# Given a quadrilateral ABCD. It is known that the angle ABC = 124 degrees, the angle ADC = 56 degrees

Given a quadrilateral ABCD. It is known that the angle ABC = 124 degrees, the angle ADC = 56 degrees, the angle BAC = 32 degrees, CAD = 54 degrees a) Is it possible to describe a circle around this quadrilateral? b) Find the angle between the diagonals of the quadrangle opposite to the side AB

a)
If the sum of opposite angles in a quadrangle is 180, then a circle can be described around it. Let’s check the angle ABC + angle ADC = 124 + 56 = 180. This means that a circle can be inscribed around the quadrilateral ABCD.
b)
Let the diagonals of the quadrilateral ABCD AB and CD meet at point O. We need to find the angle BOA. Consider a triangle ACD. The sum of the angles in the triangle is 180 degrees, so the angle ACD = 180 – CAD angle – ADC angle = 180 – 54 – 56 = 70 degrees. Since a circle can be inscribed around the quadrilateral ABCD, the angle BAD + angle BCD = 180
BAC angle + CAD angle + BCA angle + ACD angle = 180
32 + 54 + BCA angle + 70 = 180
ICA angle = 180 – 32 – 54 – 70
ICA angle = 24
BCA angle = BDA angle, since these angles are based on the same arc, so the angle BDA = 24
Angle ABD = angle ACD, since these angles rest on the same arc, so angle ABD = 70
Consider a triangle ABD. Find the corner of the BOA. BOA angle = 180 – ОВА angle – ОАВ angle = 180 – 32 – 70 = 78 degrees 