent triangles” are similar triangles whose scale factor is 1. These are triangles that are geometrically equal.
The special ratios “sine”, “cosine”, and “tangent” are all based on right triangles. (How these are used and defined is beyond the scope of this article.)
A special kind of right triangle that comes up regularly in physics and engineering is the “30-60-90” degree triangles. These triangles are all similar to the 30-60-90 triangle that has opposite side lengths 1, square root of 3, and 2, respectively.
A special kind of right triangle that comes up regularly in physics and engineering is the “45-45-90” degree triangles. These triangles are all similar to the 45-45-90 triangle that has opposite side lengths 1, 1, and square root of 2, respectively.
John H. Saxon, Jr. Algebra 2: An Incremental Development. Third Edition