GSEB Class 9 Maths Solutions Chapter 10 Circles Exercise 10.2

Question 1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles angles at their centres.
Answer:Given: \( AB \) and \( CD \) are two equal chords of congruent circles with centres \( O \) and \( O' \) separately.
To Prove: \( \angle AOB = \angle CO'D \).
Proof: Let's consider \( \triangle OAB \) and \( \triangle O'CD \).
We know \( OA = O'C \) [These are the radii of congruent circles]
Also, \( OB = O'D \) [These are also radii of congruent circles]
And \( AB = CD \) [This information is provided]
Therefore, \( \triangle OAB \cong \triangle O'CD \) [By SSS Congruence Rule]
This outcome signifies that \( \angle AOB = \angle CO'D \). [This is due to CPCT, meaning Corresponding Parts of Congruent Triangles]

In simple words: If two circles are exactly the same size, and they have chords of equal length, then the angles these chords make at the center of each circle will also be equal. This is proven by showing that the triangles formed are congruent using the SSS rule.

Exam Tip: Remember to clearly state the "Given" and "To Prove" sections in geometry proofs. Always mention the congruence rule (like SSS, SAS, ASA) used to prove triangle congruence, and then use CPCT for corresponding parts.

Question 2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Answer:Given: \( \angle AOB \) and \( \angle CO'D \) represent two equal angles formed by chords \( AB \) and \( CD \) at the centres \( O \) and \( O' \) of two matching circles respectively.
To Prove: \( AB = CD \).
Proof: Let's examine \( \triangle OAB \) and \( \triangle O'CD \).
We have \( OA = O'C \) [These are the radii of congruent circles]
Also, \( OB = O'D \) [These are also radii of congruent circles]
And \( \angle AOB = \angle CO'D \) [This information is provided]
Therefore, \( \triangle OAB \cong \triangle O'CD \) [By SAS Congruence Rule]
This result indicates that \( AB = CD \). [This is due to CPCT, meaning Corresponding Parts of Congruent Triangles]

In simple words: If two circles are congruent, and their chords form equal angles at the centers, then those chords must have the same length. This is shown by proving that the triangles formed are congruent using the SAS rule.

Exam Tip: This question is the converse of Question 1. When working with converse theorems, ensure you correctly identify what is given and what needs to be proven based on the new statement.