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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
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
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