Four Corners on a Circle
Part of the Circles section of Coddy's Geometry journey. Lesson 42 of 62.
A quadrilateral with all four corners on a circle is a cyclic quadrilateral. Its opposite angles, the ones that do not share a side, add up to 180°.
Why? The angle at A and the angle at C both stand on the diagonal BD, but from opposite sides, so they open onto the two different arcs. Their central angles are the two angles at O between OB and OD, which fill the whole turn, 360°. Each inscribed angle is half its central angle, so together they are half of 360°, which is 180°. Here 85° + 95° = 180°.
Challenge
ABCD has all four corners on circle O. Tap the angle opposite ∠1, then press Check.
Try it yourself
Find the right part on the figure
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Circles
1Parts of a Circle
Center and RadiusChords and DiametersAll Radii Are EqualDiameter Is Two RadiiRecap: Parts of a Circle4Arcs and Sectors
Arc LengthArea of a SectorFind the Central AngleThe Missing Central AngleRecap: Arcs and Sectors2Circumference
Around the CircleUsing the DiameterRadius From CircumferenceCircumference With LettersRecap: Circumference5Radii Make Isosceles
Two Radii, One TriangleBase Angles From the CenterThe Angle at the CenterIsosceles With LettersRecap: Radii and Triangles8Angles on the Same Arc
Same Arc, Same AngleUsing the Same ArcSpot the Bow TieSame Arc With LettersRecap: Same Arc3Area of a Circle
The Area FormulaArea From the DiameterRadius From the AreaCircumference to AreaRecap: Area of a Circle6Inscribed Angles
Meet the Inscribed AngleHalf the Central AngleTwice the Inscribed AngleThe Big ArcInscribed With LettersRecap: Inscribed Angles9Cyclic Quadrilaterals
Four Corners on a CircleOpposite Angles Add to 180Is It Cyclic?Cyclic With LettersRecap: Cyclic Quadrilaterals