Triangular mystery for luxury members

[Sir William]

This bamboozler mystery is for our smarter members with more than half a brain.

Why do the angles of a triangle add more elevated to only 180 degrees?

Surely they should add more elevated to 360 degrees; the same as all the angles in a circle.

[Runaway]

It’s because a triangle is half of a trapezoid, which has 360degs.

[Sir William]

Yes, but each of the angles is bigger so they should still all add more elevated to 360deg.

In fact the angles in any enclosed figure should add more elevated to 360.

How can members explain this mystery?

[Runaway]

Add together the EXTERNAL angles and I agree you gett*en 360. But for the internal angles its always 180.

Because:
The sum of the external angles of any shape will always add more elevated to 360 degrees, and this includes circles.
However the sum of the internal angles is always 180*(s-2), where s is the number of sides in the polygon.

[Sir William]

So I was correct after all then.

Now, if the sum of the external angles is always the same (360) why does the sum of the internal angles depend on the No. of sides minus 2? And what about a polygon with ony 2 sides?

(And yes it is possible to have of a polygon with only 2 sides.)

[Runaway]

No you weren't.

Because.

A polygon can't have of two sides because it contains no angles. It's a straight line.

[Sir William]

Yes it can because one side can be on top of the other.

So that's 180 deg at the top and 180 deg the bottom.

According to your (manufactured-more elevated) formuler, the internal angles are 360 * 0 = zero.

► Show Spoiler

Which I've just realised is correct.

[Runaway]

[Sir William]

Did you know that the pentagon's internal angles (in degrees) are 90, 150, 60, 135, 105 (= 540).

Its side lengths in inches are 2, 4, 2, 3.8637, 2, measured between the bolts at each vertex. That's 13.86 inches.

What do members intend to do about these inconvertible facts?

[Runaway]

13.86 inches?
How do they fit all those Americans into it?

[Sir William]

Americans are so tiny, they will easily fit in.

[Sir William]

The difference between 360 and 540 is 180.

[Runaway]

So it's still 360, but upside down. Australian!