Inscribed Quadrilateral

Inscribed Quadrilateral

How to solve inscribed quadrilateral problems: property, proof, example, and its solution.

Formula

The opposite interior angles of an inscribed quadrilateral are supplementary angles.

The opposite interior angles of an inscribed quadrilateral
are supplementary angles.

m∠(purple) + m∠(blue) = 180

Proof

Inscribed Quadrilateral: Proof

The purple angle is the inscribed angle
that intercept the purple arc.

So m∠(purple) = (1/2)⋅m[purple arc].

2m∠(purple) = m[purple arc]

The blue angle is the inscribed angle
that intercept the blue arc.

So m∠(blue) = (1/2)⋅m[blue arc].

2m∠(blue) = m[blue arc]

The purple arc and the blue arc form a circle.

So m[purple arc] + m[blue arc] = 360.

Put 2m∠(purple) = m[purple arc]
and 2m∠(blue) = m[blue arc]
into m[purple arc] + m[blue arc] = 360.

Then 2m∠(purple) + 2m∠(blue) = 360.

Divide both sides by 2.
Then m∠(purple) + m∠(blue) = 180.

Example

Find the value of x. The measures of the interior angles of an inscribed quadrilateral: x, y, 100, 70.

m∠(purple) + m∠(blue) = 180

x + 100 = 180

m∠(purple dot) + m∠(blue dot) = 180

70 + y = 180