A figure formed by joining four points in an order is called a quadrilateral. Show what APCQ is a parallelogram. Delhi - 110058. 414-3 Rhombus and Square On 1 — 2, refer to rhombus ABCD where diagonals AC and BD intersect at E. Given rho bus ABCD where diagonals AC and BD intersects at E. 4 In a parallelogram, the diagonals bisect each other. Show that:(i)     quadrilateral ABED is a parallelogram(ii)    quadrilateral BEFC is a parallelogram(iii)   AD || CF and AD = CF(iv)   quadrilateral ACFD is a parallelogram, (v)     AC = DF(vi)    ∆ABC ≅ ∆DEF. A line that intersects another line segment and separates it into two equal parts is called a bisector . Data: ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. (v)    ∵ The diagonals of a parallelogram bisect each other.∴ OB = OD∴ OB - BQ = OD - DP| ∵ BQ = DP (given)∴ OQ = OP    ...(1)Also, OA = OC    ...(2)| ∵ Diagonals of a || gm bisect each otherIn view of (1) and (2), APCQ is a parallelogram. a diagonal of a parallelogram divides it into two congruent triangles, and; the diagonals of a parallelogram bisect each other. Show what APCQ is a parallelogram. Diagonal AC of a parallelogram ABCD bisects ∠A (see figure). (ii) In ∆BDA and ∆DBC,BD = DB    | CommonDA= BC| Sides of a square ABCDAB = DC| Sides of a square ABCD∴ ∆BDA ≅ ∆DBC| SSS Congruence Rule∴ ∠ABD = ∠CDB    | C.P.C.T.But ∠CDB = ∠CBD| ∵ CB = CD (Sides of a square ABCD)∴ ∠ABD = ∠CBD∴ BD bisects ∠B.Now, ∠ABD = ∠CBD∠ABD = ∠ADB | ∵ AB = AD∠CBD = ∠CDB | ∵ CB = CD∴ ∠ADB = ∠CDB∴ BD bisects ∠D. Question 1. If BM bisects ∠B, then AM bisects ∠A as diagonals of a parallelogram bisect each other and here M is the point of intersection of the diagonals AC and BD. Result It is verified that. (i)    ∆APD ≅ ∆CQB(ii)   AP = CQ(iii)  ∆AQB ≅ ∆CPD(iv)  AQ = CP(v)   APCQ is a parallelogram. BD = 4 cm. How we can show that it is a rectangle. 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. Question 3. Diagonals of a parallelogram are perpendicular to each other. Then, AC = 2AO. 8.21). asked Feb 1, 2018 in Class IX Maths by aman28 (-872 points) quadrilaterals. Vertices A, B and C are joined to vertices D, E and F respectively.To Prove: (i) quadrilateral ABED is a parallelogram(ii)    quadrilateral BEFC is a parallelogram(iii)    AD || CF and AD = CF(iv)    quadrilateral ACFD is a parallelogram(v)     AC = DF(vi)    ∆ABC ≅ ∆DEF.Proof: (i) In quadrilateral ABED,AB = DE and AB || DE| Given∴ quadrilateral ABED is a parallelogram.| ∵    A quadrilateral is a parallelogram if a pair of opposite sides are paralleland are of equal length(ii)    In quadrilateral BEFC,BC = EF and BC || EF    | Given∴ quadrilateral BEFC is a parallelogram.| ∵    A quadrilateral is a parallelogram if a pair of opposite sides are paralleland are of equal length(iii)    ∵ ABED is a parallelogram| Proved in (i)∴ AD || BE and AD = BE    ...(1)| ∵    Opposite sides of a || gmare parallel and equal∵ BEFC is a parallelogram | Proved in (ii)∴ BE || CF and BE = CF    ...(2)| ∵    Opposite sides of a || gmare parallel and equalFrom (1) and (2), we obtainAD || CF and AD = CF. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). Given diagonal AC bisects ∠B = 180° – ∠A = 180°- 35° [∴ ∠A = 35°, given] ⇒ ∠B = 145° Ans. Special parallelograms. The same reason for