Question

Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.

Answer 1

Given : AB||DE and BC ||EF
Case I If angles are formed as in the adjoining figure
Since arms BA||ED and BC cuts them
`:. angle 1=angle3` (corresponding angles are equal)...(1)
Since BC||EF and DE cuts them
`:. angle 3= angle 2` (corresponding angles are equal)....(2)
From (1) and (2), we get
`angle1= angle=2" "....(3)`
Case II : If angle are formedas in the adjoining figure
Sicne AB||DE and BC cuts them
`:. angle 1 +angle3 =180^(@)` (sum of co-interior angle is `180^(@))`
Also, since BC||EF and DE cuts them `:. angle3 =angle 2` (corresponding angles )....(5)
`:.` From (4) and (5) , we get
`angle1+angle2=180^(@)`
From (3) and (6), we can say that
the given angles are equal or they are supplementary