Circle Theorems: Cyclic Quadrilaterals and the Same Segment
Use the equal angles subtended by the same arc, and the opposite angles of a cyclic quadrilateral adding to 180 degrees.
What a learner can do afterwards
- Find equal angles standing on the same arc in a busy diagram
- Use opposite angles of a cyclic quadrilateral to find a missing angle
- Give the theorem name with each step rather than the answer alone
1 · Read
Angles standing on the same arc are equal, but only on the same side of the chord. Points C and D on the same arc see chord AB under the same angle: if angle ACB is 42 degrees, angle ADB is 42 too.
A cyclic quadrilateral has all four corners on the circle. Opposite angles total 180 degrees. With angle A at 70 degrees, angle C is 110, since 180 minus 70 is 110. With angle Q at 115 degrees, angle S is 65.
Give the theorem name with each step, not just the answer. Say Same Segment Theorem for a shared arc step, and Cyclic Quadrilateral Theorem for an opposite angle step.
Both rules can combine in one question. In cyclic WXYZ with angle W 80 degrees, angle Y is 100 by the Cyclic Quadrilateral Theorem. Angle WXZ matches angle WMZ at 35 degrees by the Same Segment Theorem, since X and M share one arc.
Match shared arcs for equal angles, total opposite angles to 180, and name the theorem each time.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.