A transversal is a line that intersects two lines in the same plane at two different points.

In the diagram shown below, let the lines 'l_{1}' and 'l_{2}' be parallel. Because the line 'm' cuts the lines 'l_{1}' and 'l_{2}', the line 'm' is transversal.

So, the two parallel lines 'l_{1}' and 'l_{2}' cut by the transversal 'm'.

The angles formed by the two parallel lines 'l_{1}' and 'l_{2}' and the transversal 'm' in the above diagram will be as follows.

Vertically opposite angles are equal. |
∠ 1 = ∠ 3 ∠ 2 = ∠ 4 ∠ 5 = ∠ 7 ∠ 6 = ∠ 8 |

Corresponding angles are equal. |
∠ 1 = ∠ 5 ∠ 2 = ∠ 6 ∠ 3 = ∠ 7 ∠ 4 = ∠ 8 |

Alternate interior angles are equal. |
∠ 3 = ∠ 5 ∠ 4 = ∠ 6 |

Alternate exterior angles are equal. |
∠ 1 = ∠ 7 ∠ 2 = ∠ 8 |

Consecutive interior angles are supplementary. |
∠ 3 + ∠ 6 = 180° ∠ 4 + ∠ 5 = 180° |

Same side exterior angles are supplementary. |
∠ 1 + ∠ 8 = 180° ∠ 2 + ∠ 7 = 180° |

**Problem 1 :**

In the figure given below, let the lines l_{1} and l_{2} be parallel and m is transversal. If ∠F = 65°, find the measure of each of the remaining angles.

**Solution : **

From the given figure,

∠F and ∠H are vertically opposite angles and they are equal.

Then,

∠H = ∠F

∠H = 65°

∠H and ∠D are corresponding angles and they are equal.

Then,

∠D = ∠H

∠D = 65°

∠D and ∠B are vertically opposite angles and they are equal.

Then,

∠B = ∠D

∠B = 65°

∠F and ∠E are together form a straight angle.

Then, we have

∠F + ∠E = 180°

Substitute ∠F = 65°.

65° + ∠E = 180°

∠E = 115°

∠E and ∠G are vertically opposite angles and they are equal.

Then,

∠G = ∠E

∠G = 115°

∠G and ∠C are corresponding angles and they are equal.

Then,

∠C = ∠G

∠C = 115°

∠C and ∠A are vertically opposite angles and they are equal.

Then,

∠A = ∠C

∠A = 115°

Therefore,

∠A = ∠C = ∠E = ∠G = 115°

∠B = ∠D = ∠F = ∠H = 65°

**Problem 2 :**

In the figure given below, let the lines l_{1} and l_{2 }be parallel and t is transversal. Find the value of x.

**Solution : **

From the given figure,

∠(2x + 20)° and ∠(3x - 10)° are corresponding angles.

So, they are equal.

Then,

2x + 20 = 3x - 10

30 = x

**Problem 3 :**

In the figure given below, let the lines l_{1} and l_{2} be parallel and t is transversal. Find the value of x.

**Solution : **

From the given figure,

∠(3x + 20)° and ∠2x° are consecutive interior angles.

So, they are supplementary.

Then,

3x + 20 + 2x = 180°

5x + 20 = 180°

5x = 160°

x = 32°

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