Notes for June 16

The interior angles of a triangle always add up to 180 degrees°.

There are two classification systems.

Largest angle classification

The measure of an obtuse angle x° is 180° > x° > 90°.
The measure of a right angle is 90° exactly.

The measure of an acute angle x° is 90° > x° > 0°.

Every triangle is exactly one of these three things.

Relationships between angles classification

An equilateral triangle has three equal angles, all of them measure 60°.
An isosceles triangle has at least two equal angles. What this means is equilateral is a special case of isosceles, the same way a square is a special case of a rectangle.
A scalene triangle's angles are all different from one another.

Defining a triangle by side lengths
 
If all we have are the angles, we have no idea about the area. For that, we need the sides lengths. Let's call the lengths u, v and w. Any three letters will do, but we are going to have a formula for area called Heron's Formula, and for that, we will need to reserve the letters p for perimeter and s for semi-perimeter. But before we get to that we have the triangle inequality. For three numbers to be the side lengths of a triangle we need

u + v  >=  w  >=  | u - v |


The last term is the absolute value of the difference between u and v. This is the cleanest way to put the information in an equation. In plain English, this says any two sides must add up to at least as much as the third side, and the easiest way to check that all of them are true is to see if the two short sides add up to more than or equal to the longest side.


Why did we make the inequality with greater than or equal to signs? Shouldn't just strictly be greater than?


No. If we have a case where the sum of the two short sides equals the long side, we have a degenerate triangle, which is a line segment with a dot on it. If we consider the simple area formula Area = ½ bh, a degenerate triangle has a height of zero, so the area is zero.

If we have the three sides, we don't have the height, so the simplest area formula is no good to us. Instead, we will use Heron's formula. Follow this link to see a few practice problems using Heron's formula.

  

Degenerate triangles

The triangle inequality state that if the sides of a triangle are labeled a, b and c, we need two inequalities to be true a + b >= c >= |a - b|. If we get an equality, the triangle is said to be degenerate. For example, if the lengths are 9, 6 and 3, we would draw a line segment of length 9 with a point on the line splitting into two parts of length 6 and 3. To think of it in terms of base and height, the base would be 9 and the height 0, so the area would be zero.

There are some facts about degenerate triangles you should know.

a) If you plug in these values into Heron's formula, you will get an area of 0.

b) The angle sum is 180°, since there are angles of 180°, 0° and 0°. It is not truly obtuse, since 180° is a straight angle.

c) If the two short sides are equal, it qualifies as an isosceles triangle, if they are not equal, it is scalene.

d) When we define a triangle by three distinct points in the plane, we will get a degenerate triangle if the three points are all on the same line, which is known as colinear.