Question 1 - Think and Reflect (Page 1) - Definition - Chapter 9 Class 9 - Propositions and their Converses (Ganita Manjari)

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Question 1 - Think and Reflect (Page 1) Statement 1: If two sides of a triangle are equal, then the angles opposite the equal sides are equal. Statement 2: If two angles of a triangle are equal, then the sides opposite the equal angles have equal lengths We have proved the first statement in an earlier grade. Is the second statement true? Can you prove it? (Hint: Draw the altitude from the vertex containing the third angle.) Here, Statement 2 is the converse of Statement 1 We have to prove Statement 2 Statement 2: If two angles of a triangle are equal, then the sides opposite the equal angles have equal lengths Given: A triangle ABC where two angles are equal i.e. ∠B = ∠C To Prove: Sides opposite to equal angles are equal i.e. AB = AC Construction: Draw the altitude from vertex A to BC, meeting at point D ∴ 𝐴𝐷⊥𝐵𝐶 Proof: In △BAD and △CAD ∠ B = ∠ C ∠BDA = ∠CDA AD = AD ∴ △BAD ≅ △CAD Thus, by Corresponding parts of congruent triangles ∴ AB = AC Hence, sides opposite to equal angles are equal. Hence proved (Given) (Both 90° as AD ⊥ BC) (Common) (By AAS congruence rule) Thus, by Corresponding parts of congruent triangles ∴ AB = AC Hence, sides opposite to equal angles are equal. Hence proved