Math-a-Magic

The Sum of the Angles in a Triangle is 180 degrees

All of the triangles add up to 180 degrees. We can do a rigorous proof of this (see below), but if you just want to illustrate this in a visual way, you can check out my newest You Tube video!

Before doing a rigorous proof of the triangle angle sum theorem (the sum of the angles in a triangle is 180 degrees) some background knowledge of geometry is needed.

1) A straight angle is defined as 180 degrees.

2) Congruent means the same size. Also Congruent Angles have the same measure of degrees (or radians). When we refer to the actual the degree measure, we use an m in front of the angle sign.

3) Parallel lines are lines are ALWAYS equidistant (equal distance) from each other.

The red segments show that the distance is equal.

The red segments show the distance is not equal.

4) Understanding substitution: If we say a variable x = z and we also know that y = z, then where ever we see an x we can substitute a y (since they both equal z.) We can also substitute y in for x.

5. When you have a line and a point not that line there is only one line you can draw through that point that is parallel to the original line. This is called EUCLID'S PARALLEL POSTULATE.

A line and a point not on the line.

Only one parallel line can be made through the point.
The pink lines are NOT parallel to the blue.

6. Vertical Angles are congruent.

7. When parallel lines are intersected by the same line or segment (called a TRANSVERSAL) the angles created are related to each other.

Adding another transversal to our parallel lines and using that vertical angles are congruent we can show numerous angle congruences.

For this particular proof, we only need to use the alternate interior angles (you can also construct a proof using corresponding angles.)

Okay, now we are close to being ready to construct a formal proof. It is best to go through the steps informally first, and then write them down. Here is a review of the background information we will use.