Question

The angles of a cyclic quadrilateral are the consecutive terms of an arithmetic sequence and one of the angles is 450 . a) What is the measure of the angle opposite to 450 angle ? b) Find the measures of other two angles of the quadrilateral

Answer 1

Answer:

the opposite angles of a cyclic quadrilateral is supplementary i.e. they add up to 180°

so measure of angle opposite to 45° = 180-45

= 135°

now since angles are in AP

let angles are

a-3d, a-d, a+d, a+3d

sum of all angles = 360°

this gives

4a = 360°

a = 90°

now angles are

90-3d, 90-d, 90+d, 90+3d

we know one of the angles is 45°. since 45° is less than 90°, it must be either 90-d or 90-3d

case 1:

90-3d = 45

3d = 90-45 = 45

d = 15

then angles are

90-45,90-15,90+15,90+45

45,75,105,135

case 2

90-d i= 45

d = 45

then 90-3d = 90-3*45

= 90-135 = -45° which is not possible

hence, angles are

45,75,105,135