Show how three lines in a plane can intersect in no points, exactly one point, exactly two points, or exactly three points?

Draw it

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Enotes does not have the facilities for drawing figures. Therefore I will answer this question by verbal descriptions only.

We can draw three or more lines in a plane that do not intersect by making all these lines paralle to each other.

To draw three or more lines that intersect at one point, we can start by drawing any one line. Then we choose any one point on this line and draw another line passing through this point and making a positive angle with the original line. Then we draw a third line passing through this intersection point of the first two lines and making a positive angle with both the earlier lines.

To draw three lines that intersect at exactly two points, we start by drawing two parallel lines, and then drawing a third line that intersects both these lines. Alternatively we cam choose one point on each of the two parallel lines and draw a third line passing through these two points.

We can draw three lines that intersect at exactly three points by initially drawing two lines that intersect at one point, as has been done above for drawing three lines passing through only one point. Then to draw the third line we choose one point on each of the two lines, other than their point of intersection, and draw a third line passing through them.

Drawing a picture will be the easiest way to picture the four scenarios that you are describing. It will be easiest to visualize if you draw a separate diagram for each of the four scenarios.

In the first scenario you are asked to show how three lines in a plane can intersect in no points. If three lines within the same plane are parallel, they will not intersect. To draw this, you should first draw a plane, usually represented by a parallelogram. Within the plane, draw three parallel lines.

In the second scenario you are asked to show how three lines can intersect in one points. In this case, your lines will all cross in one location, similar to an asterisk.

In the third scenario you are asked to show how two lines cam intersect in two points. In this case, you will need two parallel lines and a third line that intersects both of the parallel lines.

In the fourth scenario you are asked to show how three lines can intersect in three points. In this case you need to show three lines forming a triangle with the ends of the lines extended beyond the three corners of the triangle.

I hope that these descriptions help you see the final results.

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