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72
Chapter - 8
(Quadrilaterals)
Key Concept (1) Sum of the angles of a quadrilateral is 360
0
. (2) A diagonals of a parallelogram divides it into two congruent triangles. (3) In a parallelogram (a) diagonals bisects each other. (b) opposite angles are equal. (c) opposite sides are equal (4) Diagonals of a square bisects each other at right angles and are equal, and vice-versa. (5) A line through the mid-point of a side of a triangle parallel to another side bisects the third side. (Mid point theorem) (6) The line through the mid points of sides of a
║
to third side and half of it.
Section - A
Q.1 The figures obtained by joining the mid-points of the sides of a rhombus, taken in order, is (a) a square (b) a rhombus (c) a parallelogram (d) a rectangle Q.2 The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O, if
then
is (a) 32
0
(b) 24
0
(c) 40
0
(d) 63
0
Q.3 In a square ABCD, the diagonals AC and BD bisect at 0. Then
is (a) acute angled (b) right angled (c) obtuse angled (d) equilateral
73
Q.4 ABCD is a rhombus such that
then
is (a) 40
0
(b) 45
0
(c) 50
0
(d) 60
0
Q.5 A quadrilateral ABCD is a parallelogram if (a) AD || BC (b) AB = CD (c) AB = AD (d)
Q.6 Three angles of a quadrilateral are 60
0
, 70
0
and 80
0
. The fourth angle is (a) 150
0
(b) 160
0
(c) 140
0
(d) None of these
Section - B
Q.7 In the adjoining figure QR=RS Find
Q.8 Prove that the sum of the four angles of a quadrilateral is 360
0
. Q.9 Prove that the diagonals of a parallelogram bisects each other. Q.10 The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral. Q.11 ABCD is a rectangle in which diagonal AC bisects
as well as
. Show that ABCD is a square.
74
Q.12 In the adjoining figure, ABCD is a ||gm. If
. Find
.
Section - C
Q.13 Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side. Q.14 ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus. Q.15 Prove that the straight line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides and is equal to half their difference. Q.16 In the adjoining figure, D, E and F are mid-points of the sides BC, CA and AB of
If AB = 4.3cm, BC = 5.6cm and AC = 3.5cm, find the perimeter of
Q.17 In a parallelogram ABCD, AP and CQ are drawn perpendiculars from vertices A and C on diagonal BD. Prove that
Q.18 In a parallelogram ABCD, E and F are points on AB and CD such that AE = CE.
75
Prove that ED||BF.
Section - D
Q.19 If a line is parallel to the base of a trapezium and bisects one of the non-parallel sides, then prove that it bisects either diagonal of the trapezium. Q.20 AD is a median of
and E is the mid-point of AD. BE Produced meets AC in F. Prove that
Q.21 ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that (i) D is the mid-point of AC (ii) CM =
Q.22 Show that the bisectors of angles of a parallelogram form a rectangle.
Answers
- Q.1 (d) Rectangle Q,2 (c) 40
0
Q.3 (b) Right angled Q.4 (c) 50
0
Q.5 (d)
Q.6 (a) 150
0
Q. 7
0
------

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