Transversal of Parallel Lines
The two lines which never meet with each other are called Parallel Lines. If we have a transversal on two parallel lines then-
a. All the pairs of corresponding angles are equal.
∠3 = ∠7
∠4 = ∠8
∠1 = ∠5
∠2 = ∠6
b. All the pairs of alternate interior angles are equal.
∠3 = ∠6
∠4 = ∠5
c. The two Interior angles which are on the same side of the transversal will always be supplementary.
∠3 + ∠5 = 180°
∠4 + ∠6 = 180°
Checking for Parallel Lines
This is the inverse of the above properties of the transversal of parallel lines.
- If a transversal passes through two lines so that the pairs of corresponding angles are equal, then these two lines must be parallel.
- If a transversal passes through two lines in so that the pairs of alternate interior angles are equal, then these two lines must be parallel.
- If a transversal passes through two lines so that the pairs of interior angles on the same side of the transversal are supplementary, then these two lines must be parallel.
Example: 1
If AB ∥ PQ, Find ∠W.
Solution:
We have to draw a line CD parallel to AB and PQ passing through ∠W.
∠QPW = ∠PWC = 50° (Alternate Interior Angles)
∠BAW =∠CWA = 46°(Alternate Interior Angles)
∠PWA = ∠PWC +∠CWA
= 50°+ 46°= 96°
Example: 2
If XY ∥ QR with ∠4 = 50° and ∠5 = 45°, then find all the three angles of the ∆PQR.
Solution:
Given: XY ∥ QR
∠4 = 50° and ∠5 = 45°
To find: ∠1, ∠2 and ∠3
Calculation: ∠1 + ∠4 + ∠5 = 180° (sum of angles making a straight angle)
∠1 = 180°- 50°- 45°
∠1 = 85°
PQ is the transversal of XY and QR, so
∠4 = ∠2 (Alternate interior angles between parallel lines)
∠2 = 50°
PR is also the transversal of XY and QR, so
∠5 = ∠3 (Alternate interior angles between parallel lines)
∠3 = 45°