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In this figure ∠7 and ∠3 are alternate interior angles . Alternate interior angles in two parallel lines with a transversal will always be equal . Since ∠7 is congruent to ∠3 , we can say that line m and line n are parallel lines .

Therefore :-

The lines are parallel because the alternate interior angles in them are congruent .

Given :-

m∠3 = ( 15x + 22 )°

Then , m∠3 will be equal to :-

x = 8

= 15 × 8 + 22

= 120 + 22

m∠3 = 142°

m∠1 = ( 19x - 10 )°

= 19 × 8 - 10

= 152 - 10

m∠1 = 142°

angle 1 and angle 3 are corresponding angles , which means their value will be equal if they are present on two parallel lines with a transversal . As their values are equal we can say that line m and line n are parallel .

Therefore :-

Yes , the lines are parallel because the corresponding angles in them are equal .

∠7 and ∠6 are interior angles on the same side of transversal , which means their angle su, will be equal to 180° . But since their values are equal these two lines are not parallel .

Therefore :-

∠7 and ∠6 are interior angles on the same side of transversal , so their sum will be equal to 180° . But since that is not the case in the lines shown above , those lines are not parallel lines .

Given :-

m∠2 = ( 5x + 3 )°

x = 14

= 5 × 14 + 3

= 60 + 3

= 63°

m∠3 = ( 8x - 5 )°

= 8 × 14 - 5

= 112 - 5

= 107°

We know that the sum of angle 2 and angle 3 will be equal to 180° if they are on two parallel lines as they are interior angles on the same side of transversal .

As :-

∠2 + ∠3 ≠ 180°

63° + 107° is not equal to 180° ,

Therefore :-

These lines are not parallel as the sum of their interior angles on the same side of transversal is not equal to 180°.

Given :-

m∠8 = ( 6x - 1 )°

x = 9

= 6 × 9 - 1

= 54 - 1

= 53°

m∠4 = ( 5x + 3 )°

= 5 × 9 + 3

= 45 + 3

= 48°

angle 8 and angle 2 are vertically opposite angles , which means m∠8 = m∠2 .

m∠2 = 53°

As ∠2 and ∠4 are corresponding angles their value will be equal , if they are on parallel lines .

Since angle 2 , angle 8 and angle 4 are not equal these lines are not parallel lines .

Therefore :-

The lines are not parallel as their corresponding angles are not equal .

Angle 5 and angle 7 are corresponding angles they will be congruent when on parallel lines .

As , m∠5 ≅ m∠7 these lines are parallel .

Therefore :-

These lines are parallel lines as their corresponding angles are equal .

Angle 1 and angle 7 are vertically opposite angles , which means their angle measure will be equal . As angle 7 and angle 5 are corresponding angles their angle measure will also be equal .

Since m∠5 ≅ m∠7 , these lines are parallel .

Therefore :-

These lines are parallel lines as their corresponding angles are equal .

Here , the value of x is the same .

Which means :-

m∠2 = x + 15° is greater than m∠6 = x + 10°

Angle 2 and angle 6 are alternate interior angles , their values should be equal . But since their values are not equal , the lines on which they reside are not parallel .

Therefore :-

These lines are not parallel as their alternate interior angles are not equal .

It is given that angle 1 and angle 3 are supplementary angles . It is also clear that angle 1 and angle 2 are a linear pair , which means their angle sum will be equal to 180° . Also , angle 3 and angle 2 should be equal since they are the supplement of the same angle . Angle 2 and angle 3 are also corresponding angles . Since corresponding angles in two lines with a transversal will be equal and in these two lines the corresponding angles are equal , these lines are parallel lines.

Therefore , JK // HI as their corresponding angles are equal .

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