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
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.
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
Answer: 78 degrees
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