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Two lines that are stretched into infinity and still never intersect are called coplanar lines and are said to be parallel lines. The symbol for "parallel to" is //.
If we have two lines (they don't have to be parallel) and have a third line that crosses them as in the figure below - the crossing line is called a transversal:
In the following figure:
If we draw to parallel lines and then draw a line transversal through them we will get eight different angles.
The eight angles will together form four pairs of corresponding angles. Angles F and B in the figure above constitutes one of the pairs. Corresponding angles are congruent if the two lines are parallel. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs.
Angles that are in the area between the parallel lines like angle H and C above are called interior angles whereas the angles that are on the outside of the two parallel lines like D and G are called exterior angles.
Angles that are on the opposite sides of the transversal are called alternate angles e.g. H and B.
Angles that share the same vertex and have a common ray, like angles G and F or C and B in the figure above are called adjacent angles. As in this case where the adjacent angles are formed by two lines intersecting we will get two pairs of adjacent angles (G + F and H + E) that are both supplementary.
Two angles that are opposite each other as D and B in the figure above are called vertical angles. Vertical angles are always congruent.
$$\angle A\; \angle F\; \angle G\; \angle D\;are\; exterior\; angles\\ \angle B\; \angle E\; \angle H\; \angle C\;are\; interior\; angles\\ \angle B\;and\; \angle E,\; \angle H\;and\; \angle C\;are\; consecutive\; interior\; angles\\ \angle A\;and\; \angle G,\; \angle F\;and\; \angle D\;are\; alternate\; exterior\; angles\\ \angle E\;and\; \angle C,\; \angle H\;and\; \angle B\;are\; alternate\;interior\; angles\\ \left.\begin{matrix} \angle A\;and\; \angle E,\; \angle C\;and\; \angle G\\ \angle D\;and\; \angle H,\; \angle F\;and\; \angle B\\ \end{matrix}\right\} \;are\; corresponding\; angles$$
Two lines are perpendicular if they intersect in a right angle. The axes of a coordinate plane is an example of two perpendicular lines.
In algebra 2 we have learnt how to find the slope of a line. Two parallel lines have always the same slope and two lines are perpendicular if the product of their slope is -1.
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Find the value of x in the following figure
A transversal line, in geometry, passes through two lines in the same plane at two distinct points. Transversals play a role in establishing the parallelism of two or more other straight lines in the Euclidean plane. It intersects two lines at distinct points. Intersection caused by transversal forms several angles. These are corresponding angles, alternate interior angles, alternate exterior angles, and co-interior angles.
Transversals and Transversal Lines
In geometry, a transversal is any line that intersects two straight lines at distinct points. In the following figure, L is the transversal line that cuts \(L_{1}\) and \(L_{2}\) lines at two distinct points. In the universe of parallel and transverse lines, a transversal line connects the two parallel lines.
From the diagram, we can say that 'L is a transversal, cutting the lines \(L_{1}\) and \(L_{2}\)', and thus line L is the transversal line. Here, there is no relationship between the angles formed as the lines are not parallel. Let us now see how to construct a transversal on parallel lines and what are the properties of transversal angles.
Constructing a Transversal on Parallel Lines
The construction of a transversal is easy. First, we create two parallel lines. We construct the angle (at which we want to create the transversal) on the first line (say x) as shown.
Further, we extend this constructed angle up to a point it covers both the parallel lines as done here. Here is what we get: A transversal on the two parallel lines at the desired angle (x).
Transversal Angles
When a transversal cuts two parallel lines, several angles are formed by these two intersections. Those are called transversal angles. Those types of angles on a transversal are given below:
- Corresponding angles
- Alternate Interior Angles
- Alternate Exterior Angles
- Co-interior Angles
From the given diagram, if we try to organize the angles based on the relative positions they occupy, we get the following categories of angles :
Corresponding Angles
The following pairs of angles are corresponding angles:
- ∠1 and ∠5
- ∠2 and ∠6
- ∠3 and ∠7
- ∠4 and ∠8
Alternate Interior Angles
The following pairs of angles are alternate interior angles:
Alternate Exterior Angles
The following pairs of angles are alternate exterior angles:
Co-interior Angles
The following pairs of angles are co-interior angles:
Important Notes:
- A transversal is any line that intersects two straight lines at distinct points.
