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Exercise - 3.3

Understanding Quadrilaterals

**Question-1 :-** Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(i) AD = ...... (ii) ∠ DCB = ...... (iii) OC = ...... (iv) m ∠DAB + m ∠CDA = ......

(i) AD = BC [Since opposite sides of a parallelogram are equal] (ii) ∠ DCB = ∠ DAB [Since opposite angles of a parallelogram are equal] (iii) OC = OA [Since diagonals of a parallelogram bisect each other] (iv) m ∠DAB + m ∠CDA = = 180 [Adjacent angles in a parallelogram are supplementary]

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

(i) ∠ B + ∠ C = 180° [Adjacent angles in a parallelogram are supplementary] 100° + x = 180° And Also, x = 180° - 100° = 80° z = x = 80° [Since opposite angles of a parallelogram are equal] y = 100° [Since opposite angles of a parallelogram are equal]

(ii) x + 50° = 180° [Adjacent angles in a //gm are supplementary] x = 180° - 50° = 130° z = x = 130° [Corresponding angles]

(iii) x = 90° [Vertically opposite angles] y + x + 30° = 180° [Angle sum property of triangle] y + 90° + 30° = 180° y + 120° = 180° y = 180° - 120° y = 60° z = y = 60° [Alternate angles]

(iv) z = 80° [Corresponding angles] x + 80° = 180° [Adjacent angles in a //gm are supplementary] x = 180° - 80° x = 100° y = x = 100° [opposite angles are equal in a //gm]

(v) y = 112° [opposite angles] x + y + 40° = 180° [Angle sum property of triangle] x + 112° + 40° = 180° x + 152° = 180° x = 180° - 152° x = 28° z = x = 28° [Alternate angles]

**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°?

(i)It can be , but not always as you need to look for other criteria as well. (ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram. (iii) Here opposite angles are not equal, so it 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.

ABCD is a quadrilateral in which angles ∠ A = ∠ C = 110°. Therefore, it could be a kite.

**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.

Let two adjacent angles be 3x and 2x. Since the adjacent angles in a parallelogram are supplementary. 3x + 2x = 180° 5x = 180° x = 180°/5 x = 36° one angle = 3x = 3 x 36° = 108° second angle = 2x = 2 x 36° = 72°

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

Let each adjacent angle be x. Since the adjacent angles in a parallelogram are supplementary. x + x = 180° 2x = 180° x = 180°/2 x = 90° So, each adjacent angle is 90°.

**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.

Angle opposite to y = 180° - 70° = 110° Hence, y = 40° [alternate angle of ∠ PHE] x = 180° - (110° + 40°) = 30°, [triangle’s angle sum] z = 30° [Alternate angle of a transversal]

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

(i) As opposite sides are equal in a parallelogram. so, In //gm GUNS 3y - 1 = 26 3y = 26 + 1 3y = 27 y = 27/3 y = 9 cm similarly, 3x = 18 x = 18/3 x = 6 cm

(ii) As you know diagonals bisect each other in a parallelogram. so, In //gm RUNS y + 7 = 20 [Diagonals of //gm bisects each other] y = 20 - 7 y = 13 cm similarly. x + y = 16 x + 13 = 16 x = 16 - 13 x = 3 cm

**Question-9 :-**

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

In //gm RISK, ∠ RIS = ∠ K = 120° [opposite angles of //gm are equal] ∠ m + 120° = 180° [Linear Pair] ∠ m = 180° - 120° ∠ m = 60° Now, ∠ ECI = ∠ L = 70° [Corresponding Angles] m + n + ∠ ECI = 180° [Angle sum property of triangle] 60° + n + 70° = 180° 130° + n = 180° n = 180° - 130° n = 50° Also, x = n = 50° [Vertically opposite angles]

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

Here, ∠M + ∠L = 100° + 80° = 180° [sum of interior opposite angles] NM and KL are parallel. So, KLMN is a trapezium.

**Question-11 :-** Find m∠C in Figure if AB||DC .

Here, ∠B + ∠C = 180° [AB||DC] 120° + m∠C = 180° m∠C = 180° - 120° m∠C = 60°

**Question-12 :-** Find the measure of ∠P and ∠S if SP||RQ in Figure. (If you find m∠R, is there more than one method to find m∠P?)

Here, ∠P + ∠Q = 180° [sum of co-interior angles] ∠P + 130° = 180° ∠P = 180° - 130° ∠P = 50° ∠R = 90° ∠S + 90° = 180° [Given] ∠S = 180° - 90° ∠S = 90°

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