Question

In the given figure, ABCF is a parallelogram, DME,
DGB and DFC are straight lines. The measure of
ADG – MDF is
56°
72°
52°
A B
E
C F
M
G
D
(1) 14°
(2) 5°
(3) 16°
(4) 12°

Answer 1

Answer: Option (1): 14°

Step-by-step explanation:

Given data:

ABCF is a parallelogram

∠DAF = 56°

∠GBE = 90°

∠MEB = 72°

∠FCE = 52°

To find: the measure of [∠ADG – ∠MDF]

Solution:

Step 1:

∠BGA = ∠GBE = 90° ….. [alternate interior angles] ….. (i)

Since the exterior angles of a triangle is equal to the sum of opposite interior angles.

So, considering ∆ ADG,

∴ ∠BGA = ∠DAG + ∠ADG

⇒ 90° = 56° + ∠ADG ….. [substituting value from (i) and ∠DAF = ∠DAG]

⇒ ∠ADG = 34° ….. (ii)

Step 2:

∠CEM = 180° – ∠MEB ….. [linear pairs]  

⇒ ∠CEM = 180° – 72° = 108°

Step 3:

In ∆ DEC, applying the angle sum property,

∠CED + ∠DCE + ∠EDC = 180°

⇒∠CEM + ∠FCE + ∠EDC = 180°

⇒ ∠EDC = 180° - [108°+52°] = 20° = ∠MDF….. (iii)

Step 4:

Thus, from (ii) & (iii), we get

∠ADG – ∠MDF

= 34° - 20°

= 14°