Both the proposition and its converse (“If IE = IF, then AB = AC”) are true, as equal sides yield matching bisector segments through symmetry and equal segments enforce isosceles side lengths.
Given any triangle ABC, let us bisect the angles at B and C. The bisectors meet at the incentre I of the triangle. Now extend the bisectors beyond I till they meet the opposite sides at E and F respectively, as shown. Proposition: If AB = AC, then IE = IF.
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Both the proposition and its converse (“If IE = IF, then AB = AC”) are true. When AB = AC, the triangle is isosceles, creating congruent sub-triangles and symmetry that guarantees IE = IF. Conversely, if the extended bisector segments satisfy IE = IF, geometric congruence of the resulting inner segments forces the base angles to be equal, ensuring AB = AC.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 9 Propositions and their Converses Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-9/