### Mathematics Class VII

Integers
Fractions and Decimals
Exponents and Powers
Algebraic Equations
Simple Linear Equations
Lines and Angles
Comparing Quantities
Congruence of Triangles
Rational Numbers
Perimeter and Area
Data Handling
Practical Geometry
Symmetry
Visualising Solid Shapes

# 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°

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