Telangana SCERT Solution Class IX (9) Math Chapter 8 Quadrilaterals Exercise 8.1
(1) (i) Every parallelogram is a trapezium
(ii) All parallelograms are quadrilaterals
(iii) All trapeziums are parallelograms
(iv) A square is a rhombus
(v) Every rhombus is a square
(vi) All parallelograms are rectangles
(i) False, trapezium has only two side’s parallel.
(ii) True
(iii) False, Parallel has opposite sided parallel
(iv) True
(v) False, A spare has to have all right angles bat a rhombus.
(v) False, a rectangle has to have all angles to be right angles.
(2) Complete the following table by writing (YES) if the property holds for the particular Quadrilateral and (NO) if property does not holds.
| Properties | Trapezium | Parallelogram | Rhombus | Rectangle | Square |
| a) only one pair of opposite sides are parallel | Yes | No | No | No | No |
| b) Two pairs of opposite sided are parallel | No | Yes | Yes | Yes | Yes |
| c) Opposite sides are equal | No | Yes | Yes | Yes | Yes |
| d) Opposite angles are equal | No | Yes | Yes | Yes | Yes |
| e) Consecutive angles are supplementary | No | Yes | Yes | Yes | Yes |
| f) Diagonals bisects each other | No | Yes | Yes | Yes | Yes |
| g) Diagonals are equal | No | No | No | Yes | Yes |
| h) All sides are equal | No | No | Yes | No | Yes |
| i) Each angled is right angles | No | No | No | Yes | Yes |
| j) Diagonals are ____ to each other. | No | No | Yes | No | Yes |
(3) ABCD is trapezium in which AB || CD. If AD = BC, show that ∠A = ∠B and ∠C = ∠D
Solution: Given,
ABCD is a trapezium
AB ∥ CD, AC = BD
Now,
Draw two altitudes of trapezium ABCD
AP ⊥ CD ⊥ AQ [∵ AB ∥ CD]
BQ ⊥ CD ⊥ AB [∵ AB ∥ CD]
Now, In APD & BQC
(i) AD = ABC [given]
(ii) AP = BQ
Since, AP and BQ are two altitudes of the parallel side AB and CD
(iii) APD = BQC = 90o [AP ⊥CD & BQ ⊥ CD]
∴ △APD ≅ △BQC by RHS
∴ ∠DAP = ∠CBQ [∵ corresponding angles of congruent triangles are equal]
∠BAP = ∠ABQ = 90o [∵ AP ⊥ AB & BQ ⊥ AB]
∴ ∠A = BAP + DAP
= 90 + DAP
∠B = AÅŒQ + CBQ
= 90 + CBQ
∴ ∠A = ∠B [∵∠DAP = ∠CBQ] Hence shown
C = D [∵ corresponding angles of congruent triangles are equal] Hence shown.
(4) The four angles of a quadrilateral are in the ratio 1: 2:3:4. Find the measure of each angle of the quadrilateral.
Solution: Given,
Ration of the angles of quadrilateral is 1 : 2 : 3 : 4
We know, sum of angles of a quadrilateral is 360o
Let, x be the factor that multiplies with each angle of the quadrillateral
x + 2x + 3x + 4x = 360o
Or, 10x = 360o
Or, x = 36o
∴ The measures of each angles of the quadrilateral are 1 x 36 = 36o, 2 x 36 = 72o, 3 x 36 = 108o,
4 x 36 = 144o
(5) ABCD is a rectangle AC is diagonal. Find the nature of ΔACD. Give reasons
Solution: Given,
ABCD is a rectangle
AC is the diagonal
∠B = 90o [all angles of a rectangle is 90o and opposite sides are parallel also adjacent sides are perpendicular to each other]
△ABC is a right angled triangle at ∠B = 90o with AC being the hypotenuse.
