Directions: Study the following questions and choose the right answer:
1
In a cyclic quadrilateral ∠A + ∠C = ∠B + ∠D = ?
 
» Explain it
D

The sum of opposite angles of a concyclic quadrilateral = 180°

∴  ∠A + ∠C = ∠B + ∠D = 180°

Hence, option D is correct.
 
2
If ABCD be a cyclic quadrilateral in which ∠A = 4x°, ∠B = 7x°, ∠C = 5y°, ∠D = y°, then x : y is
» Explain it
B

The sum of opposite angles of a concyclic quadrilateral is 180°.

∴  ∠A + ∠C = 180°

4x + 5y = 180°      ...(i)

∠B + ∠D = 180°

7x + y = 180°      ...(ii)

By equation (ii) × 5 – (i), we get

31x = 720

x =  720
31

From equation (ii),

7x + y = 180°
 
7 ×  720  + y = 180°
31

y = 180° –  5040  =  540
31 31

∴  x : y =  720  :  540  = 4 : 3
31 31

Hence, option B is correct.

3
A quadrilateral ABCD circumscribes a circle and AB = 6 cm, CD = 5 cm and AD = 7 cm. The length of side BC is
» Explain it
A

We know tangents drawn to a circle from same external point are equal

AM = AQ = x (let)

∴  MB = 6 – x = BN

QD = 7 – x = DP

PC = y (let) = CN

Now, CD = DP + PC = 5

⇒  7 – x + y = 5

⇒  y – x = – 2

BC = CN + BN

= y + 6 – x = y – x + 6 = – 2 + 6 = 4

Hence, option A is correct.
 
4
ABCD is a cyclic quadrilateral and AD is a diameter. If ∠BAC = 55° then value of ∠ADC is
» Explain it
C

∠BAC = 55°

∠ACB = 90°      [  Angle of semi-circle]

In ΔABC, we know that

∠ABC = 180° – 90° – 55° = 35°

∴  ∠ABC + ∠ADC = 180°

∠ADC = 180° – ∠ABC = 180° – 35° = 145°

Hence, option C is correcrt.
 
5
The difference between the exterior and interior angles at a vertex of a regular polygon is 150°. The number of sides of the polygon is
» Explain it
C
Let the number of sides of a polygon be n. Then,

180° –  360°  –  360°  = 150°
n n

180° × n – 720° = 150° × n

30° × n = 720°

n = 24

Hence, option C is correct.