Please help sum11(I)

Please help sum11(I) Prove angle of a rectangle is 900. rectangle. that it is a (iO If the angle of a quadrilateral are equal, prove (iii) If the diagonals of a rhombus are equal, prove that it is a square. (iv) Prove that every diagonal of a rhombus bisects the angles at the vertices. ABCD iS parallelogram. If the diagonal AC bisects LA, then prove that: (O AC bisects Z C (ii) A BCD is a rhombus (iii) AC -L BD. (i) Prove that bisectors of any two adjacent angles of a parallelogram are at right (ii) Prove that bisectors of any two opposite angles of a parallelogram are parallel. (ih) If the diagonals of a quadrilateral are equal and bisect each Other at right angles, then prove that it is a square. (1) If A BCD is a rectangle in which the diagonal BD bisects Z B, then show that ABCD is a square. (if) Show that if the diagonals of a quadrilateral are equal and bisect each Other at right angles, then it is a square. pand Q are points on opposite sides AD and BC of a parallelogram ABCD Such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O. Hint. Show that AOAP AOCQ. 14 (a) In fi gure (l ) gi ven below. ABCD is a parallelogram and X is mi d-point Of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that (1) the triangles ABX and QCX are congruent. (h) DC = CQ=QP (b) In figure (2) given below, points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = CQ. Show that AC and PQ bisect each other. Q c o (2) ABCD is a square. A is joined to a point P on BC and D is joined to a point Q on AB. If AP DQ prove that AP and DQ are perpendicular to each other. Hint. AABP DAQ ZBAP = ZADQ. But z BAD 900 ZPAD + ZADQ 90'. If P and Q are points of trisection of the diagonal BD Of a parallelogram ABCD, prove that CQ II AP. Hint. ABP = ACDQ. A transversal cuts two parallel lines at A and B. The two interior angles at Aare bisected and so are the two interior angles at B; the four bisectors form a quadrilateral ACBD

ABCD is a parallelogram. OA and OD are the bisectors of adjacent angles,∠A and ∠D.

ABCD is a parallelogram. 

∴ AB||DC    (Opposite sides of the parallelogram are parallel)

AB||DC and AD is the transversal,

∴ ∠BAD + ∠CDA = 180°  (Sum of interior angles on the same side of the transversal is 180°)

⇒ ∠1 + ∠2 = 90°    (AO and DO are angle bisectors ∠A and ∠D)  ...(1)

In ΔAOD,

∠1 + ∠AOD + ∠2 = 180°

⇒∠AOD + 90° = 180°  [from (1)]

⇒∠AOD  = 180° – 90° = 90°

∴ In a parallelogram, the bisectors of the adjacent angles intersect at right angle.

Cheers!

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