Parallel lines play a fundamental role in geometry, engineering and many other important fields. Learning to work with parallel lines will allow you to solve many different types of geometry problems at various levels of difficulty.
Properties of parallel lines
We can state the following about parallel lines:
Parallel lines are always coplanar.
The distance between two parallel lines is constant (never changes), meaning that they will never intersect.
We can also find parallel lines in quadrilaterals that have sides, like the following:
In parallelograms, rectangles, squares and rhombuses there are two pairs of parallel sides.
In trapezoids there is only one pair of parallel sides.
If you are interested in learning more about angles, try visiting one of the following articles:
Adjacent angles are two angles formed by the intersection of two lines (or rays).
Adjacent angles share a side.
Two adjacent angles are supplementary, i.e., the sum of their values is equal to 180º.
In the following figure :
1 and 2 are adjacent angles
2 and 3 are adjacent angles
3 and 4 are adjacent angles
4 and 1 are adjacent angles
We can therefore state that:
∢1+∢2=180°
∢2+∢3=180°
∢3+∢4=180°
∢4+∢1=180°
Angles formed by a transversal
A line that intersects two parallel lines at different points is called a transversal. When a transversal intersects two parallel lines, eight angles are formed, four at each point of intersection. In the following picture, two parallel lines l and m are intersected by transversal line s. Eight angles 1, 2, 3, 4, 5, 6, 7 and 8 are formed.
Figure 3 :
Classification of angles
Depending on their position, the angles formed can either be:
Internal angles: These are the angles that are in between the two parallel lines.
In Figure 3 angles 3, 4, 5 and 6 are internal angles.
OR
External angles: These are the angles that are not in between the parallel lines.
In Figure 3 angles 1, 2, 7 and 8 are external angles.
Two angles formed by a transversal intersecting two parallel lines can be alternate angles, conjugate angles or corresponding angles, depending on which parts of the transversal forms those angles.
In the following image the angles α y ß are corresponding angles
Two corresponding angles are on the same side of the transversal line.
One of the corresponding angles will be an external angle while the other will be an internal angle.
Two corresponding angles do not share any of their sides.
In Figure:
Angles 1 and 5 are corresponding
Angles 2 and 6 are corresponding
Angles 3 and 7 are corresponding
Angles 4 and 8 are corresponding
We can state that:
If two parallel straight lines are cut by a transversal, then the corresponding angles are equal.
Which means that in figure 3:
∢1=∢5
∢2=∢6
∢3=∢7
∢4=∢8
Parallel lines practice problems
Exercise 1: parallel lines
In the following image, be a∣∣b
Question:
What is the value of ß?
Solution:
We can see that the angles α y ß are corresponding angles. We know that when two parallel lines like a and b are cut by a transversal like c, the corresponding angles are equal and, therefore ß=40º
Here we have two parallel lines cut by a transversal. Since we know that angle ß and the angle marked 130º are corresponding angles, then we know that these angles are equal and therefore. ß=130º.
Now we have to find the value for angle ∡α . Since the angles ∡α and ∡ß are adjacent, then we know that they are supplementary, which means that they add up to 180º. Therefore,
α+ß=180º
By replacingß with its value we get the following:
α+130º=180º
Subtracting it results in
α=50º
Exercise 3: parallel lines
How many parallel lines are there in the following graph?
Explanation
In the graph you can see:
that the straight line f intersects the straight lines b and c (in dashed lines) at two points
that at both points of intersection the angle of intersection is the same (90°)
that these two angles are corresponding
Therefore the straight lines b and c are parallel.
In the following graph you can see
that the line b intersects the lines d and e (in dashed lines) in two points
that at both points of intersection the angle of intersection is the same (130°)
that these two angles are external alternate angles
Therefore, it can be said that the straight lines d and e are parallel.
Solution:
Therefore, the final answer is that the graph has 2 pairs of parallel lines.