We just proved that when you inscribe a quadrilateral inside a circle, the opposite angles are supplementary.
Inscribed Quadrilateral Theorem
A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.
Let's use this theorem to solve problems involving inscribed quadrilaterals. Work through the following tabs to practice.
Example 1
Example 2
Example 3
What two equations can you write to show that opposite angles of inscribed quadrilaterals are supplementary?
m∠A + m∠C = 180°
m∠B + m∠D = 180°
Quadrilateral ABCD is inscribed in a circle. Find the measures of angles A and B.
Start with angle A. Let's call this unknown angle y. Since angles A and C are supplementary, you can write and solve the following equation:
y° + 80° = 180°
y° = 100°
Now, follow the same steps for angle B. Let's call this unknown angle x. Since angles B and D are supplementary, you can write and solve the following equation:
x° + 70° = 180°
x° = 110°
Quadrilateral ABCD is inscribed inside a circle. If angle A measures (3x + 6)° and angle C measures (2x + 4)°, find x.
Since these angles are opposite each other, they are also supplementary. You can write and solve the following equation:
m∠A + m∠C = 180°
(3x + 6) + (2x + 4) = 180
5x + 10 = 180
5x = 170
x = 34
