Question

If below fig, ∠AOF and ∠FOG form a linear pair.

∠EOB = ∠FOC = 90° and ∠DOC = ∠FOG = ∠AOB = 30°
(i) Find the measures of ∠FOE, ∠COB and ∠DOE.
(ii) Name all the right angles.
(iii) Name three pairs of adjacent complementary angles.
(iv) Name three pairs of adjacent supplementary angles.
(v) Name three pairs of adjacent angles.

Answer 1

(i)  `∠`FOE = x, `∠`DOE = y and `∠`BOC = z sat

Since `∠`AOF , `∠`FOG is Linear pair

⇒`∠`AOF + 30° = 180°              [`∠`AOF + `∠`FOG = 180° and `∠`FOG = 30°]

⇒ `∠`AOF = 180° - 30°

⇒ `∠`AOF = 150°

⇒ `∠` AOB + `∠`BOC + `∠`COD + `∠`DOE + `∠`EOF = 150°

⇒ 30° + z + 30° + y + x = 150°

⇒ x + y + z = 150° - 30° - 30°

⇒ x + y + z = 90°                          .....(1)

Now `∠`FOC = 90°

⇒ `∠`FOE + `∠`EOD + `∠`DOC = 90°

⇒ x + y + 30° = 90°

⇒ x + y = 90° - 30°

⇒ x + y = 60°                              .....(2)

Substituting (2) in (1)

x + y + z = 90°

⇒ 60 + z = 90° Þ z = 90° - 60° = 30°

i.e., `∠`BOC = 30°

Given `∠`BOE = 90°

⇒`∠`BOC + `∠`COD + `∠`DOE = 90°

⇒ 30° + 30° + `∠`DOE = 90°

⇒ `∠`DOE = 90° - 60° = 30°

∴ `∠`DOE = x = 30°

Now, also we have

x + y = 60°

⇒ y = 60° - x = 60° - 30° = 30°

`∠`FOE = 30

(ii) Right angles are

`∠`DOG, `∠`COF , `∠`BOF , `∠`AOD

(iii) Three pairs of adjacent complementary angles are

`∠`AOB, `∠`BOD;

`∠`AOC, `∠`COD;

`∠`BOC, `∠`COE

(iv) Three pairs of adjacent supplementary angles are

`∠`AOB, `∠`BOG;

`∠`AOC, `∠`COG;

`∠`AOD, `∠`DOG.

(v) Three pairs of adjacent angles

`∠`BOC, `∠`COD;

`∠`COD, `∠`DOE;

`∠`DOE, `∠`EOF ,