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.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 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.