Transversal

Transversal

In geometry, a transversal is a line which cuts two or more lines at distinct points. In the figure given below, lines AB and CD are cut by the line EF at the points G and H respectively. So, EF is a transversal.

Types of angles formed by a transversal

When two line segments are cut by a transversal then there will form 8 angles. In the figure given above, transversal EF has cut the line segments AB and CD at points G and H respectively. The angles formed there are ∠AGE, ∠BGE, ∠AGH, ∠BGH, ∠GHC, ∠GHD, ∠CHF and ∠DHF.

Interior angles: The angles formed inside the two line segments are called interior angles. In the given figure, angles ∠AGH, ∠BGH, ∠GHC and ∠GHD are interior angles.

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Exterior angles: The angles formed outside the two line segments are called exterior angles. In the given figure, angles ∠AGE, ∠BGE, ∠CHF and ∠DHF are exterior angles.

Alternate angles

A pair of non adjacent interior angles on the opposite sides of a transversal is called alternate angles. 

In the figure the alternate angles are:

i.      ∠AGH and ∠GHD

ii.    ∠BGH and ∠GHC

Note: When the two lines cut by transversal are parallel, then alternate angles are equal.

Corresponding angles

In a pair of non-adjacent angles on the same side of a transversal, one is external and other internal are called corresponding angles. 

In the figure the corresponding angles are:

i.      ∠AGE and ∠GHC

ii.    ∠AGH and ∠CHF

iii.   ∠BGE and ∠GHD

iv.   ∠BGH and ∠DHF

Note: When the two lines cut by transversal are parallel, then corresponding angles are equal.

Co-interior angles

A pair of two interior angles on the same side of a transversal are called co-interior angles. 

In the figure the co-interior angles are:

i.      ∠AGH and ∠GHC

ii.     ∠BGH and ∠GHD

Note: When the two lines cut by transversal are parallel, then co-interior angles are supplementary i.e. their sum is 180°.

Workout Examples

Example 1: Find the values of a, b, c and d from the given figure.

Solution: Here,

                         5a + 30° + 3a – 10° = 180° ----------------> sum of co-interior angles

                or,     8a + 20° = 180°

                or,     8a = 180° - 20°

                or,     a = 160°/8

                or,     a = 20°

                         b = 5a + 30° --------------------> Corresponding angles

                            = 5×20° + 30°

                            = 100° + 30°

                            = 130°

                         c = 3a – 10° --------------------> Alternate angles

                            = 3×20° – 10° 

                            = 60° – 10°

                            = 50°

                         d = c ----------------------> Vertically opposite angles

                            = 50°

                ∴  a = 20°, b = 130°, c = 50° and d = 50°

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