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Ex 10.5, 8 - If non-parallel sides of a trapezium are equal - Cyclic quadrilaterals

Ex 10.5, 8 - Chapter 10 Class 9 Circles - Part 2

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Ex 10.5, 8 If the non-parallel sides of a trapezium are equal, prove that it is cyclic. Given: ABCD is a trapezium where AB ∥ DC & non parallel sides are equal, i.e., AD = BC To prove: ABCD is cyclic quadrilateral Construction: We draw DE ⊥ AB & CF ⊥ AB Proof: To prove ABCD is a cyclic quadrilateral, we prove that sum of one pair of opposite angles is 180° In ∆ADE & ∆ BCF ∠AED = ∠ BFC AD = BC DE = CF ∴ ∆ADE ≅ ∆ BCF So, ∠ DAE = ∠ CBF i.e., ∠ A = ∠ B Now, for parallel lines AB and DC , & AD is the transversal line ∠A + ∠D = 180 ° ∠B + ∠D = 180 ° So, in ABCD, sum of one pair of opposite angles is 180 ° Therefore, ABCD is a cyclic quadrilateral Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.