If two lines intersect, prove that the vertically opposite angles are equal.


Let us draw the figure.



Here, we can see that


AB and CD intersect each other at point E.


The two pairs of vertically opposite angles are -


1st pair - AEC and BED


2nd pair - AED and BEC


We need to prove that the vertically opposite angles are equal, i.e.,


AEC = BED, and AED = BEC


Now, we can see that the ray AE stands on the line CD. We know, if a ray stands on a line then the sum of the adjacent angles is equal to 180°.


AEC + AED = 180° (By linear pair axiom) - - - - (i)


Similarly, the ray DE stands on the line AEB.


AED + BED = 180° (By linear pair axiom) - - - - (ii)


From equations (i) and (ii), we have


AEC + AED = AED + BED


AEC = BED - - - - (iii)


Similarly, the ray BE stands on the line CED.


DEB + CEB = 180° (By linear pair axiom) - - - - (iv)


Also, the ray CE stands on the line AEB.


CEB + AEC = 180° (By linear pair axiom) - - - - (v)


From equations (iv) and (v), we have


DEB + CEB = CEB + AEC


DEB = AEC - - - - (vi)


Thus, from equation (iii) and equation (vi), we have


AEC = BED, and DEB = AEC


Therefore, it is proved that the vertically opposite angles are equal.


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