基本结构–角平分线，垂直平分线，角度为60°

大多数时候，我们使用图表来描述数学中的形状和场景。但是它们并不精确，它们只是没有适当测量的实际形状的表示。但是，当我们在建造诸如木桌之类的东西时，或要建造建筑物的地图时。它的测量需要精确。对于这种情况，我们需要学习做一些基本的构造。让我们详细研究它们。

施工概论

要绘制多边形，角度或圆形，需要一些基本的工具，这些工具在我们的几何框中提供。

几何框应包含以下仪器：

1.标尺或刻度尺：

It should have centimetres and millimetres marked off on one side and the other side should have inches and their parts marked off on it. 

2.一对三角尺：

One should have 60°, 90° and 30° as its angles and the other one should have 90°, 45° and 45°.

3.分频器

Dividers have a jointed pair of legs, each leg has a sharp point. They can be used for scribing a cricles and taking off and transferring the dimensions. 

4.指南针

Compass is used for inscribing circles or arcs. As similar to dividers, they can also be used for taking off and transferring the dimensions. 

5.Protractor

This instruments helps us in measuring the angles. It has degrees marked from 0 to 180 and from 180 to 0. It has readings from both left to right and right to left. Both the readings supplement each other. 

为了学习基本结构，我们将主要使用其中的三个。刻度尺，指南针和Protractor。让我们看一些基本的构造。

基本构造

我们知道平分线是将任何东西分成相等的两部分的线，无论是角度还是线段。让我们看看如何使用几何框中的仪器制作任意给定角度的平分线。

结构1：角平分线

假设我们给定一个角度PQR，目标是构造给定角度的等分线。

Steps for Construction: 

Step 1.  Take Q as centre and draw an arc that intersects the rays QP and QR with any radius. Let’s name the intersection points T and S. 

Step 2. Now take T and S as the Centre and with the radius that is more than half of the length of TS. Draw arcs such that they intersect, let’s call that intersection U.

Step 3. Now join QU, this is our required bisector. 

构造2：给定线的垂直平分线。

垂直平分线是将给定线平分并垂直于该线的线。给定一条直线PQ，目标是构造一个垂直平分线。

Steps for Construction: 

Step 1. Let’s take P and Q as the centre and take any radius that more than half the length of PQ. Now draw arcs on both sides of the line PQ and let them intersect at A and B respectively.

Step 2. Now Join AB and let it intersect at M on PQ. This is our required perpendicular bisector. 

结构3：角度为60°

The goal in this construction is to construct an angle of 60° from a given ray PQ.

Steps of Construction: 

Step 1. With P as the centre and some arbitrary radius, construct a circular arc that intersects PQ at S.

Step 2. Now take S as the centre and the same radius as above. Draw an arc on the arc already drawn. Let’s say the point of intersection of both arcs is T. 

Step 3. Draw a ray PR passing through T. This will give us the required angle of 60°. 

让我们看一些关于这些概念的例子。

样本问题

问题1：在下图中，∠PQR分为许多部分。

确定∠PQR的平分线。

解决方案：

We know that bisector of an angle divides it into two equal parts. 

Notice that, 

 ∠PQT = ∠SQT + ∠PQS 

 ∠PQT = 30° + 20°

∠PQT = 50°

and, 

 ∠TQR = ∠TQU + ∠UQR

∠TQR = 35° + 15°

∠TQR = 50°

This the ray QT is the bisector of the angle PQR. 

问题2：对以下陈述陈述是对还是错。

1. 垂直平分线将线段一分为二。
2. 角平分线将角分为两个相等的部分。

解决方案：

Statement 1: False 

According to the definition of perpendicular bisector, it is a line that divides a line segment into two equal parts and is also perpendicular to it. 

Statement 2: True 

An angle bisector divides an angle into two equal parts. 

问题3：使用上述构造技术构造30°的角度。

解决方案：

Steps of Construction: 

Step 1. Using the construction technique mentioned above for creating 60° angle. Let’s call that angle ∠CAB.

Step 2. Now, since the angle is 60° and out goal is to construct an angle of 30°. We need to bisect this angle using the technique mentioned above.

Now the ray AE gives us the angle of 30°.

问题4：构造一个135°的角度。

解决方案：

An angle of 135° can be made using one 90° and one 45°. 

Let’s make a straight line PQ with a point R in between. 

Now we have made two 90° angles. We need to bisect on the of angles to make a 45°. 

Thus, ∠PRE gives us the required angle 135°.