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Formation of Angles by Intersecting Lines

Found in topics: Angles , Lines
Maths Query > Unit > Geometry > Fundamentals of Geometry

Relationship between Lines and Angles

We have learnt the basics of angles, their measures and various types of angles in the chapter Angle, its Measures and Types of Angle. As we have seen, angles are formed only when two lines intersect at some point. So, there are many more case where different types of angles are defined when lines intersect each other.
Such angles are formed by a transversal line when it intersects two or more parallel or non parallel lines.

Vertically Opposite angles

When two straight lines intersect each other, they form four angles at the point of intersection. Out of four angles the two angles which are directly opposite to each other are called vertically opposite angles. These two vertically opposite angles are always equal.

Example of Vertically Opposite angles
Example of Vertically Opposite angles

Here, in the above diagram, we can see ∠AOD, ∠BOC, ∠AOC and ∠BOD are the four angles formed at point O when two lines AB and CD intersect at point O.

The angles ∠AOD and ∠BOC are directly opposite to each other, therefore they are called vertically opposite angles and ∠AOD = ∠BOC

Similarly, the angles ∠AOC and ∠BOD are also directly opposite to each other, therefore they are also called vertically opposite angles and ∠AOC = ∠BOD
Moreover, the sum of each pair of adjacent angles is alway equal to 180°.

∠AOC + ∠BOC = 180°
∠AOC + ∠AOD = 180°
∠BOC + ∠BOD = 180°
∠AOD + ∠BOD = 180°

Exterior angles

Exterior angles are formed at points on the exterior sides of the two lines where a transversal line intersects these two lines.
Let’s understand it by an example.

Example of Exterior angles
Example of Exterior angles

∠1, ∠2, ∠7 and ∠8 are formed at points A and B on the exterior sides of two non parallel lines l and m respectively, when a transversal line n cuts through them. Therefore, ∠1, ∠2, ∠7 and ∠8 are called exterior angles.

Interior angles

Interior angles are formed at points on the interior sides of the two lines where a transversal line intersects these two lines.
Let’s understand it by an example.

Example of Interior angles
Example of Interior angles

∠3, ∠4, ∠5 and ∠6 are formed at points A and B on the interior sides of two non parallel lines l and m respectively, when a transversal line n cuts through them. Therefore, ∠3, ∠4, ∠5 and ∠6 are called interior angles.

Alternate angles

Alternate angles again are formed when a transversal line cuts through two parallel or non parallel lines. These angles are a pair of angles which can exist on the interior or exterior sides of the two lines.
We can understand it more precisely with the following example.

Example of Alternate angles
Example of Alternate angles

Here, a transversal line n cuts thru two lines l and m and forms total of eight angles at points A and B which are:

∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8.
If we select the pairs of angles in the following way, then these pairs are the alternate angles.

∠1 and ∠7
∠2 and ∠8
∠3 and ∠6
∠4 and ∠5

Depending upon where the pair exists on the two lines l and m we can name them alternate interior angles and alternate exterior angles.
e.g. here, the pairs of alternate angles ∠1 and ∠7, ∠2 and ∠8 lie outside the lines i.e. exterior sides of the two lines, such type of alternate angles are called alternate exterior angles.

Similarly, the pairs of alternate angles ∠3 and ∠6, ∠4 and ∠5 lie inside the lines i.e. interior sides of the two lines, such type of alternate angles are called alternate interior angles.

Note

The pair of alternate angles formed with parallel lines are always equal.

In the above example, lines l and m are parallel lines. So, we can see that the following alternate angles pairs are equal
∠1 = ∠7
∠2 = ∠8
∠3 = ∠6
∠4 = ∠5

Corresponding angles

A pair of angles is called corresponding angles in which one arm of both angles is on the same side of transversal and their other arms are directed in the same sense.
If lines l and m are parallel, then pairs of corresponding angles are equal.
Let’s understand it with the following example.

Example of Corresponding angles
Example of Corresponding angles

So, from the above diagram, the following pair of angles are corresponding angles that are formed on two parallel lines l and m.

∠1 and ∠6
∠2 and ∠5
∠3 and ∠7
∠4 and ∠8

As said above, these pairs of angles, also called as corresponding angles, are always equal.
Therefore, we can write them as:
∠1 = ∠6
∠2 = ∠5
∠3 = ∠7
∠4 = ∠8

More, in parallel lines l and m, sum of interior angles of the same side of transversal is 180° i.e. ∠4 + ∠6 = 180°
∠3 + ∠5 = 180°

Linear pair of angles

When sum of two adjacent angles is 180°, they are called linear pair of angles.
Or, we can say when supplementary angles formed on a straight line, those angles are called a linear pair of angles.

