0 votes
ago in Circle by (33.7k points)
edited ago by

Prove that, if two lines containing chords of a circle intersect each other outside the circle, then the measure of angle between them is half the difference in measures of the arcs intercepted by the angle.

Please log in or register to answer this question.

1 Answer

0 votes
ago by (34.9k points)
edited ago by

Given: Chord AB and chord CD intersect at E in the exterior of the circle. 

To prove: ∠AEC = 1/2 [m(arc AC) – m(arc BD)] 

Construction: Draw seg AD.


∠ADC is the exterior angle of ∆ADE. 

∴ ∠ADC = ∠DAE + ∠AED [Remote interior angle theorem] 

∴ ∠ADC = ∠DAE + ∠AEC [C – D – E] 

∴ ∠AEC = ∠ADC – ∠DAE ……(i) 

∠ADC = 1/2m(arc AC) (ii) [Inscribed angle therom]

∠DAE = 1/2 m(arc BD) (iii) [A – B – E, Inscribed angle theorem] 

∴ ∠AEC = 1/2 m(arc AC) – 1/2 m (arc BD) 

[From (i), (ii) and (iii)] 

∴ ∠AEC = 1/2 m(arc AC) – m (arc BD)

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.