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Example 4 - Two parallel lines l and m are intersected - Opposite sides of parallelogram

Example 4 - Chapter 8 Class 9 Quadrilaterals - Part 2
Example 4 - Chapter 8 Class 9 Quadrilaterals - Part 3

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Example 4 Two parallel lines l and m are intersected by a transversal p. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle. Given: line l ∥ m transversal p intersects l & m at A & C resp. Bisector of ∠ PAC & ∠QCA meet at B & Bisector of ∠ SAC & ∠RCA meet at D To prove: ABCD is a rectangle Proof: We know that a rectangle is a parallelogram with one angle 90° First we will prove ABCD is a parallelogram, For l ∥ m & transversal p ∠ PAC = ∠ ACR So ,1/2 ∠ PAC = 1/2 ∠ ACR i.e., ∠ BAC = ∠ ACD For lines AB and DC with AC as transversal ∠ BAC & ∠ ACD are alternate angles, and they are equal So, AB ∥ DC Similarly, for lines BC & AD, with AC as transversal ∠ BAC & ∠ ACD are alternate angles, and they are equal ∴ BC ∥ AD Now, In ABCD, AB ∥ DC & BC ∥ AD As both pairs of opposite sides are parallel, ABCD is a parallelogram. Also for line l , ∠ PAC + ∠ CAS = 180° Multiplying both sides by half 1/2 ∠ PAC + 1/2 ∠ CAS = 1/2 × 180° ∠ BAC + ∠ CAD = 90° ∠ BAD = 90° So, ABCD is a parallelogram in which one angle is 90°. ∴ ABCD is a rectangle.

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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.