In a cyclic quadrilateral ABCD, the opposite interior angles are given as linear expressions: ∠A=(4y+20)
∘
, ∠B=(3y−5)
∘
, ∠C=(4x)
∘
, and ∠D=(7x+5)
∘
. Find the values of the ordered pair (x,y).

A core property of a cyclic quadrilateral states that the sum of opposite interior angles is always equal to 180
∘
.
Therefore, the opposite angle pairs satisfy:

∠A+∠C=180
∘


∠B+∠D=180
∘


Let's set up the system of linear equations using the given expressions:

Equation 1:

(4y+20)+4x=180
4x+4y=160⟹x+y=40— (i)
Equation 2:

(3y−5)+(7x+5)=180
7x+3y=180— (ii)
Now, solve the simultaneous equations. From equation (i), express y in terms of x:

y=40−x
Substitute this value of y into equation (ii):

7x+3(40−x)=180
7x+120−3x=180
4x=60⟹x=15
∘

Substitute x=15
∘
back into the expression for y:

y=40−15=25
∘

Thus, the ordered pair solution (x,y) is (15
∘
,25
∘
).