Two Roads to One Angle
Part of the Circles section of Coddy's Geometry journey. Lesson 54 of 62.
A circle chase often has more than one road. Different rules can lead to the same angle, and any correct road is fine. Seeing two roads is also a good check: if both give the same number, you can trust it.
Here AC is a diameter, ∠BAC = 40°, and the question is ∠BDC. Road one: ∠BDC stands on the chord BC from the same side as ∠BAC, so it is 40° at once, by Angles on the same arc. Road two goes through the center: OA and OB are radii, so ∠OBA = 40°, the exterior angle ∠BOC is 40° + 40° = 80°, and the inscribed angle ∠BDC is half of that, 40°. Same answer, so it holds.
Challenge
AC is a diameter of circle O, and B and D are on the circle on opposite sides of it. Find ∠ADB, then press Check.
Try it yourself
Find the marked part of 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 Arc11Circle Chases
Tangent Meets ChordTwo Roads to One AngleEnough to Find It?Cyclic Meets IsoscelesRecap: Circle Chases3Area 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 Angles