Exterior Angle Theorem Part 1
- Author:
- Kayla Moore
The exterior angle theorem states that if an exterior angle is created by extending a line from a triangle then the exterior angle created is greater than the two other interior angles of the triangle.Proof: Let there exist a triangle ABC. Form an exterior angle by extending side AC by adding a point D. Create midpoint F on side BC by using Proposition 10. Draw a line BG so that BF FG.
From Postulate 5, we know that:
ABC + BCA + CAB = 180
BEA + EAB + ABE = 180 and BCE + CEB + EBC =180.
From Proposition 13, we know:
BCA + ECD = 180.
From Proposition 15, we know:
BEA + AEF = 180 and
BEC + CEF = 180.
By Common Notion 1, notice:
BEA + AEF = BEA + EAB + ABE
-BEA -BEA By Common Notion 3
So, AEF = EAB + ABE.
Similarly,
BEC + CEF = BEC + BCE + EBC.
So, CEF = BCE + EBC.
When looking at the whole triangle ABC, we see through a similar process that:
BCA + ECD = ABC + BCA + CAB by Common Notion 1.
By Common Notion 3, we can subtract BCA from both sides which leaves us with:
ECD = ABC + CAB.
By Common Notion 5, we know that the whole is greater than the parts. In this case, the whole is ECD and the parts are ABC and CAB. Therefore, ECD, the exterior angle, is greater than ABC and CAB which are the other interior angles of the triangle.