Question

The angle between the two altitudes of a parallelogram through the vertex of an obtuse

angle of the parallelogram is 45 degrees.Find the angles of the parallelogram.​

Answer 1

ANSWER

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=60)

To find out

The angles of the parallelogram ABCD.

Solution-

∠DOM+∠NOM=180

(linear pair)

⟹60

+∠NOM=180 ⟹∠NOM=120

(i)

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

∴ In the quadrilateral AMON,

∠AMO+∠DNO=180

∴∠NOM+∠MAN=360

−180

=180 (sum of the angles of a quadrilateral =360)

⟹∠MAN=180

−120 (from i )

i.e ∠A=60

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

∴∠D+∠B=360

−120

=240

∴∠D=∠B=

2

240

=120 (opposite angles of a paralleogram)

Ans- ∠A=∠C=60

,∠D=∠B=120

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