**Parallel lines and transversals**

A transversal is a line that intersects two lines in the same plane at two different points. Transversal and lines l₁ and l₂ form eight angles.

More clearly,

Here, l₁ and l₂ are parallel lines.

From the above figure, we have the following important points.

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° |

**Example 1 :**

In the figure given below, let the lines l₁ and l₂ 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°

Plug ∠F = 65°

∠F + ∠E = 180°

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°

**Example 2 :**

In the figure given below, let the lines l₁ and l₂ 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, we have

2x + 20 = 3x - 10

30 = x

Hence, x = 30°

**Example 3 :**

In the figure given below, let the lines l₁ and l₂ 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, we have

3x + 20 + 2x = 180°

5x + 20 = 180°

5x = 160°

x = 32°

Hence, x = 32°

After having gone through the stuff given above, we hope that the students would have understood "Parallel lines and transversals".

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