- A transversal that cuts the lines L1 and L2 is the transversal line.
- Intersection caused by transversals forms several angles. These are : a) Corresponding Angles b) Alternate Interior Angles c) Alternate Exterior Angles
d) Co-interior Angles
Related Articles on Transversal
Check these interesting articles on transversal. Click to know more!
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Example 1: Consider the following figure, in which M and N are parallel lines. What is the value of ∠C?
Solution: Through C, draw a line segment parallel to M and N, as shown below:
Now, AC and BC act as transversal interesting two parallel lines.
We have ∠y = ∠β = 60º (alternate interior angles) and ∠x = 180º – 120º (co-interior angles) ⇒ ∠x = 60º. Thus, we get ∠C = ∠x + ∠y = 120º. ∴ ∠C = 120º. -
Example 2: Consider the following figure, in which L, M, and N are parallel lines. The line AC is perpendicular to N, as shown. What are the values of ∠x, ∠y, and ∠z?
Solution: Since AC ⊥ N, it will be perpendicular to each of the other two parallels as well. AC is the transversal line to all the three parallel lines L, M, and N. This means that ∠x = 90º – 60º = 30º (as ∠LAC and ∠ACN are corresponding angles which are equal) AN is also a transversal intersecting the three given parallel lines. Now, ∠y = 180º – ∠x (co-interior angles)
⇒ ∠y = 150º
Also, ∠z = ∠y = 150º (corresponding angles)
∴ The values of x, y, and z are 30°, 150°, and 150° respectively.
Example 3: Consider the following figure, in which M || N. The angle bisectors of ∠BAX and ∠ABY intersect at C, as shown. Find the value of ∠ACB.
Solution: In the given figure, L is the transversal on two parallel lines M and N. We note that ∠BAX and ∠ABY are co-interior angles, which means that their sum is 180°. ⇒ ∠BAX + ∠ABY = 180º ⇒ 2(∠1 + ∠2) = 180º
⇒ ∠1 + ∠2 = 90º
Now, in ΔACB, we apply the angle sum property: ∠1 + ∠2 + ∠ACB = 180º ⇒ 90º + ∠ACB = 180º ⇒ ∠ACB = 90º
Therefore, the value of ∠ACB is 90 degrees.
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FAQs on Transversal
Yes, transversals are straight lines that intersect two or more lines at different points. A transversal line meets the other line at one point which forms four angles around the point of intersection.
How Many Angles are Formed by the Transversal?
The transversal forms several types of angles. Some of those angles are:
- Corresponding Angles
- Alternate Interior Angles
- Alternate Exterior Angles
- Co-interior Angles
What does Transversal Mean in Geometry?
In geometry, a transversal is a line, ray, or line segment that intersects other lines, rays, or line segments on a plane at different intersecting points. When it intersects parallel lines, there formed several angles that share a common property, on the other hand when a transversal intersects two or more non-parallel lines, there is no relationship between the angles formed.
How Many Angles are Formed by the Transversal?
When two parallel lines are intersected by the transversal, eight angles are formed. The eight angles include corresponding angles, alternate interior and exterior angles, vertically opposite angles, and co-interior angles.
Are Vertical Angles Formed by a Transversal Equal?
Vertical angles are always equal to each other in measure. When a transversal intersects a line, two pairs of vertical angles are formed.
Do Transversal Lines have to be Parallel?
No. Transversal lines can be at any angle to the given parallel lines. Thus, any 2 transversals need not be parallel.
What are Corresponding Angles in a Transversal?
Two or more angles that are on the same side of the transversal when it cuts two or more parallel lines are called corresponding angles. When a transversal intersects two parallel lines, four pairs of corresponding angles are formed.