Sum

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

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#### Solution

Consider be angles AOB and ACB

Given OA ⊥ AC, OB ⊥ BC

To prove: `∠`AOB = `∠`ACB (or)

`∠`AOB + `∠`ACB = 180°

Proof:- In a quadrilateral [Sum of angles of quadrilateral]

⇒`∠`A + `∠`O + `∠`B + `∠`C = 360°

⇒ 180 + `∠`O + `∠`C = 360°

⇒ `∠`O + `∠`C = 360 -180 = 180°

Hence, `∠`AOB + `∠`ACB = 180° ......(i )

Also,

`∠`B + `∠`ACB = 180° ......(i )

Also,

`∠`B + `∠`ACB = 180° ......(i )

Also,

`∠`B + `∠`ACB = 180°

⇒ `∠`ACB = 180° - 90°

⇒`∠`ACB = 90° .....(ii)

From (i) and (ii)

∴`∠`ACB = `∠`AOB = 90°

Hence, the angles are equal as well as supplementary

Concept: Pairs of Angles

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