Question

if the arms of one angle are respectively parallel to the arms of another angle, show that the two angles are either equal or supplementary​

Answer 1

Given :-

$Two \: angles \angle ABC and \angle DEF \\ such \: that \: BA||E \: and \: BC||EF$

TO PROVE

$\angle ABC= \angle DEF \\ or \angle ABC+ \angle DEF=180°$

PROOF

The arms of the angles may be parallel in the same sense or in the opposite sense.

So, three cases arise.

Case 1 :-

when both pairs of arms are parallel in same sence [ fig.( i ):

in the case , BA||ED and BC is transversal.

$:.. \angle ABC = \angle 1 ( corres. \angle s)$

Again,

BC||EF and DE is transversal.

$.: \angle1 = \angle DEF ( corres. \angle s)$

$Hence, \angle ABC= \angle DEF$

Case 2 :-

when both pairs of arms are parallel in opposite sence [ fig. (ii) ]

in this case, BA|| ED is the transversal

$:.. \angle ABC = \angle 1 ( corres. \angle s)$

Again,FE ||BC and ED is transversal.

$.:. \angle DEF = \angle 1( alt. int. \angle s)$

$hence, \angle ABC= \angle DEF.$

Case 3

When one pair of arms are parallel in same sense and other pair parallel in opposite sense: [Fig. (iii)]

In this case, A||ED and BC is the transversal.

$\angle EGB= \angle ABC \\ [Alt. Int. \angle \: s] .$

Now, BC|| EF and DE is transversal

$.:. \angle DEF + \angle EGB= 180° \\ [Co.Int. \angle s]$

$.:. \angle DEF + \angle ABC= 180° \\ [ \angle EGB= \angle ABC]$

$hence, \angle ABC and \angle DEF \\ are \: supplementary$