三角形的內角和總等於１８０°嗎？

三角形內角和在歐幾里德幾何是180度
怎樣測出來?這像1加1=2一樣約定俗成
三角形內角和可以大於180度,亦可以小於
這豈不是和第一行矛盾?
第一行中,是歐幾里德幾何,是平面幾何
但三角形內角和大於180度是黎曼幾何,是畫在球面上的三角形
所以第1,2條問題已被否定,不是一定,可以是其他度數

根據歐幾里得的<幾何原理>第五個假定條件描述:如果一條直線與另外兩條直線相交,使得同一邊的內角小於兩個直角(180度),那麼把這兩條線無限制延長,將會在內角和小於兩直角那側的某處相交.這才是描述三角形的內角總和會等於兩個直角相加,也就是180度的根本來源,而四邊形的內角和是根據三角形的內角和所推論出來的.不過這只有在歐式平面裡才成立的.

三角形ABC-->畫輔助線即可!!

由A畫一平行於BC的線,
由於不能畫圖所以以文字表示:

D　　　Ａ　　　Ｅ
　　　
　　Ｂ　　　Ｃ

自行把點連起來,直線ＤＡＥ平行於ＢＣ
則角ＤＡＢ＝角ＡＢＣ　,角ＥＡＣ＝角ＡＣＢ
又ＤＡＥ為直線所以角ＤＡＢ＋角ＢＡＣ＋角ＥＡＣ＝１８０度

綜合上面的即角ＡＢＣ＋角ＢＡＣ＋角ＡＣＢ＝１８０度
所以三角形內角合為１８０度

不過我們日常用的三角形內角和大部分都是１８０度~~~~~~~~~

平面就一定係

在曲面未必

im going to take the special right triangles to make my examples,

to your first qusetion. its a yes, every triangles angles have to make up a 180 degree, because thats what it takes to connect the 3 dots togethe, anything less will not allow you to connect the 3 cots for the triangles.

according to the converse of the Pythagoras Theorem, each 2 sides have to add up so that its greater then the 3rd side. To determine the legth of a side of the triangle, i have to also talk about angles on a Unit Circle. In the unit circle, the radius is always going to be 1, whicih allows you to easily define the trig functions like sin and cos, which is based on the coeffient of 1. the idea of 180degree triangle is from the unit circle.

im going to take 45degrees, which is pi/4. on a unit circle, if you add cos(r/x)45degrees, sin(r/y)45degree, and tan(y/x)45degree, you'll get a radian measurment of pi, which in degrees in 180degrees. this is the originial definiation of sin cos and tan, and of course, the idea of a 180degree triangles.

一定係等於１８０度
你試下用紙撕成三角形，然後撕出三隻角，再慢慢拼湊成一條水平線。

係的，這是三角形的其中一個特性Principle,no explanation