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Last updated at Aug. 4, 2018 by Teachoo

Transcript

Ex 3.1, 3 What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!) Convex Quadrilateral In Quadrilateral ABCD Sum of Angles of ABCD = ∠ A + ∠ B + ∠ C + ∠ D Let’s divide Quadrilateral ABCD into two triangles – Δ ABC & Δ ADC Now, We know that Sum of angles of a triangle is 180° For Δ ABC ∠ BAC + ∠ B + ∠ BCA = 180° For Δ ADC ∠ DAC + ∠ D + ∠ DCA = 180° Adding (1) & (2) ∠ BAC + ∠ B + ∠ BCA + ∠ DAC + ∠ D + ∠ DCA = 180° + 180° (∠ BAC + ∠ DAC) + ∠ B + (∠ BCA + ∠ DCA) + ∠ D = 360° ∠ A + ∠ B + ∠ C + ∠ D =360° So, Sum of angles of convex quadrilateral = 360° Let’s try for non-convex quadrilateral Non-Convex Quadrilateral In Quadrilateral ABCD Sum of Angles of ABCD = ∠ A + ∠ B + ∠ C + ∠ D Let’s divide Quadrilateral ABCD into two triangles – Δ ABC & Δ ADC Now, We know that Sum of angles of a triangle is 180° For Δ ABC ∠ BAC + ∠ B + ∠ BCA = 180° For Δ ADC ∠ DAC + ∠ D + ∠ DCA = 180° Adding (1) & (2) ∠ BAC + ∠ B + ∠ BCA + ∠ DAC + ∠ D + ∠ DCA = 180° + 180° (∠ BAC + ∠ DAC) + ∠ B + (∠ BCA + ∠ DCA) + ∠ D = 360° ∠ A + ∠ B + ∠ C + ∠ D =360° So, Sum of angles of non-convex quadrilateral = 360° Hence, sum of angles is same for both convex and non-convex quadrilaterals.

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.