the angle between the two altitudes through the vertex of an obtuse angle is 50 degree. find the angle of parallelogram.
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Answer:
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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
o
)
To find out-
The angles of the parallelogram ABCD.
Solution-
∠DOM+∠NOM=180
o
...(linear pair)
⟹60
o
+∠NOM=180
o
⟹∠NOM=120
o
.........(i)
Now, ∠AMO=∠DNO=90
O
...(both are perpendiculares on AD & AB, respectively.)$$
∴ In the quadrilateral AMON,
∠AMO+∠DNO=180
O
∴∠NOM+∠MAN=360
o
−180
O
=180
o
(sum of the angles of a quadrilateral =360
o
)
⟹∠MAN=180
O
−120
o
...(from i )
i.e ∠A=60
o
∴∠A=∠C=60
o
....(opposite angles of a paralleogram)
∴∠D+∠B=360
o
−120
o
=240
o
∴∠D=∠B=
2
240
o
=120
o
...(opposite angles of a paralleogram)
Ans- ∠A=∠C=60
o
,∠D=∠B=120
o