Is it true? Through a point not on a line, one and only one line can be drawn parallel to the given line.
Solution:
Answer is true.
The given statement is called the Parallel postulate, which is one of the five postulates, or axioms of Euclidean geometry.
The following are the Euclidean postulate:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
This is depicted in the figure (a) below where the sum of angle ⍺ and angle ꞵ is less than 180°, so line 1 and line 2 will intersect on one side when extended.
As shown in figure (b) sum of angles ⍺ and ꞵ is equal to 180° then lines 1 and 2 are parallel.
The Parallel postulate states that, “if a point and a line are given, the point being not on the line, then a number of lines can be drawn passing through that line but one and only one line can be drawn parallel to the given line”.
From the diagram above, A is the given line and P a point on which four lines are drawn namely, l1, l2, l3 and l4 we observe that there is only one line l₄ which is passing through the given point and parallel to the given line.
Is it true? Through a point not on a line, one and only one line can be drawn parallel to the given line.
Summary:
It is true that, through a point not on a line, one and only one line can be drawn parallel to the given line.