What is transversal?
The transversal is a straight line that passes across any set of straight lines(at least two) in the same plane, intersecting each line at one distinct point.While doing so, it forms various angles between the lines as shown.It forms corresponding angles, vertical angles, adjacent angles and alternate angles.
In the above figure, AB and CD are any two lines in a plane and EF is a transversal.
It cuts the lines forming 8 angles.
Angles lying side by side are adjacent angles.They are not necessarily equal.
1 and 2 ,1 and 4, 2 and 3, 3 and 4 are all adjacent angles.
Angles lying vertically opposite are vertical angles.They are always equal.
1 and 3, 2 and 4, 5 and 7, and 6 and 8 are vertical angles.
1 and 5, 4 and 8, 2 and 6, 3 and 7 are corresponding angles.
1 and 7, 2 and 8 are alternate exterior angles.
4 and 6, 3 and 5 are alternate interior angles.
Only, when the lines AB and CD are parallel, each pair of corresponding angles ,alternate interior angles, alternate exterior angles, become equal.
Note that interior angles on the same side of the transversal, become supplementary. i.e. sum is 180. i.e angles 4 and 5, and 3 and 6.
The converse is also true that whenever corresponding or alternate angles are equal, or when the sum of interior angles on the same side of the transversal is 180, then the lines are parallel.
Sample Math Problems
Question 1: Solve for x in the following figure if AB||CD and EF transversal
Answer:
Sum of interior angles on same side of transversal is 180
2x -15 + x +27 = 180
3x = 180 -12 = 168
x = 56
Question 2: In the figure below - find all the angles:
Answer:
BGH = 2x - 15 = 112 - 15 = 97
GHD = x +27 = 56 + 27 = 83
AGE = BGH = 97 ( vertical angles)
EGB = 83 ( EGB and BGH form a linear pair whose sum is 180)
AGH = EGB = 83 (vertical angles)
CHG = BGH = 97 (alternate angles)
CHF = GHD = 83 (vertical angles)
DHF = BGH = 97 (corresponding angles)
Question 3: In the figure below, what is the sum of AGH and CHG? why?
Answer: Sum of angles AGH and CHG is 180 because interior angles on the same side of the transversal are supplementary.
Question 4:
In the above figure can AB be transversal of EF? Check if angle AGE= DHF
Answer:
No. Because a transversal has to cut at least 2 lines.
Angle AGE = Angle DHF (alternate exterior angles). The value is also found to be 97 degree (Q2)
