Solution (1) :-
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 .
Solution (2) :-
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 .
Solution (3) :-
∠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 .
Solution (4) :-
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°.
Solution (5) :-
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 .
Solution (6) :-
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 .
Solution (7) :-
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 .
Solution (8) :-
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 .
Solution (9) :-
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 .
