In figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

$angle ROS equals 1 half left parenthesis angle QOS minus angle POS right parenthesis$

Solution

∵ Ray OR is perpendicular to line PQ.
∴ ∠QOR = ∠POR = 90°    ...(1)
∠QOS = ∠QOR + ∠ROS    ...(2)
∠POS = ∠POR - ∠ROS    ...(3)
From (2) and (3),
∴ ∠QOS - ∠POS = (∠QOR - ∠POR) + 2∠ROS = 2∠ROS | Using (1)
$rightwards double arrow space angle ROS equals 1 half left parenthesis angle QOS minus angle POS right parenthesis$

Some More Questions From Lines and Angles Chapter

Prove “if two lines intersect each other, then the vertically opposite angles are equal.”

In figure, OP bisects ∠AOC, OQ bisects ∠BOC and OP ⊥ OQ. Show that the points A, O and B are collinear.

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Lines And Angles

In figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

$angle ROS equals 1 half left parenthesis angle QOS minus angle POS right parenthesis$

Some More Questions From Lines and Angles Chapter

Prove “if two lines intersect each other, then the vertically opposite angles are equal.”

In figure, OP bisects ∠AOC, OQ bisects ∠BOC and OP ⊥ OQ. Show that the points A, O and B are collinear.

In figure, if y = 20°, prove that the line AOB is a straight line.

In the given figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex angle EOC.

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

In figure, ray OS stands on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS and ∠SOQ, respectively. If ∠POS = x, find ∠ROT.

If the complement of an angle is one-third of its supplement, find the angle.

In figure, find the value of x.

In figure, if ∠POR and ∠QOR form a linear pair and a - b = 80° then find the value of a and b.

What value of x would make XOY a line if ∠XOZ = 7x + 20° and ∠YOZ = 3x in figure.

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