**Understanding Quadrilaterals**

**Question 1.****Given a parallelogram ABCD. Complete each statement along with the definition with the definiton or property used.(i)** AD = ………

**(ii)**∠DCB = …………….

**(iii)**OC = ……………….

**(iv)**m∠DAB + m∠CDA = …………..

**Solution.****(i)** AD = BC : In a parallelogram, opposite sides are equal.**(ii)** ∠DCB = ∠DAB : In a parallelogram, opposite angles are equal.**(iii)** OC = OA : The diagonals of a parallelogram bisect each other.**(iv)** m∠DAB + m∠CDA = 180° : In a parallelogram, the stun of any two adjacent angles is 180°.

**Question 2.****Consider the following parallelograms. Find the values of the unknowns x, y, z.**

**Solution.**

**Question 3.**

Can a quadrilateral ABCD be a parallelogram if**(i)** ∠D + ∠B = 180° ?**(ii)** AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm**(iii)** ∠A = 70° and ∠C = 65°?

**Solution.****(i)** If in a quadrilateral ABCD, ∠D + ∠B = 180°, then it is not necessary that ABCD is a parallelogram.**(ii)** Since AD ≠ BC, i.e., the opposite sides are unequal, so ABCD is not a parallelogram.**(iii)** Since ∠A ≠ ∠C, i.e., the opposite angles are unequal, so ABCD is not a parallelogram.

**Question 4.****Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.**

**Solution.**

A rough figure of a quadrilateral ABCD that is not a parallelogram has been drawn with exactly two opposite angles of equal measure such that ∠A – ∠C which is a kite as an example.

**Question 5.****The measures of two adjacent angles of a parallelogram are in the ratio 3 :2. Find the measure of each of the angles of the parallelogram.**

**Solution.**

Let two adjacent angles A and B of ||gm ABCD be 3x and 2x respectively.

Since the adjacent angles of a parallelogram are supplementary.

Since the opposite angles are equal in a parallelogram, therefore,

∠C = ∠A = 108° and ∠D = ∠B = 72°

Hence, ∠A = 108°, ∠B = 72°, ∠C = 108° and ∠D = 72°.

**Question 6.****Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.**

**Solution.**

Let two adjacent angles A and B of parallelogram ABCD be x each.

Since the adjacent angles of a parallelogram are supplementary.

**Question 7.****The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.Solution.**

Since HOPE is a parallelogram, therefore, HE || OP and HO || EP.

Now, HE || OP and transversal HO intersects them.

Hence, x = 110°, y = 40° and z = 30°

**Question 8.****The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)**

**Solution.****(i)** Since GUNS is a parallelogram, therefore, its opposite sides are equal.**(ii)** In a parallelogram, diagonals bisect each other, therefore,

**Question 9.**

In the below figure both RISK and CLUE are parallelograms. Find the value of x.

**Solution.**

**Question 10.****Explain how this figure is a trapezium. Which of its two sides are parallel ?Solution.**

Since ∠KLM + ∠NML = 180° i.e., the pair of consecutive interior angles are supplmentary.

Therefore, KL || NM and so KLMN is a trapezium.

**Question 11.****Find m∠C in the figure, if || .Solution.**

Since AB || DC and transversal BC intersects

them.

∠B + ∠C= 180°

[∵ Sum of interior angles is 180°]

⇒ 120°+ ∠C =180° A

⇒ ∠C = 180°- 120° = 60°

Hence, m∠C =60°

**Question 12.****Find the measure of ∠P and ∠S, if || in the figure. (If you find mZ R, is there more than one method to find m∠P ?)Solution.**

Since SP || RQ and PQ is a transversal intersecting them at P and Q.

∴ ∠P + ∠Q =180°

[∵ Sum of interior angles is 180°]