MathematicsChapter 10: Tangents and Angles of a Circle
Cyclic Quadrilaterals
A cyclic quadrilateral has all four vertices on one circle.
Learning objectives
- Explain cyclic quadrilaterals using the correct definitions, conditions, and notation.
- Apply all four on a circle, opposite angles sum to 180 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
A cyclic quadrilateral has all four vertices on one circle. Opposite angles are supplementary: A+C=180 deg and B+D=180 deg. Converse (named): if opposite angles of a quadrilateral sum to 180, it is cyclic. If angle A=110 then angle C=70. If A=70 then C=110.
Mathematical form
C=180 deg and B+D
A=110 then angle C
A=70 then C
A = 110 deg. Find angle C. opposite angles o
Where / symbols
- CYCLIC QUADRILATERALS
- A cyclic quadrilateral has all four vertices on one circle. Opposite angles are supplementary: A+C=180 deg and B+D=180 deg. Converse (named): if opposite angles of a quadrilateral sum to 180, it is cyclic. If angle A=110 then angle C=70. If A=70 then C=110.
- THE THEOREM
- A+C=180, B+D=180. If A=110, C=70. If B=115, D=65. Adjacent angles need not be supplementary unless a side is a diameter story or something else is given.
- CONVERSE AND CHECK
- A rectangle is cyclic (each opposite pair 90+90). A random kite may not be. To show cyclic, show one pair of opposite angles sums to 180, or use equal segment angles.
Overview
Supplementary opposites:
The Theorem
A+C=180, B+D=180. If A=110, C=70. If B=115, D=65. Adjacent angles need not be supplementary unless a side is a diameter story or something else is given.
Examples 110 + C = 180 so C=70 · B=115 -> D=65 · Not each angle 90 unless a rectangle
Converse and Check
A rectangle is cyclic (each opposite pair 90+90). A random kite may not be. To show cyclic, show one pair of opposite angles sums to 180, or use equal segment angles.
Examples Rectangle is cyclic · P=85, Q=95 does not force R without the cyclic fact · If cyclic and P=85 then R=95
WHAT — CYCLIC QUADRILATERALS
A cyclic quadrilateral has all four vertices on one circle. Opposite angles are supplementary: A+C=180 deg and B+D=180 deg. Converse (named): if opposite angles of a quadrilateral sum to 180, it is cyclic. If angle A=110 then angle C=70. If A=70 then C=110. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Cyclic Quadrilaterals matters because tangent and circle-angle theorems explain the geometry at a point of contact. They support proofs, numerical angle questions, and practical circle constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE THEOREM, then CONVERSE AND CHECK. Worked: cyclic ABCD, angle A = 110 deg. Find angle C. opposite angles of a cyclic quadrilateral are supplementary angle A + angle C = 180 110 + angle C = 180 angle C = 70 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- cyclic ABCD, angle A = 110 deg. Find angle C.
- opposite angles of a cyclic quadrilateral are supplementary
- angle A + angle C = 180
- 110 + angle C = 180
- angle C = 70 deg
In simple words
Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
Key points
- all four on a circle
- opposite angles sum to 180
- A+C=180
- converse is a test
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Cyclic Quadrilaterals
Sticky Notes · Quick Revision
Core Idea
- A cyclic quadrilateral has all four vertices on one circle. Opposite angles are supplementary: A+C=180 deg and B+D=180 deg. Converse (named): if opposite angles of a quadrilateral sum to 180, it is cyclic. If angle A=110 then angle C=70. If A=70 then C=110.
THE THEOREM
- A+C=180, B+D=180. If A=110, C=70. If B=115, D=65. Adjacent angles need not be supplementary unless a side is a diameter story or something else is given.
CONVERSE AND CHECK
- A rectangle is cyclic (each opposite pair 90+90). A random kite may not be. To show cyclic, show one pair of opposite angles sums to 180, or use equal segment angles.
Worked Method
- Worked: cyclic ABCD, angle A = 110 deg. Find angle C.
- opposite angles of a cyclic quadrilateral are supplementary
- angle A + angle C = 180
- 110 + angle C = 180
- angle C = 70 deg
Common Misconception
- Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
FBISE Exam Check
- Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
60-second revision
- A cyclic quadrilateral has all four vertices on one circle.
- THE THEOREM: A+C=180, B+D=180.
- CONVERSE AND CHECK: A rectangle is cyclic (each opposite pair 90+90).
- Worked-method reminder: angle C = 70 deg
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