Understanding the Mathematical Subject Triangles

A triangle is a polygon formed by three connecting straight lines. The sum of the angles of a triangle in plane geometry is equal to 180 degrees. Examples are 90+30+60, 30+30+120, and 45+45+90. The area of a triangle is one-half the base times the height (1/2 bh). An equilateral triangle has all sides equal. An isosceles triangle has two sides equal. A scalene triangle does not have any equal sides. An obtuse triangle has one angle that is greater than 90 degrees. An acute triangle has one angle that is less than 90 degrees. The perimeter of a triangle is the sum of the sides. Triangles are composed of three connected segments. These three connected segments form three interior angles. A diagonal of a rectangle or square forms two triangles. The triangle inequality states that the sum of any two sides of a triangle is always greater than the third side.

A right triangle has 90 degrees as one of its angles. The side opposite this angle is called the hypotenuse. The Pythagorean Theorem (discovered by the ancient Greek Pythagoras) states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, or a^2+b^2=c^2. The right triangle with side a equal to 3 and side b equal to 4 has a hypotenuse equal to 5 because  3^2+4^2=5^2 (9+16=25). This means all triangles with side a equal to 3 and side b equal to 4 have hypotenuses equal to 5. The diagonal of a square forms two right triangles.


Euclid made many observations about triangles in his Elements. Two triangles are congruent if their corresponding sides and angles are equal. There are several congruence theorems. The Side-Side-Side Theorem (SSS) states that if all three sides of a triangle are equal, then the triangles are congruent. The Side-Angle-Side Theorem (SAS) states that two triangles are congruent if two sides and the included angle are equal. The Angle-Side-Angle Theorem (ASA) states that two triangles are congruent if two angles and the included side are equal. The Angle-Angle-Side Theorem states that two triangles are congruent if two angles and a NOT included side are equal.

Pascal’s Triangle is not a triangle; it is just the shape of a triangle. It is a configuration of numbers that has many uses like the Binomial Theorem, coefficients of powers of algebraic expressions, and determining whether or not a number is prime. The rows are configurated by starting with 1 for each row, then 1 1 for row 2. Next, add the row above (1+1=2), then 1 again. The first five rows are:

1 1
1 2 1

1 3 3 1

1 4 6 4 1