什么是三角形？
在数学中，有过多的形状和大小，而三角形被认为是最重要的形状之一。现实生活中的三角形可以在很多东西中看到，例如，交通信号灯、金字塔的前后视图、百慕大三角形等。三角形非常重要，因为它有支撑底座，可用于建造底座和桁架。
定义
三角形是由三条线相交形成的三边封闭多边形。在日常生活中经常遇到。它是几何的基本形状之一。它有三个边、三个角和三个顶点。我们每天都会遇到三种类型的三角形——等边三角形、等腰三角形和不等边三角形。下图显示了一个三角形 ABC，它有顶点和角 A、B 和 C。这个三角形的边是 AB、AC 和 BC。
三角形的性质：
- The sum of all the angles of a Triangle is 180°
- The sum of any two sides of a Triangle will always be greater than the third.
- The difference of any two sides will always be smaller than the third side. (this can be easily deduced by previous property).
- The side which is present opposite to the greatest angle is longest in all three sides.
- The side which is present opposite to the smallest angle is shortest in all three sides. (similar to previous property).
- According to the Exterior Angle property of the triangle, The exterior angle is equal to the sum of the opposite interior angles.
三角形的类型
三角形分为两类：一类是基于它们的边，另一类是基于它们的角度。三角形按边分为等边三角形、等腰三角形和不等边三角形。根据其角度，三角形分为锐角、钝角和直角三角形。
基于三角形的边
- 等边三角形
Triangles having all sides and all angles equal are known as equilateral triangle. Since, all the angles are equal, each angle is equal to 60° and the other name of equilateral triangle is Equiangular triangle.
- 等腰三角形
The triangles having two sides equal and the third one is not equal to the rest two. The angles opposite to the equal sides of the triangle are also equal.
- 不等边三角形
A Scalene triangle is the one which has none of its sides equal to each other and also, none of the angles are equal to each other, but the general properties of the triangle are applied to scalene triangle as well. Hence, the sum of all the interior angles are always equal to 180°
基于三角形的角度
- 锐角三角形
An Acute angled Triangle is the one where all the interior angles of the Triangle is less than 90°. For Instance, an Equilateral Triangle is an acute angled triangle (all angles are less than 90°).
An Acute angled triangle
- 直角三角形
A Right Angled Triangle is the one where one of the angles is always equal to 90°. Pythagoras Theorem is derived for Right angled triangles, Which states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of base and perpendicular.
A right-angled triangle
- 钝角三角形
An obtuse angled triangle has one of the sides more than 90°, In this case, since one of the three angles is more than 90°, the rest of the two angles is less than 90°.
An obtuse angled triangle
三角形的中位数
中值将一条边分成两个相等的部分。在三角形的上下文中，从一个顶点到另一边的中线将边分成两部分。每个三角形都有三个源自每个顶点的中线。下图表示从顶点 A 到边 BC 的中线。请注意，中线不需要垂直，但绝对需要平分另一侧。
三角形中线的性质：
1. 无论三角形的形状如何，任何三角形的三个中线总是在一个点相交。所有三个中线在三角形中相遇的点称为质心。
2. 三角形的中线将三角形分成面积相等的两个较小的部分。
3. 假设三角形的边长是“a”、“b”和“c”，中线的长度是l 1 、l 2和l 3 。
a 2 + b 2 + c 2 = l 2 1 + l 2 2 + l 2 3
三角形内角和
有四个边的多边形称为四边形，它们的角之和为 360°。我们知道三角形的角和是180°。这些属性是事实，适用于所有类型的四边形和三角形。这称为角度和属性。本节介绍了该属性的证明。
Consider the given triangle ABC, in any triangle the sum of all of its angles is 180°. In the given triangle,
∠A + ∠B + ∠C = 180°
Proof:
In the figure below, draw a line through A which is parallel to BC. Now, these angles are alternate angles which are equal. Thus, from the figure you can see,
∠A + ∠B + ∠C = 180°
示例问题
问题1：三角形ABC有∠A = 60°，∠B = 60°。求角∠C。
解决方案：
We know that sum of the angles in the triangle is 180°.
∠A + ∠B + ∠C = 180°
⇒ ∠A + ∠B + ∠C = 180°
⇒ 60 + 60 + ∠C = 180°
⇒ 120° + ∠C = 180°
⇒ ∠C = 60°
问题2：找出给定图中缺失的角度。
解决方案：
Let the two missing angles be x and y. Now we can see in the figure that opposite sides of the missing angles are equal in length. That means that the angles must be equal.
So, x = y.
From the angle sum property of the triangle.
x + y + 100° = 180°
⇒ 2x + 100° = 180°
⇒ 2x = 80°
⇒ x = 40°
问题3：从给定的数字中找出∠ACD的值。
解决方案：
We know that, ∠ACB and ∠BCD are supplementary angles.
From the angle sum property of the triangle,
∠A + ∠B + ∠C = 180°
⇒ 45° + 30° + ∠C = 180°
⇒ 75° + ∠C = 180°
⇒ ∠C = 105°
∠BCD = 180° – 105°
⇒ ∠BCD = 75°
问题 4：在给定的三角形 PQR 中，找到∠PSQ 的值。
解决方案：
Considering triangle QSR.
∠SRQ = ∠SQR
⇒ ∠SQR = 40°
Due to angle sum property of the triangle.
∠SQR + ∠SRQ + ∠QSR = 180°
⇒ 40° + 40° + ∠QSR = 180°
⇒ 80° + ∠QSR = 180°
⇒ ∠QSR = 100°
We know,
∠QSR + ∠QSP = 180°
⇒ ∠QSP + 100° = 180°
⇒ ∠QSP = 80°
问题 5：三角形 ABC 的角比例为 1:2:3。找出所有角度的值。
解决方案：
The interior angles are in the ratio 1:2:3. Thus, they also be written as,
x : 2x: 3x
Using the angle sum property of the triangle.
x + 2x + 3x = 180°
⇒ 6x = 180°
⇒ x = 30°
Thus, the angles in are,
30°, 60° and 90°.