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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.
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Cyclic Quadrilaterals
What is it?
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.
In simple words
- 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
Important terms
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.
What you should learn
- 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.
Examples
- 110 + C = 180 so C=70
- B=115 -> D=65
- Not each angle 90 unless a rectangle
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.
Recommended for Class 10
Full school-level explanation
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.
Supplementary opposites:
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.
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.
Remember
- A+C=180
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.
Remember
- converse is a test
Examples
- Rectangle is cyclic
- P=85, Q=95 does not force R without the cyclic fact
- If cyclic and P=85 then R=95
Remember
- all four on a circle
- opposite angles sum to 180
- 70 deg
- sum 180, not equal
- A+C=180
- converse is a test
- all four on a circle
- opposite angles sum to 180
How the ideas connect
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
Deeper understanding
Go beyond memorising Cyclic Quadrilaterals. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
THE THEOREM
CONVERSE AND CHECK
Worked Method
Exam focus
Quick revision
- all four on a circle
- opposite angles sum to 180
- 70 deg
- sum 180, not equal
- CYCLIC QUADRILATERALS: A cyclic quadrilateral has all four vertices on one circle. Opposite angles are supplementary: A+C=1
- 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
Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
For long answers on Cyclic Quadrilaterals: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- all four on a circle
- opposite angles sum to 180
- 70 deg
- sum 180, not equal
Important formulas
- all four on a circle
- opposite angles sum to 180
- 70 deg
Common exam traps
- Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
- Not each angle 90 unless a rectangle
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
- Mistake
Not each angle 90 unless a rectangle
- Mistake
Do not treat THE THEOREM and CONVERSE AND CHECK as the same step; each has a different purpose.
- Mistake
Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
all four on a circle
opposite angles sum to 180
70 deg
sum 180, not equal
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
A+C=180
- 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
- Formula
- A+C=180
- Example
- 110 + C = 180 so C=70
- Critical warning
- Supplementary means sum 180, not equal. 110 and 110 would be 220, which is not this theorem.
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Exam Practice
Define
Q1. Define Cyclic Quadrilaterals.
Explain
Q2. Explain: THE THEOREM.
Example
Q3. Give one clear example of Cyclic Quadrilaterals and say why it fits.
Mistake
Q4. State one common mistake students make about Cyclic Quadrilaterals.
Exam note
Q5. Write a short exam note on Cyclic Quadrilaterals: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.