Parallel lines

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.

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:

• Angle notation
• Sides, vertices, and angles
• Bisector
• Right angle
• Acute angle
• Obtuse angle
• Plane angles
• Perpendicular lines

On the Tutorela blog you will find a variety of interesting articles about mathematics.

When two straight lines intersect, four angles are formed. In the following image two straight lines $c$ and $d$ intersect, resulting in angles $1, 2, 3, 4$.

• Opposite angles are two angles directly opposite eachother across the vertex (where the two lines intersect).
• The intersection of two straight lines results in two pairs of opposite angles.
• Opposite angles are non-adjacent, meaning that they can not be two angles that are next to eachother.
• Opposite angles are equal.

In the following figure:

• 1 and 3 are opposite angles.
• 2 and 4 are opposite angles.

We can therefore affirm that:

• $\sphericalangle1=\sphericalangle3$
• $\sphericalangle2=\sphericalangle4$

• 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:

• $\sphericalangle1+\sphericalangle2=180°$
• $\sphericalangle2+\sphericalangle3=180°$
• $\sphericalangle3+\sphericalangle4=180°$
• $\sphericalangle4+\sphericalangle1=180°$

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.

• Two angles are alternate angles if they are on opposite sides of the transversal line.
• Two alternate angles can either be both external angles or both internal angles.
• Two alternate angles do not share any of their sides.

In Figure 3:

• Angles 4 and 6 are internal alternates.
• Angles 3 and 5 are internal alternates.
• Angles 1 and 7 are external alternates.
• Angles 2 and 8 are external alternates.

In the following image we can see two pairs of internal alternate angles, one highlighted in red and the other in blue.

We can state that:

If two parallel lines are cut by a transversal, then the pairs of internal alternate angles are equal.

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are equal.

Which means that in Figure 3:

• $\sphericalangle4=\sphericalangle6$
• $\sphericalangle3=\sphericalangle5$
• $\sphericalangle1=\sphericalangle7$
• $\sphericalangle2=\sphericalangle9$

• Conjugate angles are on the same side of the transversal line.
• Conjugate angles can be both external or both internal.

In Figure 3:

• Angles 3 and 6 are internal conjugates.
• Angles 4 and 5 are internal conjugates.
• Angles 2 and 7 are external conjugates.
• Angles 1 and 8 are external conjugates.

We can state that:

If two parallel lines are cut by a transversal then the pairs of internal conjugate angles are supplementary, i.e. their sum equals 180 degrees.

If two parallel lines are cut by a transversal, then the pairs of external conjugate angles are supplementary.

Which means that in Fig:

• $\sphericalangle3+\sphericalangle6=180°$
• $\sphericalangle4+\sphericalangle5=180°$
• $\sphericalangle2+\sphericalangle7=180°$
• $\sphericalangle1+\sphericalangle9=180°$

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:

• $\sphericalangle1=\sphericalangle5$
• $\sphericalangle2=\sphericalangle6$
• $\sphericalangle3=\sphericalangle7$
• $\sphericalangle4=\sphericalangle8$

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 $ß=40º$

In the following image $a||b$

Question:

What are the values of $α$ and $ß$?

$A\Vert B$

Observe the plane and solve:

• $ß=?$
• $α=?$

Solution:

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 replacing $ß$ with its value we get the following:

$α+130º=180º$

Subtracting it results in

$α = 50º$

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.

How many degrees do we have to add to angle $β$ so that there will be another parallel line in the following graph?

Explanation

By adding $4°$ degrees to angle $∡β$we will get an angle of $90°$ degrees, and by doing so we will create another line parallel to the two below it.

$86°+4°=90°$

Solution:

The correct answer is: 4°

This question is divided into several parts:

1. How many degrees is angle $∡ABC$ and what kind of angle is it in relation to $∡CBF$?
2. How many degrees is angle $∡BDE$ and what kind of angle is it in relation to $∡ADC$?

Answer 1:

A. Angle $∡ABC$ is equal to $180º-130º=50º$

B. Angle $∡ABC$ is adjacent to angle $∡CBF$ .

Answer 2:

1. Angle $∡BDE$ is equal to $90º$ because it is the opposite angle of angle $∡ADC=90º$