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What is the measure of an angle which is equal to 5 times its supplement?
Let the measure of angle be x°
∴ X° = 5(180°  x)
⇒ 6x = 5 × 180°
⇒ x = 150°
Two complementary angles are in the ratio 2 : 7. The measure of a smaller angle is:
Let the measure of angles be 2x° and 7x° respectively
∴ 2x° + 7x° = 90°
⇒ 9x = 90°
⇒ x = 10°
∴ Measure of smaller angle
= 2x = 2 × 10° = 20°
OP bisects ∠BOC and OQ, ∠AOC. Find the measure of ∠POQ.
∵ AOB is a straight line
∴ ∠AOB = 180°
∠AOC + ∠BOC = 180°
⇒ ∠COQ +∠POC = 90°
⇒ ∠POQ = 90°
∠AOC +∠ BOC = 180°
⇒ 7x + 3y = 180° ...(i), and
y  x = 10°
⇒ x = y  10° ...(ii)
Using (ii) and (i)
7(y  10°) + 3y = 180°
⇒ 10y = 250°
⇒ y = 25°
POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR, then ∠POS is equal to:
Let the measure of ∠POS be x°
∵ ∠ POQ = 180°
⇒ ∠POS + ∠ROS + ∠QOR = 180°
⇒ ∠POS + ∠ROS + 90° = 180°
⇒ ∠ROS = (90°  x).
⇒ ∠QOS = 90° + (90°  x) = 180°  x
∴ ∠POS = x = ∠QOS  2∠ROS.
q = 5p, r = 3p and
∵ ∠POQ = 180°
⇒ p + q + r = 180°
⇒ p + 5p + 3p = 180°
⇒ 9p = 180° ⇒ p = 20°
r = 3p = 3 × 20° = 60°
∴ r + p = 60° + 20° = 80°
In the adjoining figure, ∠AOQ : ∠AOP = 5 : 7, then the measure of ∠BOQ is:
∵ ∠POQ = 180° [PQ is a straight line]
⇒ ∠AOQ + ∠AOP = 180°
⇒ 5k + 7k = 180°
⇒ 12k = 180°
⇒ k = 15°
∴ ∠BOQ = ∠AOP = 7 × 15° = 105°
[vertically opposite ∠s]
In the adjoining figure, ∠AOC + ∠BOE = 70° and ∠BOD = 40°, then measure of reflex ∠BOE is
∠BOD = ∠AOC = 40°
[vertically opposite ∠s]
∵ ∠SOB is a straight angle
∴ ∠AOB = 180°
⇒ ∠AOC + ∠COE + ∠BOE = 180°
⇒ ∠COE = 180°  (∠AOC + ∠BOC)
= 180°  70° = 110°
⇒ ∠AOC + ∠BOE = 70°
∠AOE = 70°  40° = 30°
∴ reflex (∠BOC) = 360°  30 = 330°
∵ ∠COD = 180°
⇒ ∠BOC + ∠BOQ + ∠DOQ = 180°
⇒ 55° + 60° + z = 180°
⇒ z = 65°
Similarly ∠BOC = ∠AOD
⇒ 2x + 3° = 55°
⇒ x = 26°
and,
∠AOP = ∠BOQ [vertically opposite ∠s]
⇒ 5y  10° = 60°
⇒ y = 70°/5 = 14°
∴ x  y + z = 26°  14° + 65° = 77°
In the adjoining figure, AB  CD and, PQ ⊥ AB, find the measure of ∠PCM.
∠ANP + ∠PNO = 180° [Straight angle]
⇒ ∠PNO = 180°  120° = 60°
∴ ∠PNO = ∠PCM = 60° [corresponding ∠S]
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