Mathematics
In the figure, given alongside, AM bisects angle A and DM bisects angle D of parallelogram ABCD. Prove that : ∠AMD = 90°.
Rectilinear Figures
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Answer
In a parallelogram,
Sum of consecutive angles equal to 180°.
∴ ∠A + ∠D = 180° …….(1)
Given,
AM bisects angle A and DM bisects angle D of parallelogram ABCD.
∴ ∠MDA = and ∠DAM =
In △ AMD,
By angle sum property of triangle,
⇒ ∠MDA + ∠DAM + ∠AMD = 180°
⇒ + ∠AMD = 180°
⇒ + ∠AMD = 180°
⇒ + ∠AMD = 180° [From equation (1)]
⇒ 90° + ∠AMD = 180°
⇒ ∠AMD = 180° - 90° = 90°.
Hence, proved that ∠AMD = 90°.
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Related Questions
If the diagonals of a quadrilateral bisect each other at right angle, the quadrilateral is :
square
rhombus
parallelogram
rectangle
State, 'true' or 'false' :
(i) The diagonals of a rectangle bisect each other.
(ii) The diagonals of a quadrilateral bisect each other.
(iii) The diagonals of a parallelogram bisect each other at right angle.
(iv) Each diagonal of a rhombus bisects it.
(v) The quadrilateral, whose four sides are equal, is a square.
(vi) Every rhombus is a parallelogram.
(vii) Every parallelogram is a rhombus.
(viii) Diagonals of a rhombus are equal.
(ix) If two adjacent sides of a parallelogram are equal, it is a rhombus.
(x) If the diagonals of a quadrilateral bisect each other at right angle, the quadrilateral is a square.
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