Question 47 of 50intermediate💡 UnderstandShort Answer3 marks

Let ABCD be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g., ∠CDE=∠ABC\angle CDE = \angle ABC, where E is a point on the extension of side CD).

Correct Answer

Let ABCD be a cyclic quadrilateral and E be a point on the extension of side CD. We must show that exterior angle ∠CDE equals interior opposite angle ∠ABC.

Since E lies on the extension of CD, ∠CDE and ∠CDA form a linear pair. Thus, ∠CDE + ∠CDA = 180° ... (1). By Theorem 11, the sum of opposite angles of a cyclic quadrilateral is 180°. So, ∠ABC + ∠CDA = 180° ... (2).

From equations (1) and (2), we get ∠CDE + ∠CDA = ∠ABC + ∠CDA. Subtracting ∠CDA from both sides, we get ∠CDE = ∠ABC. Hence, the exterior angle at any vertex equals the interior opposite angle.

Exercise: END-OF-CHAPTER EXERCISES | Q: 21 | (Page 24)
For More Understanding

Explanation

The proof uses Theorem 11 from the context, which states that the sum of opposite angles in a cyclic quadrilateral is 180°. By combining this with the linear pair property of angles on a straight line, we establish the equality of the exterior angle and the interior opposite angle.

Solution Steps

  1. Step 1: State the linear pair equation: ∠CDE + ∠CDA = 180°

  2. Step 2: State Theorem 11 for opposite angles: ∠ABC + ∠CDA = 180°

  3. Step 3: Equate the two expressions to get ∠CDE + ∠CDA = ∠ABC + ∠CDA

  4. Step 4: Subtract ∠CDA from both sides to conclude ∠CDE = ∠ABC

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