Class 9 Maths Lines and Angles Parallel Lines & a Transversal

Parallel Lines & a Transversal

A line a line which intersects two or more lines at distinct points is called a transversal. Line l intersects lines m and n at points P and Q respectively. Therefore, line l is a transversal for lines m and n. Observe that four angles are formed at each of the points P and Q. ∠ 1, ∠ 2, ∠ 7 and ∠ 8 are called exterior angles, while ∠ 3, ∠ 4, ∠ 5 and ∠ 6 are called interior angles

Corresponding angles: (i) ∠ 1 and ∠ 5 (ii) ∠ 2 and ∠ 6 (iii) ∠ 4 and ∠ 8 (iv) ∠ 3 and ∠ 7

 Alternate interior angles: (i) ∠ 4 and ∠ 6 (ii) ∠ 3 and ∠ 5

 Alternate exterior angles: (i) ∠ 1 and ∠ 7 (ii) ∠ 2 and ∠ 8

 Interior angles on the same side of the transversal: (i) ∠ 4 and ∠ 5 (ii) ∠ 3 and ∠ 6

Interior angles on the same side of the transversal are also referred to as consecutive interior angles or allied angles or co-interior angles.

Now, let us find out the relation between the angles in these pairs when line m is parallel to line n.

Measure any pair of corresponding angles and find out the relation between them. You may find that: ∠ 1 = ∠ 5, ∠ 2 = ∠ 6, ∠ 4 = ∠ 8 and ∠ 3 = ∠ 7. From this, you may conclude the following axiom.

Axiom 3: If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. This Axiom is also referred to as the corresponding angles axiom

Let’s check if the converse is also true.

Converse Statement:  If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel.

It can be verified by observations/construction that the converse statement is true. Thus we have one more Axiom.

 

Axiom 4: If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

This Axiom is also referred to as the corresponding angles axiom.

 

Lets now use this the corresponding angles axiom, to prove relationship between the alternate interior angles, ∠ 4 and ∠ 5 , ∠ 3 and ∠ 6 when a transversal intersects two parallel lines.

Theorem 2: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Since this is a theorem, let’s try to prove it.

To prove: ∠ 4 = ∠ 5, given l is parallel to m.

Proof:

 ∠ 4 = ∠ 7             (Corresponding angle Axiom) – (i)

∠5 = ∠ 7               (Vertical Opposite angle Theorem) – (ii)

Using I & ii , we get  ∠ 4 = ∠ 5

 

Converse of theorem 2 is also true. It is stated in Theorem 3.

 

Theorem3: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

Since this is a theorem, let’s try to prove it.

To prove: n is parallel to m, given ∠ 4 = ∠ 5

Proof:

 ∠ 4 = ∠ 5             (Given) – (i)

∠5 = ∠ 7               (Vertical Opposite angle Theorem) – (ii)

Using I & ii , we get  ∠ 4 = ∠ 7, but these are corresponding angles. So as per Corresponding angle axion, line n is parallel to line m.

 

Similarly we have theorems for interior angle on same side of Transversal.

 

Theorem 4: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

 

Theorem 5: If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

 

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