Question

20. The angle between two altitudes of a parallelogram through the vertex
of an obtuse angle of the parallelogram is 60°. Find the angles of the
parallelogram.​

Answer 1

Step-by-step explanation:

Given-

□ABCD is a parallelogram. 

∠B & ∠D are obtuse. 

BM & DN are altitudes from B & D on AD & AB, respectively, intersecting at O. 

∠DOM=BON        ...(vertically opposite angles=60o) 

To find out-

The angles of the parallelogram ABCD.

Solution-

∠DOM+∠NOM=180o      ...(linear pair)

⟹60o+∠NOM=180o 

⟹∠NOM=120o   .........(i)

Now, ∠AMO=∠DNO=90O       ...(both are perpendiculares on AD & AB, respectively.)$$

∴ In the quadrilateral AMON,

∠AMO+∠DNO=180O 

∴∠NOM+∠MAN=360o−180O=180o

 (sum of the angles of a quadrilateral =360o) 

⟹∠MAN=180O−120o     ...(from i )

i.e ∠A=60o 

∴∠A=∠C=60o     ....(opposite angles of a paralleogram)

∴∠D+∠B=360o−120o=240o 

∴∠D=∠B=2