Question

In the given figure ∠OPQ =30 degrees and ∠ORQ = 57 degrees Then, the measure of ∠POR is

Answer 1

Given :

• ∠OPQ = 30°
• ∠ORQ = 57°

To find :

• Measure of ∠POR

According to the question,

∠OQP = ∠ORQ

∠ORQ = 57°

Reason : Radius are equal. So, the angles are also equal.

In △QOR :

∠ORQ + ∠OQR + ∠QOR = 180°

Reason : Sum of all angles of a triangle is 180°

57° + 57° + ∠QOR = 180°

114° + ∠QOR = 180°

∠QOR = 180° - 114°

∠QOR = 66°

• So,the measure of ∠QOR = 66°.

Now,

∠OPQ = ∠OQP

∠OQP = 30°

Reason : Radius are equal. So, the angles are also equal.

In △POQ

∠OPQ + ∠OQP + ∠POQ = 180°

Reason : Sum of all angles of a triangle is 180°

30° + 30° + ∠POQ = 180°

60° + ∠POQ = 180°

∠POQ = 180° - 60°

∠POQ = 120°

So,

∠POR = ∠POQ - ∠QOR

∠ POR = 120° - 66°

∠POR = 54°

• So, option C) 54° is correct.

Answer 2

$\Large\bf{\color{indigo}GiVeN,}$

• In the attached picture,

• $\bf{\angle{OPQ}\:=\:30°}$

• $\bf{\angle{ORQ}\:=\:57°}$

☆ OP, OQ & OR are the radius of the circle, i.e.

• OP = OQ = OR

$\Large\bf{In\:\triangle{POQ},}$

$\red\checkmark\:\:\bf{OP\:=\:OQ}$

$\bf\pink{We\:know\:that,}$

• Opposite angles of two equal sides is equal.

$:\implies\:\:\bf{\angle{OPQ}\:=\:\angle{OQP}\:=\:30°}$

$\Large\bf{In\:\triangle{OQR},}$

$\red\checkmark\:\:\bf{OQ\:=\:OR}$

$\bf\pink{We\:know\:that,}$

• Opposite angles of two equal sides is equal.

$:\implies\:\:\bf{\angle{OQR}\:=\:\angle{ORQ}\:=\:57°}$

$\Large\bf{In\:\triangle{OQR},}$

$\longmapsto\:\:\bf{\angle{OQR}\:+\:\angle{ORQ}\:+\:\angle{ROQ}\:=\:180°\:}$

$\longmapsto\:\:\bf{57°\:+\:57°\:+\:\angle{ROQ}\:=\:180°\:}$

$\longmapsto\:\:\bf{\angle{ROQ}\:=\:180°\:-\:114°\:}$

$\longmapsto\:\:\bf{\angle{ROQ}\:=\:66°\:}$

$\Large\bf{In\:\triangle{OQS},}$

$\longmapsto\:\:\bf{\angle{OQS}\:+\:\angle{OSQ}\:+\:\angle{SOQ}\:=\:180°\:}$

$\longmapsto\:\:\bf{30°\:+\:\angle{OSQ}\:+\:66°\:=\:180°\:}$

$\longmapsto\:\:\bf{\angle{OSQ}\:=\:180°\:-\:96°\:}$

$\longmapsto\:\:\bf{\angle{OSQ}\:=\:84°\:}$

$\bf\pink{We\:know\:that,}$

• The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

$\longmapsto\:\:\bf{\angle{OSQ}\:=\:\angle{OPS}\:+\:\angle{POS}\:}$

$\longmapsto\:\:\bf{84°\:=\:30°\:+\:\angle{POS}\:}$

$\longmapsto\:\:\bf{\angle{POR}\:=\:84°\:-\:30°\:}$

$\longmapsto\:\:\bf\green{\angle{POR}\:=\:54°\:}$

$\orange\bigstar\:\:{\boxed{\boxed{\blue{\bf{Correct\:Option\:\longrightarrow\:(c)\:54°\:}}}}}$