Please help sum11(ii)

Please help sum11(ii) 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

Dear student

ABCD is a parallelogram

the bisectors of ∠ADC and ∠BCD meet at point E and the bisectors of ∠BCD and ∠ABC meet at F

we have to prove that the ∠CED = 90º and ∠CFG = 90º

this way we will be able to prove that DE and CE intersect at right angles and BG and ED are parallel

∠ADC + ∠BCD = 180º  (sum of adjacent angles of a parallelogram)

⇒∠ADC/2 +∠ BCD/2 = 90º

⇒∠EDC + ∠ECD = 90º

in triangle ECD sum of angles = 180º

⇒∠EDC + ∠ECD + ∠CED = 180º

⇒ ∠CED = 90º

Hence the first condition is proved that in a parallelogram the bisectors of angles intersect at 90º

 

Similarly taking triangle BCF it can be proven that ∠BFC=90º

Also ∠BFC+∠CFG = 180º  (adjacent angles on a line)

⇒∠CFG = 90º

 

 

Now since ∠CFG = ∠CED = 90º it means that lines DE and BG are parallel

 

Hence it is proved that bisectors of opposite angles in a parallelogram are parallel.
Regards

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