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.

To do:

 We have to 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.

Solution:

Let in two angles $\angle ABC$ and $\angle DEF$,

$AB \perp DE$ and $BC \perp EF$

Produce the sides $DE$ and $EF$ of $\angle DEF$, to meet the sides of $\angle ABC$ at $H$ and $G$.

From the figure,

$BGEH$ is a quadrilateral

$\angle BHE = 90^o$ and $\angle BGE = 90^o$

Sum of the angles in a quadrilateral is $360^o$

Therefore,

$\angle HBG + \angle HEG = 360^o - (90^o + 90^o)$

$= 360^o - 180^o$

$= 180^o$

$\angle ABC$ and $\angle DEF$ are supplementary.

In quadrilateral $BGEH$,

$\angle BHE = 90^o$ and $\angle HEG = 90^o$

$\angle HBG + \angle HEG = 360^o - (90^o + 90^o)$

$= 360^o- 180^o$

$= 180^o$.......…(i)

$\angle HEF + \angle HEG = 180^o$......…(ii)

From equations (i) and (ii),

$\angle HEF = \angle HBG$

This implies,

$\angle DEF = \angle ABC$

Hence, $\angle ABC$ and $\angle DEF$ are either equal or supplementary.

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Updated on: 10-Oct-2022

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