In below fig, ray OS stand on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS

and ∠SOQ respectively. If ∠POS = x, find ∠ROT.

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#### Solution

Given,

Ray *OS *stand on a line *POQ*

Ray *OR *and Ray *OT *are angle bisectors of Ð*POS *and Ð*SOQ *respectively

`∠`*POS *= *x*

*`∠`POS *and `∠`*QOS *is linear pair

Ð*POS *+ Ð*QOS *= 180°

*x *+ Ð*QOS *= 180°

*`∠`QOS *= 180 - *x*

Now, ray or bisector `∠`*POS*

*∴ `∠`ROS *= `1/2` `∠`*POS*

= `1/2`× *x [ ∵ `∠`POS = x]*

*`∠`ROS *= *x*

Similarly ray OT bisector `∠`*QOS*

* ∵`∠`T**OS *= `1/2` `∠`*QOS*

= `180/2` - *x [∵ `∠`QOS = 180 - x]*

=` 90/2` - *x*

∴ `∠`*ROT *= `∠`*ROS *+ `∠`*ROT*

= `x/2`+ 90 -`x/2`

= 90°

∴`∠`*ROT *= 90°

Concept: Pairs of Angles

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