**Transversal lines properties :**

When a transversal cuts two parallel lines, there are certain relationships among the angles at the points of intersection of transversal and parallel lines.

The relationship among the angles are called as properties of transversal lines.

In the above figure, let l₁ and l₂ be parallel lines and m is transversal.

From the above figure, we can have the following following transversal lines properties.

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° |

**Problem 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°**

**Problem 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°**

**Problem 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°**

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