In the given figure, ABCD is a cyclic quadrilateral. Then the ordered pair (x,…

In a cyclic quadrilateral (a quadrilateral whose four vertices lie on a circle), every pair of opposite interior angles is supplementary — each such pair sums to 180°.

1. From the figure, angle A = (3y − 5)°, angle B = 4x°, angle C = (7x + 5)°, and angle D = (4y + 20)°.

2. Angles A and C are opposite, so they are supplementary: (3y − 5) + (7x + 5) = 180, which simplifies to 7x + 3y = 180.

3. Angles B and D are opposite, so they are supplementary too: 4x + (4y + 20) = 180, which simplifies to 4x + 4y = 160, i.e. x + y = 40, so y = 40 − x.

4. Substituting y = 40 − x into 7x + 3y = 180 gives 7x + 3(40 − x) = 180, i.e. 7x + 120 − 3x = 180, i.e. 4x = 60, so x = 15.

5. Then y = 40 − 15 = 25, giving the ordered pair (x, y) = (15°, 25°).

Check: angle A = 3(25) − 5 = 70° and angle C = 7(15) + 5 = 110°, and 70° + 110° = 180°. Also, angle B = 4(15) = 60° and angle D = 4(25) + 20 = 120°, and 60° + 120° = 180°. All four angles also add up to 360° (70 + 60 + 110 + 120 = 360), confirming the quadrilateral's angle-sum property as well.

So the ordered pair (x, y) = (15°, 25°).