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In figure, lines PQ and RS intersect each other at point O. If $∠ P O R$ : $∠ R O Q = 5 : 7,$ find all the angles.

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∠ P O R = 75 °, ∠ Q O S = 75 °, ∠ R O Q = 15 °, a n d ∠ P O S = 15 °

∠ P O R = 75 °, ∠ Q O S = 75 °, ∠ R O Q = 105 °, a n d ∠ P O S = 105 °

∠ P O R = 23 °, ∠ Q O S = 23 °, ∠ R O Q = 105 °, a n d ∠ P O S = 105 °

∠ P O R = 75 °, ∠ Q O S = 75 °, ∠ R O Q = 56 °, a n d ∠ P O S = 56 °

answer is B.

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Detailed Solution

Given a figure in which lines PQ and RS intersect each other at point O and

∠ P O R : ∠ R O Q = 5 : 7

In the figure, lines PQ and RS intersect each other at point O. If ∠ POR: ∠ ROQ= 5:7. Find all the angles.

As,

∠ P O R

and

∠ R O Q

form a linear pair as POQ is a line, therefore,

∠ P O R + ∠ R O Q = 180 °

---(1)
Ratio is

5 : 7

.
Assume

∠ P O R = 5 x

and

∠ R O Q = 7 x

.
Substitute the values in (1) and evaluate,

5 x + 7 x = 180 ° ⇒ 12 x = 180 °

⇒ x = 180 ° 12 ⇒ x = 15 °

Evaluate angles by substituting the values of x,

∠ P O R = 5 x ⇒ P O R = 5 (15 °)

⇒ ∠ P O R = 75 °

∠ R O Q = 7 x ⇒ ∠ R O Q = 7 (15 °) ⇒ ∠ R O Q = 105 °

∠ P O R

and

∠ Q O S

are vertically opposite angles, therefore,

∠ P O R = ∠ Q O S = 75 °

.
As,

∠ R O Q

and

∠ P O S

are vertically opposite angles, therefore,

∠ R O Q = ∠ P O S = 105 °

.
Hence,

∠ P O R = 75 °, ∠ Q O S = 75 °, ∠ R O Q = 105 °, a n d ∠ P O S = 105 °

.
Hence, option 2 is correct.

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