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

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