Example of Linear pair of angles
Example of Linear pair of angles

In the above diagram, ∠ABO + ∠CBO = 180°, therefore, they form a linear pair of angles.

Solved Examples

1) Find the value of x.

Chapter: lines and angles, Solved example 1

Solution
Since, ∠AOC + ∠BOC = 180° (linear pair)
3x + 2x = 180°
5x = 180°
x = 180° 5
x = 36°

2) Find the ∠POR.

Chapter: lines and angles, Solved example 2

Solution
∠POR + ∠QOR = 180° (linear pair)
2x + x = 180°
3x = 180°
x = 180° 3
x = 60°
∠POR = 2x
∠POR = 2 × 60
∠POR = 120°

3) Find the value of x.

Chapter: lines and angles, Solved example 3

Solution
Since, ∠POR & ∠MON are vertically opposite angles
∴ ∠POR = ∠MON
4x = 120°
x = 120° 4
x = 30°

4) Find the value ∠1, ∠2 and ∠3, if ∠4 = 30°.

Chapter: lines and angles, Solved example 4

Solution
∠1 & ∠4 are vertically opposite angles
∴ ∠1 = ∠4
∴ ∠1 = 30°
Also, ∠1 + ∠2 = 180°
30° + ∠2 = 180°
∠2 = 180° - 30°
∠2 = 150°
Also, ∠2 = ∠3      because vertically opposite angles
∴ ∠3 = 150°

5) If p || q and r is transversal. Find the value ∠1, ∠2 and ∠3, if ∠4 = 110°.

Chapter: lines and angles, Solved example 5

Solution
∠3 & ∠4 are vertically opposite angles
∴ ∠3 = 110°
∠2 & ∠3 are corresponding angles
∴ ∠2 = ∠3
∴∠2 = 110°
Also, ∠1 + ∠2 = 180° (∵ linear pair)
∠1 + 110° = 180°
∠1 = 180° - 110°
∠1 = 70°

6) If p || q, ∠1 and ∠2 and are in the ratio of 2 : 3. Find angles ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8.

Chapter: lines and angles, Solved example 6

Solution
∠1 : ∠2 = 2 : 3
Also ∠1 + ∠2 = 180° [because ∠1 and ∠2 are linear pair]
2x + 3x = 180°
5x = 180°
x = 180 5
x = 36°
∠1 = 2x = 2 × 36° = 72°
∠2 = 3x = 3 × 36° = 108°
∠1 = ∠3 (because vertically opposite angles)
∴ ∠3 = 72°
Also, ∠2 = ∠4 (because vertically opposite angles)
∠4 = 108°
Also, ∠1 = ∠5 (because corresponding angles)
∠5 = 72°
Also, ∠2 = ∠6 (because corresponding angles)
∠6 = 108°
Also, ∠6 = ∠8 (because vertically opposite angles)
∠8 = 108°
∠5 = ∠7 (because vertically opposite angles)
∠7 = 72°

7) If AB || CD, find the values of x and y.

Chapter: lines and angles, Solved example 7

Solution
Since, x and 120° form a linear pair.
∴ ∠x + 120° = 180°
∠x = 180° - 120°
∠x = 60°
Also, AB || CD
∴ ∠ACD + ∠CAB = 180°
Because sum of interior angles on the same side of a transversal is 180°
60° + ∠y = 180°
∠y = 180° - 60°
∠y = 120°

8) In the figure, p || q, r || s and ∠1 = 105°. Find the other angles.

Chapter: lines and angles, Solved example 8

Solution
Since, ∠1 and ∠4 are vertically opposite angles.
∴ ∠1 = ∠4
∠4 = 105°
Since, r || s and p is transversal to r and s.
∠1 and ∠2 form corresponding angles.
∴ ∠1 = ∠2
∠2 = 105°
Also, ∠2 + ∠3 = 180°
105° + ∠3 = 180°
∠3 = 180° - 105°
∠3 = 75°
Now p || q and r is transversal to p and q.
∠4 + ∠5 = 180° (because sum of interior angles on the same side of transversal is 180°)
∠4 + ∠5 = 180°
105° + ∠5 = 180°
∠5 = 180° - 105°
∠5 = 75°
Since p || q and s is also transversal to p and q.
∴; ∠3 and ∠6 are corresponding angles.
∴ ∠3 = ∠6
∠6 = 75°

9) In the following figure FG || DE, ∠B = 30° and ∠C = 50°. Find values of x, y and z.

Chapter: lines and angles, Solved example 9

Solution
Since, FG || DE and AB is transversal to FG and DE.
∠x = 30°
Also, FG || DE and AC is transversal to FG and DE.
∴ 50° and y are alternate interior angles.
∠y = 50°
Also, ∠x + ∠y + ∠z = 180° (because x, y and z form linear pair)
30° + 50° + ∠z = 180°
80° + ∠z = 180°
∠z = 180° - 80°
∠z = 100°

10) In the given figure PQ || SR and PS || QR. Find the values of x, y and z.

