Mathematics
Gautam had two sticks of equal length. He named them AB and CD. He then placed these two sticks in three different ways and joined their four vertices as shown below.
(i) What type of quadrilateral is in :
(a) Fig (i) ?
(b) Fig (ii) ?
(c) Fig (iii) ?
(ii) What is the sum of consecutive angles in Fig (iii) ?
(iii) If ∠BAD = 70° in Fig (iii), then what is the measure of ∠CDA ?
Rectilinear Figures
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Answer
(i)
(a) In figure (i),
One pair of opposite sides are equal in length (AB = CD).
The diagonals are equal and they bisect each other.
If one pair of opposite sides of a quadrilateral are equal, and its diagonals are equal and bisect each other, the quadrilateral is a rectangle.
Hence, the quadrilateral in figure (i) is a rectangle.
(b) In figure (ii),
The diagonals of quadrilateral are equal and bisect each other. (AB = CD)
If the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral is a rectangle.
Hence, the quadrilateral in figure (ii) is a rectangle.
(c) In figure (iii),
Sides AB and CD are equal and parallel.
The diagonals of quadrilateral bisect each other.
Since, one pair of sides are equal and parallel and diagonals bisect each other, thus the quadrilateral is a parallelogram.
Hence, the quadrilateral in figure (iii) is a parallelogram.
(ii) In a parallelogram, sum of consecutive angles is equal to 180°.
Hence, the sum of consecutive angles in figure (iii) is 180°.
(iii) Sum of adjacent angles in a parallelogram is 180°.
Thus, in figure (iii),
⇒ ∠BAD + ∠CDA = 180°
⇒ 70° + ∠CDA = 180°
⇒ ∠CDA = 180° - 70° = 110°.
Hence, ∠CDA = 110°.
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