Q. 104.0( 22 Votes )

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

Answer :

Consider the angles,

*∠**AOB and* *∠**ACB*

Given that,

OA perpendicular AO and OB perpendicular BO

To prove: *∠**AOB =* *∠**ACB* or,

*∠**AOB +* *∠**ACB = 180 ^{o}*

Proof: In a quadrilateral

*∠**A +* *∠**O +* *∠**B +* *∠**C = 360 ^{o}*(Sum of angles of a quadrilateral)

180^{o} + *∠**O +* *∠**C = 360 ^{o}*

*∠**O +* *∠**C = 180 ^{o}*

Hence, *∠**AOB +* *∠**AOC = 180 ^{o} (i)*

Also,

*∠**B +**∠**ACB = 180 ^{o}*

*∠**ACB = 180 ^{o} – 90^{o}*

*∠**ACB = 90 ^{o} (ii)*

From (i) and (ii), we get

*∠**ACB =* *∠**AOB = 90 ^{o}*

Hence, the angles are equal as well as supplementary.

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