**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 = 90^{o} [AP ⊥CD & BQ ⊥ CD]

∴ △APD ≅ △BQC by RHS

∴ ∠DAP = ∠CBQ [∵ corresponding angles of congruent triangles are equal]

∠BAP = ∠ABQ = 90^{o} [∵ AP ⊥ AB & BQ ⊥ AB]

∴ ∠A = BAP + DAP

= 90 + DAP

∠B = AŌQ + 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 360^{o}

Let, x be the factor that multiplies with each angle of the quadrillateral

x + 2x + 3x + 4x = 360^{o}

Or, 10x = 360^{o}

Or, x = 36^{o}

∴ The measures of each angles of the quadrilateral are 1 x 36 = 36^{o}, 2 x 36 = 72^{o}, 3 x 36 = 108^{o},

4 x 36 = 144^{o}

**(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 = 90^{o} [all angles of a rectangle is 90^{o} and opposite sides are parallel also adjacent sides are perpendicular to each other]

△ABC is a right angled triangle at ∠B = 90^{o} with AC being the hypotenuse.

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Yep 3rd is also totally wrong

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Is there any mistake in representation of 3 answer

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