Important Extra Theorem: Exterior Angle of Cyclic Quadrilateral
Theorem: Prove that the exterior angle of a cyclic quadrilateral is equal to its opposite interior angle.
1. Given:
- A circle with center O.
- ABCD is a cyclic quadrilateral with vertex A at top and base side BC at bottom extended to point E to form an exterior angle ∠DCE.
2. To Prove:
- Exterior Angle ∠DCE = ∠DAB (Opposite Interior Angle)
Figure: Exterior Angle ∠DCE
3. Proof:
| Statements | Reasons |
|---|---|
| 1. ∠BCD + ∠DCE = 180° | 1. Angles on a straight line (Linear Pair). |
| 2. ∠BCD + ∠DAB = 180° | 2. Opposite angles of a cyclic quadrilateral are supplementary. |
|
3. ∠BCD + ∠DCE = ∠BCD + ∠DAB ⇒ ∠DCE = ∠DAB |
3. From statements 1 and 2 (Subtracting ∠BCD from both sides). |
Proved.
Experimental Verification
📌 Note (Special Examination Instruction):
- To prove the exterior angle theorem with base BC extended to E, write: ∠DCE = ∠DAB (Exterior Angle = Opposite Interior Angle).
1. Figures
Figure 1 (r = 3 cm)
Figure 2 (r = 3.5 cm)
2. To be Experimented
∠DCE = ∠DAB
3. Table
| Fig No. | Exterior Angle (∠DCE) | Opposite Interior Angle (∠DAB) | Remarks |
|---|---|---|---|
| Figure 1 | 85° | 85° | ∠DCE = ∠DAB |
| Figure 2 | 92° | 92° | ∠DCE = ∠DAB |
4. Conclusion
Hence, it is experimentally verified that the exterior angle of a cyclic quadrilateral is equal to its opposite interior angle.
Dynamic Demonstration
Drag vertices A, B, C, or D. Observe how extending Base BC to E creates Exterior Angle ∠DCE, which dynamically updates to always remain equal to Opposite Interior Angle ∠DAB.
Exterior Angle (∠DCE) = 0° |
Opposite Interior Angle (∠DAB) = 0°
Result: ∠DCE = ∠DAB
Result: ∠DCE = ∠DAB
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