Why 180?
Have you ever been told that the interior angles of every triangle adds up to 180? Let's check that claim! In Task 1, move the points around to make your own triangles and check the sum of the interior angles. Does every sum total 180?
Task 1
We can see that the three interior angles of any triangle added together will equal 180.
Why is this the case? Why is the sum always 180? How can we prove that?
A good starting point is to consider where else we have seen a 180 angle?
What does this have to do with the interior angles of a triangle?
Well let's assume we have two parallel lines (or two parallel straight angles), and crossing through those two parallel lines is a green line. We can call this green line a "transversal".
What do you notice about the two pairs of interior angles created by the transversal?
Task 2
The sum of each pair of angles created by the transversal equals 180 !
The transversal intersects each of the straight angles and splits it into two parts. Each of those new angles is known as a supplementary angle. Supplementary angles are any two angles whose sum totals 180.
You may have noticed that there are four interior angles, but only two different angle measurements. There are two pairs of congruent (or equal) angles that are opposite one another. These angles are known as alternate angles.
Alternate angles are created by the intersection of one line (the transversal) through at least two other lines. Alternate angles are congruent when they are formed by a transversal passing through parallel lines. (like in Task 2)
BUT WHERE ARE THE TRIANGLES???? Well let's make one.
Like the other tasks feel free to move the points of the triangle anywhere along the parallel lines to create new triangles. Do you see any relationships between any of the angles?
Task 3
You may have seen there are two pairs of alternate angles created by the transversals AB and AC. The alternate angle pairs, b and e, and, c and d, are congruent.
We already know the sum: a + b + c = 180. What about the sum of a and the alternate angles of b and c?
a + e + d = 180