Chapter: lines and angles, Solved example 10

Solution
Since, PQ || SR and PS is transversal to PQ and SR.
∠S and ∠P are co-interior angles
∴ ∠x + 60° = 180° (because co-interior angles are supplementary)
∠x = 180° - 60°
∠x = 120°
Also, PS || QR and PQ is transversal to PS and QR.
∠P and ∠Q are co-interior angles
∴ 60° + ∠z = 180° (because co-interior angles are supplementary)
∠z = 180° - 60°
∠z = 120°
Similarly, PS || QR and SR is transversal to PS and QR.
∠S and ∠R are co-interior angles
∴ ∠x + ∠y = 180° (because co-interior angles are supplementary)
120° + ∠y = 180°
∠y = 180° - 120°
∠y = 60°

Worksheet 1

By observing the following figure, fill the blanks in.

p||q, r as transversal

1) If p || q and ∠1 = ∠5, then these are ___________ angles.

2) If p || q then ∠5 + ∠6 = ___________.

3) If ∠4 = ∠6, then line p must be ___________ to line q.

4) If p || q and ∠1 = ∠3, then these are ___________ angles.

5) If p || q and ∠2 = ∠8, then these are ___________ angles.

Help iconHelp box
parallel
vertically opposite
corresponding
alternate exterior
180°

Worksheet 2

Write True or False in the boxes.

1)

If two lines in the same plane do intersect then these lines must be parallel.

2)

If a line intersects two lines then the line is parallel to the two lines.

3)

When two straight lines intersect each other at a point then the vertically opposite angles formed at the point of intersection are unequal.

4)

The sum of two adjacent angles on a same line is 180°.

5)

The distance between two parallel lines remain same.

6)

If two parallel lines are cut by a transversal line then their corresponding angles must be equal.

7)

The interior alternate angles are equal, if non parallel lines are cut by a transversal.

8)

If line m is parallel to n and n is parallel to p then m is parallel to p.

9)

The distance between two intersecting lines is zero.

10)

When a transversal cuts two lines such that a pair of alternate exterior angles are unequal then the two lines must be non parallel to each other.

Worksheet 3

Match the following.

l||m, n as transversal
1)alternate interior anglesa)∠2, ∠4

2)alternate exterior anglesb)∠1, ∠5

3)vertically opposite anglesc)∠3, ∠5

4)corresponding anglesd)∠7, ∠8

5)linear paire)∠1, ∠7

Worksheet 4

In the following figure, find the missing values.

1)

Find angles x and y

2)

Find angles p, q and s

3)

Find angles p, q and r

4)

Find angles x, y

5)

Find angles a, b, c, d, e, f and g

Worksheet 5

Multiple choice questions

By observing the figure below, where p || q and r || s, choose the correct option for the following questions.

l||m and r||s, r and s are transversal

1) Which pair of lines are parallel?

  1. p and q
  2. p and r
  3. q and r
  4. q and s

2) Which pair of angles do form vertically opposite angles?

  1. ∠3 and ∠7
  2. ∠5 and ∠7
  3. ∠1 and ∠2
  4. ∠1 and ∠3

3) Which pair of angles do form corresponding angles?

  1. ∠1 and ∠2
  2. ∠2 and ∠3
  3. ∠5 and ∠7
  4. ∠3 and ∠4

4) Which pair of angles do form alternate interior angles?

  1. ∠2 and ∠11
  2. ∠2 and ∠4
  3. ∠2 and ∠3
  4. ∠6 and ∠11

5) Which pair of angles do form linear pair of angles?

  1. ∠1 and ∠9
  2. ∠1 and ∠5
  3. ∠6 and ∠7
  4. ∠3 and ∠4

6) Which pair of angles do form alternate exterior angles?

  1. ∠5 and ∠11
  2. ∠1 and ∠2
  3. ∠5 and ∠8
  4. ∠5 and ∠7

7) What is the sum of exterior angles ∠1 and ∠10?

  1. 360°
  2. 180°
  3. 270°
  4. 90°

8) What is the sum of interior angles ∠6 and ∠11?

  1. 90°
  2. 180°
  3. 270°
  4. 360°

9) What is the sum of of linear pair of angles ∠7 and ∠11?

  1. 270°
  2. 360°
  3. 90°
  4. 180°

10) When two lines intersect each other at point O, these two lines form

  1. corresponding angles
  2. vertically opposite angles
  3. alternate exterior angles
  4. alternate interior angles
MCQ Answer Key Hide Show
1. a
2. c
3. c
4. c
5. a
6. c
7. b
8. b
9. d
10. b
Last updated on: 11-11-2024