Q34 of 186 Page 4

Find a cyclic quadrilateral ABCD, ∠A = (2x + 4)°, ∠B = (y + 3)°, ∠C = (2y + 10) and ∠D = (4x — 5)°. Find the four angles.


We know that, in a cyclic quadrilateral, the sum of two opposite angles is 180°


∴A + C = 180° and B + D = 180°


⇒ 2x + 4 + 2y + 10 = 180 and y + 3 + 4x – 5 = 180


⇒ 2x + 2y = 180 – 14 and 4x + y – 2 = 180


⇒ x + y = 83 and 4x + y = 182


So, we get pair of linear equation i.e.


x + y = 83 …(i)


4x + y = 182 …(ii)


On subtracting Eq.(i) from (ii), we get


4x + y – x – y = 182 – 83


⇒ 3x = 99


⇒ x = 33


On putting the value of x = 33 in Eq. (i) we get,


33 + y = 83


⇒ y = 83 – 33 = 50


On putting the values of x and y, we calculate the angles as


A = (2x + 43)° = 2(33) + 4 = 66 + 4 = 70°


B = (y + 3)° = 50 + 3 = 53°


C = (2y + 10)° = 2(50) + 10 = 100 + 10 = 110°


and D = (4x — 5)° = 4(33) – 5 = 132 – 5 = 127°


Hence, the angles are A = 63°, B = 57°, C = 117°, D = 123°


More from this chapter

All 186 →
32

Five years ago, A was thrice as old as B and ten years later, A shall be twice as old as B. What are the present ages of A and B?

 33

Ten years hence, a man's age will be twice the age of his son. Ten years ago, the man was four times as old as his son. Find their present ages.

 35

Find the four angles of a cyclic quadrilateral ABCD in which ∠A = (2x — 3)°,

∠B = (y + 7)°, ∠C = (2y + 17)° and ∠D = (4x — 9)°.


∠A = 63°, ∠B = 57°, ∠C = 117°, ∠D = 123°.

 36

In a ΔABC, ∠C = 3 ∠B = 2 (∠A + ∠B). Find the three angles.

∠A = 20°, ∠B = 40°, ∠C = 120°.