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In the given figure, PQRS is a parallelogram in which $∠ Q P O = 35 °$ , $∠ S P O = 40 °$ and $∠ R O S = 105 °$ . Then the measure of $∠ R Q S =$ ____.

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Detailed Solution

In the given figure, PQRS is a parallelogram in which

∠ Q P O = 35 °

∠ S P O = 40 °

and

∠ R O S = 105 °

. Then the measure of

∠ R Q S =

°

.
We are given a parallelogram PQRS.
Consider,

Δ P Q R

and

Δ Q P S

.
Here,

∠ P Q R

and

∠ Q P S

are alternate angles for transversal QS.
Use the property that the adjacent angles of parallelogram are supplementary.

∠ P Q R + ∠ Q P S = 18 0 °

Let us find the value of

∠ P Q R

∠ P Q R = 18 0 ° − ∠ Q P S

…(i)
Here,

∠ Q P S = ∠ S P O + ∠ Q P O = 30 ∘ + 45 ∘ = 75 ∘

Now substitute this value in (i).

∠ P Q R = 18 0 ° − 75 ° = 10 5 °

Let us now find the value of

∠ R Q S

∠ R Q S = ∠ P Q R − ∠ P Q S

(

∠ P Q S = ∠ P Q O

)

∠ R Q S = 10 5 ° − 4 0 ° = 65 °

Hence, the value of

∠ R Q S is 65 °

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