Maharashtra Board Class 9 Maths Part II Chapter 2 Parallel Lines PDF Download

Parallel Lines

Let's Study

Properties of angles formed by parallel lines and its transversal

Tests of parallelness of two lines

Use of properties of parallel lines

Let's Recall

Parallel lines : The lines which are coplanar and do not intersect each other are called parallel lines.

Hold a stick across the horizontal parallel bars of a window as shown in the figure. How many angles are formed ?

Do you recall the pairs of angles formed by two lines and their transversal ?

In figure 2.1, line n is a transversal of line l and line m.

Here, in all 8 angles are formed. Pairs of angles formed out of these angles are as follows :

Pairs of corresponding angles

(i) \(\angle d, \angle h\)

(ii) \(\angle a, \angle e\)

(iii) \(\angle c, \angle g\)

(iv) \(\angle b, \angle f\)

Pairs of alternate interior angles

(i) \(\angle c, \angle e\) (ii) \(\angle b, \angle h\)

Pairs of alternate exterior angles

(i) \(\angle d, \angle f\)

(ii) \(\angle a, \angle g\)

Pairs of interior angles on the same side of the transversal

(i) \(\angle c, \angle h\)

(ii) \(\angle b, \angle e\)

Some important properties :

(1) When two lines intersect, the pairs of opposite angles formed are congruent.

(2) The angles in a linear pair are supplementary.

(3) When one pair of corresponding angles is congruent, then all the remaining pairs of corresponding angles are congruent.

(4) When one pair of alternate angles is congruent, then all the remaining pairs of alternate angles are congruent.

(5) When one pair of interior angles on one side of the transversal is supplementary, then the other pair of interior angles is also supplementary.

Teacher's Note

Parallel lines are like railway tracks. They never meet each other. When a train track crosses both lines, it is like a transversal.

Exam Trick

Remember: Corresponding angles are in the same position on each parallel line. Like twin brothers standing in the same way - they look the same!

Points to Remember

Parallel lines never meet or intersect each other.
A transversal is a line that crosses two other lines.
Eight angles are formed when a transversal crosses two parallel lines.
Corresponding angles are equal when lines are parallel.
Alternate angles are also equal when lines are parallel.

Let's Learn

Properties of Parallel Lines

Activity

To verify the properties of angles formed by a transversal of two parallel lines.

Take a piece of thick coloured paper. Draw a pair of parallel lines and a transversal on it. Paste straight sticks on the lines. Eight angles will be formed. Cut pieces of coloured paper as shown in the figure, which will just fit at the corners of \(\angle 1\) and \(\angle 2\). Place the pieces near different pairs of corresponding angles, alternate angles and interior angles and verify their properties.

Teacher's Note

You can do this activity in class with coloured paper and sticks. It helps students understand angles better by doing, not just reading.

Exam Trick

The activity proves properties are true. In exams, remember: If angles fit perfectly in one place, they must be equal!

Points to Remember

Activities help us see and touch mathematical ideas.
Coloured paper pieces show which angles are equal.
Corresponding angles match in size and position.
Alternate angles are on opposite sides of the transversal.
Doing activities makes learning easier and fun.

Let's Learn

We have verified the properties of angles formed by a transversal of two parallel lines. Let us now prove the properties using Euclid's famous fifth postulate given below.

If sum of two interior angles formed on one side of a transversal of two lines is less than two right angles then the lines produced in that direction intersect each other.

Interior Angle Theorem

Theorem : If two parallel lines are intersected by a transversal, the interior angles on either side of the transversal are supplementary.

Given : line l ∥ line m and line n is their transversal. Hence as shown in the figure, a, b are interior angles formed on one side and c, d are interior angles formed on other side of the transversal.

To Prove : \(a + b = 180°\)

\(d + c = 180°\)

Proof : Three possibilities arise regarding the sum of measures of a and b.

(i) \(a + b < 180°\) (ii) \(a + b > 180°\) (iii) \(a + b = 180°\)

Let us assume that the possibility (i) \(a + b < 180°\) is true.

Then according to Euclid's postulate, if the line l and line m are produced will intersect each other on the side of the transversal where a and b are formed.

But line l and line m are parallel lines ..........given

\(\therefore a + b < 180°\) impossible ..........(I)

Now let us suppose that \(a + b > 180°\) is true.

\(\therefore a + b >180°\)

But \(a + d = 180°\)

and \(c + b = 180°\) . . . . . angles in linear pairs

\(\therefore a + d + b + c = 180° +180° = 360°\)

\(\therefore c + d = 360° - (a + b)\)

If \(a + b >180°\) then \([360° - (a + b)] < 180°\)

\(\therefore c + d < 180°\)

\(\therefore\) In that case line l and line m produced will intersect each other on the same side of the transversal where c and d are formed.

\(\therefore c + d < 180\) is impossible.

That is \(a + b >180°\) is impossible...... (II)

\(\therefore\) the remaining possibility,

\(a + b = 180°\) is true......from (I) and (II)

\(\therefore a + b = 180°\) Similarly, \(c + d = 180°\)

Note that, in this proof, because of the contradictions we have denied the possibilities \(a + b >180°\) and \(a + b <180°\).

Therefore, this proof is an example of indirect proof.

Teacher's Note

This theorem is important. It tells us that interior angles on the same side always add to 180 degrees when lines are parallel.

Exam Trick

Remember: Interior angles on same side = 180°. Like two pieces that fit together perfectly to make a straight line!

Points to Remember

Interior angles are between the two parallel lines.
Interior angles on the same side of transversal add to 180°.
This is called supplementary angles.
180° means a straight line.
Indirect proof means proving by showing wrong ideas are impossible.

This is a preview of the first 3 pages. To get the complete book, click below.