**Parallel lines cut by a transversal :**

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

**Corresponding angles : **

Angles lie on the same side of the transversal t, on the same side of lines a and b.

Example : ∠ 1 and ∠ 5

**Alternate interior angles :**

Angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 6

**Alternate exterior angles :**

Angles lie on opposite sides of the transversal t, outside lines a and b.

Example : ∠ 1 and ∠ 8

**Same-side interior angles :**

Angles lie on the same side of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 5

Identify the pairs of angles in the diagram. Then make a conjecture about their angle measures.

**Corresponding Angles : **

∠CGE and ∠AHG, ∠DGE and ∠BHG, ∠CGH and ∠AHF, ∠DGH and ∠BHF ; congruent.

**Alternate interior angles :**

∠CGH and ∠BHG, ∠DGH and ∠AHG ; congruent.

**Alternate exterior angles :**

∠CGE and ∠BHF, ∠DGE and ∠AHF ; congruent.

**Same-side interior angles :**

∠CGH and ∠AHG, ∠DGH and ∠BHG ; supplementary.

**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 cut by a transversal".

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