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Transversal Lines and Angles

The lines and line segments are the most basic concepts in geometry. Recall that the difference between a line and a line segment is that a line has endless length, while a line segment has two endpoints. The lines and line segments may be of different kinds, such as - parallel lines, perpendicular lines, intersecting lines, concurrent lines, linear pair lines etc.

Parallel lines are those two or more lines which have same distance throughout. They do not meet one another even if extended in any direction. The lines (especially parallel ones) and transversals are closed related to each other. A line segment cutting across any two or more lines is called a transversal. It plays an extremely important role in geometry. Through this article, we shall throw light on transversal lines and angles made by them.

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Transversal and Pair of Lines

A transversal is defined as a line or line segment which cuts across two or more distinct lines or segments, generally all lines lie in the same plane. i.e. a transversal line intersects other lines at different points. In the image below, a transversal "l" is shown which intersects two lines at two different points.

It is to be noted that the area between two (or even more) lines is known as "interior", while the area outside, i.e. above and below the lines is called "exterior". Also, there are two sides of transversal - same sides (either left or right) and alternate sides (one left and another right).

Upon the intersection of a transversal with other lines, several angles are formed. Most of them are called by certain names so that one may recall by the name itself that the particular angle would be located in particular position. The relation between these angles may describe whether given lines are parallel or not. In the section below, we shall learn about the angles made by a transversal.

Angles of a Transversal

The angles formed by a transversal are seen in geometry quite often. When two parallel lines are intersected by a transversal, it looks as shown below :

The angles formed by a pair of parallel lines and a transversal are discussed below.

Let us represent the angles made at different corners by names (A, B, C...) for our convenience.

Corresponding Angles
The angles above and below both lines on the same side of the transversal are known as corresponding angles. The pairs of corresponding angles are $\angle$A and $\angle$E, $\angle$B and $\angle$F, $\angle$C and $\angle$G, $\angle$D and $\angle$H. When two lines l and m are parallel, these angles must be equal. i.e.

$\angle$A = $\angle$E

$\angle$B = $\angle$F

$\angle$C = $\angle$G

$\angle$D = $\angle$H
Alternate Angles
The angle above and below both lines on the alternate side of the transversal are said to be alternate angles. Here, the pairs $\angle$C and $\angle$F, $\angle$D and $\angle$E are alternate interior angles; while $\angle$A and $\angle$H, $\angle$B and $\angle$G are called alternate exterior angles. In case, l $\parallel$ m

$\angle$C = $\angle$F

$\angle$D = $\angle$E

$\angle$A = $\angle$H

$\angle$B = $\angle$G
Consecutive Interior Angles
The interior angles on the same side of the transversal are known as consecutive interior angles or co-interior angles. Here, these are $\angle$C and $\angle$E, $\angle$ D and $\angle$F. For two parallel lines l and m, the sum of consecutive interior angles is supplementary. i.e.

$\angle$C + $\angle$E = 180$^{\circ}$

$\angle$D + $\angle$F = 180$^{\circ}$

Examples

For better understanding, have a look at the following examples.

Example 1 : A transversal cuts two parallel lines as in the diagram below.

Find the value of z.

Solution : Angles shown in the image are consecutive interior angles. Since given lines are parallel, their sum will be supplementary.

$(4z + 16)^{\circ} + 64^{\circ} = 180^{\circ}$

4z + 16 + 64 = 180

4z + 80 = 180

4z = 180 - 80

4z = 100

z = $\frac{100}{4}$

z = 25$^{\circ}$
Example 2 : Look at the adjacent figure.

Given m $\parallel$ n. Find the value of t and hence calculate the measure of both angles.

Solution : In the above image, the given angles are alternate interior angles.
Since m $\parallel$ n, both angles will be equal in measure.

Hence,

2 t - 10 = 65 - t

2 t  + t = 65 + 10

3t = 75

t = $\frac{75}{3}$

t = 25$^{\circ}$

Measure of both angles are :

(2 t - 10)$^{\circ}$

= (2 $\times$ 25 - 10)$^{\circ}$

= (50 - 10)$^{\circ}$

= 40$^{\circ}$

= (65 - t)$^{\circ}$

= (65 - 25)$^{\circ}$

= 40$^{\circ}$
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