Cyclic Quadrilateral Theorem
Theorem: The opposite angles of a cyclic quadrilateral are supplementary. Prove that.
1. Given:
- A circle with center O.
- ABCD is a cyclic quadrilateral with vertices A, B, C, and D lying on the circumference.
2. To Prove:
- ∠DAB + ∠BCD = 180°
- ∠ABC + ∠ADC = 180°
Figure: Cyclic Quadrilateral ABCD
3. Proof:
| Statements | Reasons |
|---|---|
| 1. ∠DAB = ½ BCD | 1. Inscribed angle and its opposite arc. |
| 2. ∠BCD = ½ DAB | 2. Inscribed angle and its opposite arc. |
|
3. ∠DAB + ∠BCD = ½ BCD + ½ DAB ⇒ ∠DAB + ∠BCD = ½ (BCD + DAB) ⇒ ∠DAB + ∠BCD = ½ × 360° ⇒ ∠DAB + ∠BCD = 180° |
3. Adding statements 1 and 2, and total degree measure of a complete circle is 360°. |
| 4. Similarly, ∠ABC + ∠ADC = 180° | 4. Same as above. |
Proved.
Experimental Verification
📌 Note (Special Examination Instruction):
- To prove opposite angles are supplementary, write: ∠A + ∠C = 180° and ∠B + ∠D = 180°
1. Figures
Figure 1 (r = 3 cm)
Figure 2 (r = 3.5 cm)
2. To be Experimented
∠A + ∠C = 180° and ∠B + ∠D = 180°
3. Table
| Fig No. | ∠A | ∠C | ∠B | ∠D | Remarks |
|---|---|---|---|---|---|
| Figure 1 | 85° | 95° | 92° | 88° | ∠A + ∠C = 180° ∠B + ∠D = 180° |
| Figure 2 | 100° | 80° | 105° | 75° | ∠A + ∠C = 180° ∠B + ∠D = 180° |
4. Conclusion
Hence, it is experimentally verified that the opposite angles of a cyclic quadrilateral are supplementary.
Dynamic Demonstration
Drag Point A, Point B, Point C, or Point D to alter the vertices. Notice how all individual angle values change in real time while their opposite sums always stay 180.0°.
∠A = 0° |
∠C = 0° |
∠B = 0° |
∠D = 0°
∠A + ∠C = 180.0° | ∠B + ∠D = 180.0°
∠A + ∠C = 180.0° | ∠B + ∠D = 180.0°