**Notes**
**Ncert Solutions**
**Assignments**
**Revision sheet**
Given below are the **Class 9 Maths** Assignments for Quadrilaterals

(a) Concepts questions

(b) Calculation problems

(c) Multiple choice questions

(d) Long answer questions

(e) Fill in the blanks

**Question 1.** If the diagonals of parallelogram are equal, then show that it is a rectangle.

**Question 2.**. Show that if the diagonals of the quadrilateral bisect each other at right angles, then it is a rhombus.

**Question 3.** Show that the diagonals of a square are equal and bisect each other at right angles.

**Question 4.** Show that if the diagonals of the quadrilateral are equal and bisect each other at night angles, then it is a square.

**Question 5.** ABCD is a rectangle in which diagonal AC bisect $\angle A$ as well as $\angle C$. Show that-

- ABCD is a square
- Diagonal BD bisects $\angle B$ a well as $\angle D$.

**Question 6.** l, m and n are three parallel lines intersected by transversal's p and q such that l, m and n cut off equal intercepts AB and BC on p. Show that l, m and n cut off equal intercepts DE and EF and q also.

**Question 7.** ABCD is a rhombus and P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

**Question 8.** ABCD is a rectangle and P, Q, R and S are mid points of the sides AB, BC, CD and DA respectively. Show that quadrilateral PQRS is a rhombus.

**Question 9.** Show that the line segments joining the mid- point of the opposite sides of a quadrilateral bisect each other.

**Question 10.** ABC is a triangle right angles at C. A line through the mid -point M of hypotenuse AB and parallel to BC intersects AC at D. Show that

- D is mid- point of AC
- $CM = MA = \frac {1}{2} AB$

**Question 11.** In a quadrilateral ABCD, CO and DO are bisectors of $\angle C$ and $\angle D$ respectively. Prove that $\angle COD = \frac {1}{2} (\angle A + \angle B)$

**Question 12.** In a parallelogram ABCD, $\angle D = 135^0$, determine the angles measures of $\angle A \; and \; \angle B$

**Question 13.** ABCD is a parallelogram in which $\angle A = 70^0$. Compute $\angle B \; ,\; \angle C \;and \; \angle D$

**Question 14.** ABCD is a parallelogram in which $\angle DAB = 75^0$ and $\angle DBC = 60^0$. Compute $\angle CDB \; and \; \angle ADB$.

**Question 15.** ABCD is a parallelogram and x, y are the mid- points of sides AB and DC respectively. Show that in ABCD, AXCY is a parallelogram.

**Question 16.** The sides AB and CD of a parallelogram ABCD are bisected t E and F. Prove that EBFD is a parallelogram.

**Question 17.** ABCD is a square E, F, G and H are points on AB, BC, CD and DA respectively. Such hat AE = BF = CG = DH. Prove that EFGH is a square.

**Question 18.** ABCD is a rhombus, EABF is a straight line such that EA = AB = BF. Prove that ED and FC when produced meet at right angles.

**Question 19.** ABCD is a parallelogram, AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC.

**Question 20.** ABCD is a parallelogram P is a point on AD such that AP = 1/3 AD and Q is a point on BC such that CQ = 1/3 BC. Prove that AQCP is a parallelogram.

**Question 21.** P is the mid- point of side AB of a parallelogram ABCD. A line through B parallel to PD meets DC at Q and AD produced at R. Prove that

- AR = 2BC
- BR = 2BQ.

**Question 22.** ABCD is a kite having AB = AD and BC = CD. Prove that the figure formed by joining the mid- points of the side, in order, is a rectangle.

**Question 23.** ABC is a triangle. D is a point on AB such that AD = ¼ AB and E is the point on A such that AE = ¼ AC. Prove that DE = ¼ BC.

**Question 24.** If the diagonals of a parallelogram are equal, then show that it is a rectangle.

**Question 25.** In a parallelogram, show that the angle bisectors of two adjacent angles intersect at right angles.

**Question 26.** Show that the line segment joining the mid- point of any two sides of a triangle is parallel to third side and is equal to half of it.

**Question 27.** A diagonal of a rectangle is inclined to one side of a rectangle at $25^0$. Find the acute angles between the rectangle diagonals.

**Question 28.** P, Q, R and S are respectively the mid- point of sides AB, BC, CD and DA of a quadrilateral ABCD such that AC BD. Prove that PQRS is a square.

**Question 29.** Show that the quadrilateral formed by joining the mid- points of adjacent sides of rectangle is a rhombus.

**Question 30.** Show that the diagonals of a square are equal and bisect each other at right angles.

**Question 31.** Prove that the quadrilateral formed (if possible) by the internal angular bisectors of any quadrilateral is cyclic.

**Question 32.** PQ and RS are to equal and parallel line segments. Any point M not lying on PQ or RS is joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove that the line segments MN and PQ are equal and parallel to each other.

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