Three corners of a quadrilateral inscribed in a circle, taken in the order of the following, are in the ratio 2: 6: 7.

Let a quadrilateral ABCD be given, inscribed in a circle. If x is a coefficient of proportionality, then ∠A = 2 * x, ∠B = 6 * x, ∠C = 7 * x.

1. Only a quadrilateral in which the sums of opposite sides are pairwise equal can be inscribed into a circle, that is, in the given quadrilateral ABCD, the equality must be fulfilled:

∠A + ∠C = ∠B + ∠D.

It is known that the sum of all angles of a quadrilateral is 360 °, then:

∠A + ∠B + ∠C + ∠D = 360 °.

Let’s substitute the data on the value condition in both expressions:

2 * x + 7 * x = 6 * x + ∠D;

2 * x + 6 * x + 7 * x + ∠D = 360 °.

We have obtained systems of linear equations in two variables.

Let us present similar terms in the first equation and express ∠D:

2 * x + 7 * x – 6 * x = ∠D;

∠D = 3 * x.

Let us present similar terms in the second equation and express ∠D:

∠D = 360 ° – 2 * x – 6 * x – 7 * x;

∠D = 360 ° – 15 * x.

Let’s equate both expressions:

3 * x = 360 ° – 15 * x;

3 * x + 15 * x = 360 °;

18 * x = 360 °;

x = 360 ° / 18;

x = 20 °.

2. Find the degree measures of angles:

∠A = 2 * x = 2 * 20 ° = 40 °.

∠B = 6 * x = 6 * 20 ° = 120 °.

∠C = 7 * x = 7 * 20 ° = 140 °.

∠D = 3 * x = 3 * 20 ° = 60 °.