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Question

Complete each of the following statements by means of one of those given in brackets against each: (i) If one pair of opposite sides are equal and parallel, then the figure is ........................ (parallelogram, rectangle, trapezium) (ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is ................ (square, rectangle, trapezium) (iii) A line drawn from the mid-point of one side of a triangle .............. another side intersects the third side at its mid-point. (perpendicular to parallel to, to meet) (iv) If one angle of a parallelogram is a right angle, then it is necessarily a ................. (rectangle, square, rhombus) (v) Consecutive angles of a parallelogram are ................... (supplementary, complementary) (vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ............... (rectangle, parallelogram, rhombus) (vii) If opposite angles of a quadrilateral are equal, then it is necessarily a .................... (parallelogram, rhombus, rectangle) (viii) If consecutive sides of a parallelogram are equal, then it is necessarily a .................. (kite, rhombus, square)

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Solution

(i) If one pair of opposite sides are equal and parallel, then the figure is parallelogram. Reason: In and, (Given) (Common) (Because , Alternate interior angles are equal) So, by SAS Congruence rule, we have Also, (Corresponding parts of congruent triangles are equal) But, these are alternate interior angles, which are equal. Thus, and Hence, quadrilateral ABCD is parallelogram (ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is trapezium. (iii) A line drawn from the mid-point of one side of a triangle parallel to another side intersects the third side at its mid-point. Reason: This is a theorem. (iv) If one angle of a parallelogram is a right angle, then it is necessarily a rectangle. Reason: Let ABCD be the given parallelogram. We have, In a parallelogram, opposite angles are equal. Therefore, Similarly, Also, Thus, a parallelogram with all the angles being right angle and opposite sides being equal is a rectangle. (v) Consecutive angles of a parallelogram are supplementary. Reason: Let ABCD be the given parallelogram. Thus, . Therefore, Consecutive angles and are supplementary. (vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a parallelogram. Reason: ABCD is a quadrilateral in which and . We need to show that ABCD is a parallelogram. In and , we have (Common) (Given) (Given) So, by SSS criterion of congruence, we have By corresponding parts of congruent triangles property. â€¦â€¦ (i) And Now lines AC intersects AB and DC at A and C,such that (From (i)) That is, alternate interior angles are equal. Therefore, . Similarly, . Therefore, ABCD is a parallelogram. (vii) If opposite angles of a quadrilateral are equal, then it is necessarily a parallelogram. Reason: ABCD is a quadrilateral in which and . We need to show that ABCD is a parallelogram. In quadrilateral ABCD, we have Therefore, â€¦â€¦ (i) Since sum of angles of a quadrilateral is From equation (i), we get: Similarly, Now, line AB intersects AD and BC at A and B respectively Such that That is, sum of consecutive interior angles is supplementary. Therefore, . Similarly, we get . Therefore, ABCD is a parallelogram. (viii) If consecutive sides of a parallelogram are equal, then it is necessarily a rhombus. We have ABCD, a parallelogram with . Since ABCD is a parallelogram. Thus, And But, Therefore,all four sides of the parallelogram are equal, then it is a rhombus.

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