Maharashtra Board Class 8 Maths part 1 Chapter 2 Parallel lines and transversal PDF Download

Parallel Lines And Transversal

Let's Recall

The lines in the same plane which do not intersect each other are called parallel lines.

'Line l and line m are parallel lines,' is written as 'line l || line m'.

Let's Learn

Transversal

In the adjoining figure, line l intersects line m and line n in two distinct points. Line l is a transversal of line m and line n.

If a line intersects given two lines in two distinct points then that line is called a transversal of those two lines.

Angles Made By A Transversal

In the adjoining figure, due to the transversal, there are two distinct points of intersection namely M and N. At each of these points, four angles are formed. Hence there are 8 angles in all. Each of these angles has one arm on the transversal and the other is on one of the given lines. These angles are grouped in different pairs of angles. Let's study the pairs.

Corresponding Angles

If the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of the transversal, then it is called a pair of corresponding angles.

Interior Angles

A pair of angles which are on the same side of the transversal and inside the given lines is called a pair of interior angles.

Teacher's Note

When two roads cross a railway line, the angles formed are like corresponding angles. Imagine two streets cut by one straight train track.

Exam Trick

Remember: Corresponding angles are in the same position at each intersection. Like soldiers standing at the same place on two different lines.

Points To Remember

A transversal is a line that cuts two other lines at two different points.

Eight angles are formed when a transversal cuts two lines.

Corresponding angles are on the same side of the transversal.

Interior angles are between the two lines.

Alternate Angles

Pairs of angles which are on the opposite sides of transversal and their arms on the transversal show opposite directions is called a pair of alternate angles.

In the figure, there are two pairs of interior alternate angles and two pairs of exterior alternate angles.

Interior alternate angles (Angles at the inner side of lines)

(i) \(\angle PMN\) and \(\angle MNS\)

(ii) \(\angle QMN\) and \(\angle RNM\)

Exterior alternate angles (Angles at the outer side of lines)

(i) \(\angle AMP\) and \(\angle TNS\)

(ii) \(\angle AMQ\) and \(\angle RNT\)

Pairs of corresponding angles in the given figure:

(i) \(\angle AMP\) and \(\angle MNR\)

(ii) \(\angle PMN\) and \(\angle RNT\)

(iii) \(\angle AMQ\) and \(\angle MNS\)

(iv) \(\angle QMN\) and \(\angle SNT\)

Pairs of interior angles in the given figure:

(i) \(\angle PMN\) and \(\angle MNR\)

(ii) \(\angle QMN\) and \(\angle MNS\)

Teacher's Note

Alternate angles look like the letter 'Z' in a diagram. When you see a Z-pattern, those angles are alternate angles and they are equal.

Exam Trick

To find alternate angles, look for the Z-shape in the diagram. The two angles at the corners of the Z are alternate angles.

Points To Remember

Alternate angles are on opposite sides of the transversal.

Interior alternate angles are between the two lines.

Exterior alternate angles are outside the two lines.

Alternate angles have arms pointing in opposite directions.

Practice Set 2.1

1. In the adjoining figure, each angle is shown by a letter. Fill in the boxes with the help of the figure.

Corresponding angles.

(1) p and ________ (2) q and ________

(3) r and ________ (4) s and ________

Interior alternate angles.

(5) s and ________ (6) w and ________

2. Observe the angles shown in the figure and write the following pair of angles.

(1) Interior alternate angles

(2) Corresponding angles

(3) Interior angles

Teacher's Note

In Indian classrooms, we often see two parallel railway tracks cut by a crossing road. The angles formed are like the angles in this chapter.

Exam Trick

Write down which angles go together. Mark them with the same symbol like a tick or dot to remember them easily.

Points To Remember

Practice set questions help you find pairs of angles from diagrams.

Each pair has two angles in different positions.

Use the definition of each angle type to identify them correctly.

Draw small marks on equal angles to remember them.

Properties Of Angles Formed By Two Parallel Lines And A Transversal

Activity (I)

As shown in the figure (A), draw two parallel lines and their transversal on a paper. Draw a copy of the figure on another blank sheet using a trace paper, as shown in the figure (B). Colour part I and part II with different colours. Cut out the two parts with a pair of scissors.

Note that the angles shown by part I and part II form a linear pair. Place part I and part II on each angle in the figure A.

Which angles coincide with part I?

Which angles coincide with part II?

We see that, \(\angle b \cong \angle d \cong \angle f \cong \angle h\), because these angles coincide with part I.

\(\angle a \cong \angle c \cong \angle e \cong \angle g\), because these angles coincide with part II.

(1) \(\angle a \cong \angle e\), \(\angle b \cong \angle f\), \(\angle c \cong \angle g\), \(\angle d \cong \angle h\)

(These are pairs of corresponding angles.)

(2) \(\angle d \cong \angle f\) and \(\angle e \cong \angle c\) (These are pairs of interior alternate angles.)

(3) \(\angle a \cong \angle g\) and \(\angle b \cong \angle h\) (These are pairs of exterior alternate angles.)

(4) \(m\angle d + m\angle e = 180°\) and \(m\angle c + m\angle f = 180°\)

(These are interior angles.)

Let's Discuss

When two parallel lines are intersected by a transversal eight angles are formed. If the measure of one of these eight angles is given, can we find measures of remaining seven angles?

Teacher's Note

This activity uses tracing paper to show that angles are equal. In real life, builders use this property when making parallel walls in houses.

Exam Trick

If you know one angle measure, you can find all other angles using these properties. All alternate angles are equal, so learn one and you know many.

Points To Remember

Corresponding angles from parallel lines are congruent (equal).

Alternate angles from parallel lines are congruent (equal).

Interior angles from parallel lines add up to 180 degrees.

If one angle is known, all eight angles can be found.

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