Quadrilateral ABCD is inscribed in a circle with m∠A=(x2)° , m∠B=(9x−2)° , and m∠C=(11x)° . Find m∠D .

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m∠D = ​
​°

Step-by-step explanation:

is inscribed in a circle, then its opposite angles sum up to 180�.

In other words, If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

I think if the teacher gave you this problem, you should be familiar with this fact.

If not, you can read about it in the lesson A property of the angles of a quadrilateral inscribed in a circle in this site.

OK. Based on this fact, you can write an equation for the opposite angles A and C

x%5E2+%2B+11x = 180, or

x%5E2+%2B+11x+-+180 = 0.

Factor left side:

(x-9)*(x+20) = 0.

The only positive root is x = 9.

Hence, angle A is 81� and angle C is 11*9� = 99�. (which is not so important for us now).

What is really important, it is the fact that angle B is 9*9�-2� = 79�.

Then the opposite to B angle D = 180� - 79� = 101�.

Answer. Angle D has the measure of 101�.