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Properties of Transversal Lines

If a line which crosses a two or more other lines on the same plane then that line is called as transversal line.

If a traversal t that intersects three lines l, m and n in three distinct points A, B and C, then the line segment of AB and BC are called the intercepts made by a lines l, m and n on the transversal t.

Properties of Transversal Lines

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Properties of a Parallel Lines Cut by a Transversal

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When a transversal line cut the parallel lines, eight angles are formed as shown in the figure given below:

Corresponding angles
Parallel Lines Cut by a Transversal

Alternate angles - interior and exterior

Adjacent angles

Here,

$\angle$1 = $\angle$3 =$\angle$5=$\angle$7

$\angle$2=$\angle$4= $\angle$6= $\angle$8.

Adjacent angles:

When parallel lines are cut by a transversal line, then the adjacent angles sum up to $180^{0}$ degree

$\angle$1 + $\angle$2 = $180^{0}$

$\angle$3 + $\angle$4 = $180^{0}$

$\angle$5 + $\angle$6 = $180^{0}$

$\angle$7 + $\angle$8 = $180^{0}$

Here, the adjacent angles always adds up to $180^{0}$

Corresponding Angles:

The angle in the same position at the intersection of transversal and parallel lines is said to be corresponding angle.

When a transversal line cuts parallel lines, then corresponding angles are equal

$\angle$1 = $\angle$6

$\angle$2 = $\angle$5

$\angle$3 = $\angle$8

$\angle$4 = $\angle$7

Alternate angles:

Alternate angles are equal

$\angle$3 = $\angle$5

$\angle$4 = $\angle$6

$\angle$2 = $\angle$8

$\angle$1 = $\angle$7

Examples on Transversal Lines Properties

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Below are some examples on transversal lines properties

Example 1: Find the angle 2 of nature parallel lines, when the $\angle$1 = 110?
Solution:

Here $\angle$1 and $\angle$2 are adjacent to each other. When parallel lines is cut by a transversal line, the adjacent angles sum up to $180^{0}$

So $\angle$1+ $\angle$2 = $180^{0}$

$\implies$ 110 + $\angle$2 = $180^{0}$

$\implies$ $\angle$2 = $180^{0} - 110^{0}$ = $70^{0}$.

Example 2: Find the $\angle$4 of the parallel lines cut by a transversal line, when the $\angle$3 = 105?

Solution:

Here $\angle$3 and $\angle$4 are adjacent to each other.

$\angle$1= $\angle$3

$\angle$3 = $105^{0}$

When parallel lines are cut by a transversal line, the adjacent angles sum up to $180^{0}$

So, $\angle$3+ $\angle$4 = $180^{0}$

$105^{0} + \angle4$= $180^{0}$

$\angle$4 = $180^{0} - 105^{0}$ = $75^{0}$..

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