Triangles Part 1: What is a Triangle?

Photo by Erwan Hesry on Unsplash

A triangle is a 3-sided polygon with interior angles that add to 180 degrees. It’s a remarkable shape because of its stability: when the shape is under stress, it distributes that stress across its sides. With an equilateral triangle, the stress is distributed evenly, making it a popular choice for architects and structural engineers. Not only are triangles physically stable, they also can provide metaphorical stability in arts and design.

In any triangle, the longer an individual side is, the greater the measure of its corresponding angle (the angle across from it). That means that the largest angle will be across from the longest side, and the smallest angle will be across from the shortest side (if 2 or more angles are equal, their corresponding sides will be equal, and vice versa). In the triangle below, the longest side is across from the 132 degree angle (its corresponding angle). The other two sides are equal in length and are each across from a 24 degree angle.

A triangle can have sides that are all equal in length (equilateral), sides that are 2 different lengths (isosceles), or 3 different side lengths (scalene). The sides can’t be just any set of lengths, though. For all 3 sides of a triangle to meet, each side has to be less than the sum of the other two sides. Each side must also be greater than the difference between the other two sides. For example, a triangle with side lengths of 3, 4, and 5 will close, but not one with sides of 1, 2, and 3 (1 + 2 is not greater than 3). This video has an in-depth explanation of the rule, which is called the Triangle Inequality Theorem. 

The exterior angles of a triangle (and of any polygon) add to 360 degrees. Each exterior angle is equal to the sum of the two interior angles across from it. As you can see in the examples below, there are two sets of exterior angles you can draw on a triangle (use one set at a time, not both simultaneously).

Symbols

To be able to understand diagrams of triangles and other shapes, there are some symbols and conventions that are useful to know.

Congruent (\(\cong\)): The congruent symbol means that two values (angles, side lengths, shapes) are identical.

Similar (\(\sim\)): The similar symbol means that two shapes have the same corresponding angle measurements and sides that are proportional. An example of this would be two triangles with angle measures of 30, 60, and 90 where one triangle has side lengths of \(1, \sqrt{3}, 2\) and the other has side lengths of \(3, 3\sqrt{3}\), and \(6\) (the second triangle is 3 times larger than the first.

Side and Angle Conventions: If two sides or angles of a shape are marked in the same way, they are understood to be congruent. In the diagram below, congruent sides are marked with the same number of tick marks, and congruent angles are marked with the same number of angle symbols.

Types of Triangles

Equilateral: A triangle with 3 equal sides and 3 equal angles of 60 degrees.

Isosceles: A triangle with 2 equal sides with equal corresponding angles (the angle across from a side is its corresponding angle).

Scalene: A triangle with no equal sides and no equal angles.

Pythagorean Triples: 

Three of the most common Pythagorean triples are (3, 4, 5); (5, 12, 13); and (8, 15, 17). Keep in mind that multiples of these numbers can also be Pythagorean triples. For example, if you see a 6, 8, 10 triangle, it has the same special side relationships as a 3, 4, 5 triangle (the sides have just been multiplied by 2).

 

Pythagorean Theorem:

The Pythagorean theorem is \(a^2 + b^2 = c^2\). This formula states that the square of the hypotenuse (c) of a right triangle is equal to the sum of the squares of the other two sides (a and b). In the example below, 5 is the hypotenuse (c), and 3 and 4 are the other two sides (a and b). \(5^2 = 3^2 + 4^2\), so the triangle is a right triangle, This formula is useful for determining if a triangle is right and for finding a missing side of a right triangle. Here is a great animation showing a visual proof of the Pythagorean theorem!

Obtuse: A triangle with one angle that is greater than 90 degrees. To see if a triangle is obtuse, solve the Pythagorean theorem (\(a^2 + b^2 = c^2\)). If \(c^2\) is greater than \(a^2 + b^2\), then the triangle is obtuse.

Right: A triangle with one angle that equals 90 degrees. *Note about trig. Being the study of right triangles.* The side across from the 90 degree angle is called the hypotenuse and is always the longest side. To see if a triangle is right, solve the Pythagorean theorem (\(a^2 + b^2 = c^2\)). If \(c^2 \) equals \(a^2 + b^2\), then the triangle is right.

Acute: A triangle with 3 angles less than 90 degrees. To see if a triangle is acute, solve the Pythagorean theorem (\(a^2 + b^2 = c^2\)). If \(c^2\) is less than \(a^2 + b^2\), then the triangle is acute.

45 – 45 – 90 Triangle: A 45 – 45 – 90 triangle is an isosceles right triangle with two angles that equal 45 degrees and one that equals 90 degrees. The relationships between the sides of this triangle are \(1x : 1x : \sqrt{2}x\).

30 – 60 – 90 Triangle: A 30 – 60 – 90 triangle is a scalene right triangle with angle measures of 30, 60, and 90 degrees. The relationships between the sides of this triangle are \(1x : \sqrt{3}x : 2x\).

Similar Triangles: Similar triangles are ones that have equal corresponding angles (the angles in the same places of each triangle are equal) as well as a common ratio between the sides.

Perimeter and Area

Perimeter: To find the perimeter of a triangle – or the outline of the triangle – add all three of the sides together. For the triangle below, the perimeter is 3 + 4 + 5 = 12

Area: To find the area of a triangle – or the space inside the triangle – there are a few formulas you can use.

If you know the base and the height of the triangle, use 1/2bh, where b stands for the base of the triangle and h stands for the height. The height is always perpendicular to the base, so it might not be one of the sides of the triangle. The examples below show the height as a side of the first triangle but not of the second.

If you know two sides of a triangle and the angle in between, use \(\frac{1}{2}a*b*sin{c}\).

If you know all three sides of the triangle (a, b, and c), use \(A = \sqrt{p(p-a)(p-b)(p-c)}\) where p is half the perimeter: \(p = \frac{a+b+c}{2}\).

Congruence Postulates

If two triangles are congruent, all of their corresponding sides and corresponding angles are equal.

The triangle congruence postulates give you five ways to ensure that two triangles are congruent. Check out this link for an explanation of these postulates.

Side-Side-Side (SSS): If all of the corresponding sides of two or more triangles are equal, then the triangles are congruent.

Side-Angle-Side (SAS): If two corresponding sides and the angle between them are congruent across two or more triangles, then the triangles are congruent.

Angle-Side-Angle (ASA): If two corresponding angles and the side between them are congruent across two or more triangles, then the triangles are congruent.

Angle-Angle-Side (AAS): If two corresponding angles and a side not between them are congruent across two or more triangles, then the triangles are congruent.

Hypotenuse-Leg (HL): If the hypotenuse and leg of two right triangles are congruent, then the triangles are congruent.

Two triangles are not guaranteed to be congruent if you know all three angles (AAA).

Two triangles are also not guaranteed to be congruent if you know two sides and an angle that is not between the two sides (SSA or ASS, otherwise known as “The Donkey Theorem”).