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Practical Geometry

Construct Parallel Lines Using Paper Folding

There are many methods to construct parallel lines on a paper. Let us start with the most basic method to draw parallel lines.

The steps we use to draw parallel lines are as follows.

First of all, we take a paper and fold it to make a straight line. Let this line be line1. Now let us take any point outside this line and mark it as P. This is the point from which we will draw the line parallel to line1. From this point, we draw a line which is perpendicular to line1. Let this line be line2. Now we draw a line that is perpendicular to line2 and that passes through point P. Let this line be line3. What do you see?

You can easily figure out that line1 and line3 are parallel to each other.

Now which property of parallel lines can we extract from this method of drawing parallel lines?

We can easily say, “If we draw two lines which are perpendicular to the same line, then the two lines will be parallel to each other.”

We can also say, “A line which is perpendicular to one of the two parallel lines will also be perpendicular to the second one”.

Suppose if someone asks us to draw a triangle. The first question that strikes us is that what are the lengths of the sides of the triangle which is to be drawn?

Therefore, if the three sides of a triangle are given to us, then can we draw the triangle?

If we try to draw the triangle only with the help of a ruler, then it is not possible to draw it. With the help of a ruler, we can draw two sides of the triangle very easily. However, when we try to draw the third side, it may or may not intersect the third side.

Let us assume that we are asked to draw a triangle and the sides of the triangle are 8 cm, 7 cm, and 4 cm. Firstly, we draw the two sides of the triangle, which are 8 cm and 7 cm and then the third side of length 4 cm as shown in the following figure. In these figures, we can see that a triangle is not formed.

Thus, we cannot draw a triangle only with the help of a ruler, but by using the ruler and compass.

Now, let us see the construction of a triangle using ruler and compass.

Before constructing a triangle, we should check whether the triangle is possible with the given sides or not.

 In a triangle, the sum of any two sides must be greater than the third side.

For example: Can we draw a triangle with sides of length 6 cm, 9 cm, and 2 cm?

Here, 6 cm + 2 cm = 8 cm < 9 cm

i.e., the sum of the lengths of two sides is less than the length of the third side.

Therefore, we cannot draw a triangle with sides of given lengths.

Let us solve some examples based on the construction of triangles.

Example 1:

Construct an isosceles triangle such that the two equal sides are of lengths 9 cm each and the unequal side is of …

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