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The following are circle theorems on arc and angle. |
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1) ∠POM = ∠OPM + ∠ORP | 1) By exterior angle theorem (∠POM is exterior angle) |
2) OP = OR | 2) Radii of same circle |
3) ∠OPR = ∠ORP | 3) In a Δ two sides are equal then the angle opposite to them are also equal. |
4) ∠POM = ∠ORP + ∠ORP | 4) Substitution property. From (1) |
5) ∠POM = 2∠ORP | 5) Addition property. |
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1) ∠POM = ∠OPR + ∠ORP | 1) Exterior angle theorem. |
2) ∠POM = ∠ORP + ∠ORP | 2) As,OP = OR = radius. ∴∠ORQ = ∠ORP |
3) ∠POM = 2∠ORP | 3) Substitution and addition property. |
4) ∠QOM = ∠ORQ + ∠OQR | 4) Exterior angle theorem. |
5) ∠QOM = ∠ORQ + ∠ORQ | 5) As, OQ =OR = radius, ∴ ∠ORQ = ∠OQR |
6) ∠QOM = 2∠ORQ | 6) Substitution and addition property. |
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1) ∠POM = ∠OPR + ∠ ORP | 1) By exterior angle theorem. |
2) ∠POM = 2∠ORP | 2) As, OP = OR = radius, ∴∠OPR = ∠ORP and by addition property |
3) ∠ QOM = ∠ORQ + ∠OQR | 3) By exterior angle theorem in ΔQOR |
4) ∠QOM = 2∠ORQ | 4) As, OP = OR = radius, ∴∠ORQ = ∠OQR and by addition property |
5) ∠POM + ∠QOM = 2(∠ORP + ∠ORQ) | 5) From (2) and (4) |
6) Reflex ∠POQ = 2∠PRQ | 6) Reflex of ∠POQ =∠POM + ∠QOM |
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1) ∠POQ = 2∠PRQ | 1) Angle subtended by an arc of a circle at its center is twice the angle formed by the same arc. |
2) 180 0 = 2∠PRQ | 2) As POQ is a straight line |
3) ∴ ∠PRQ = 90 0 | 3) Division property |
Russia-Ukraine crisis update - 3rd Mar 2022
The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.