ABCD is an inscribed quadrilateral. AD, BC are produced to meet at X. If angle B= 80 degrees and angle X= 30 degrees, find angle XCD.
I will assume you mean "inscribed in a circle"
In a cyclic quadrilateral, the opposite angles are supplementary, so angle D = 100°
Also angle ADC is an exterior angle in triangle DCX, so
angle DCX + angle CXD = angle ADC
angle DCX + 30 = 100
angle DCX = 70°
30
50 degrees
To find angle XCD, we need to use a property of inscribed quadrilaterals. In an inscribed quadrilateral, the opposite angles are supplementary, which means they add up to 180 degrees.
In this case, angle A and angle C are opposite angles. Since we don't have the measures of angle A and angle C directly, we need to find them using the given information.
Let's start by finding angle A.
We know that angle B is 80 degrees and angle X is 30 degrees. Angle A and angle B are adjacent angles, and the sum of adjacent angles in a straight line is 180 degrees. Therefore, angle A = 180 degrees - angle B.
Angle A = 180 degrees - 80 degrees
Angle A = 100 degrees
Now, let's find angle C.
Angle X and angle C are opposite angles. Since angle X is given as 30 degrees, angle C = 180 degrees - angle X.
Angle C = 180 degrees - 30 degrees
Angle C = 150 degrees
Now that we have the measures of angle A and angle C, we can find angle XCD.
Since angle XCD and angle A are adjacent angles, their sum is 180 degrees.
Angle XCD + Angle A = 180 degrees
Angle XCD + 100 degrees = 180 degrees
Therefore, rearranging the equation, we find:
Angle XCD = 180 degrees - 100 degrees
Angle XCD = 80 degrees
So, angle XCD is 80 degrees.