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Exercise - 3.1

Understanding Quadrilaterals

**Question-1 :-**

Classify each of them on the basis of the following.

(a) Simple curve (b) Simple closed curve (c) Polygon (d) Convex polygon (e) Concave polygon

(a) Simple curve: 1, 2, 5, 6, 7 (b) Simple closed curve: 1, 2, 5, 6, 7 (c) Polygon: 1, 2, 4 (d) Convex polygon: 1 (e) Concave polygon: 1, 4

**Question-2 :-** How many diagonals does each of the following have?

(a) A convex quadrilateral (b) A regular hexagon (c) A triangle

(a) A convex quadrilateral : Two Diagonals. (b) A regular hexagon : Nine Diagonals. (c) A triangle : No Diagonals.

**Question-3 :-** What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles. ∠A + ∠B + ∠C + ∠D = ∠1 + ∠6 + ∠5 + ∠4 + ∠3 + ∠2 = ( ∠1 + ∠2 + ∠3) + ( ∠4 + ∠5 + ∠6) = 180° + 180° [By Angle sum property of triangle] = 360° Hence, the sum of measures of the triangles of a convex quadrilateral is 360°. Yes, if quadrilateral is not convex then, this property will also be applied. Let ABCD is a non-convex quadrilateral and join BD, which also divides the quadrilateral in two triangles. Using angle sum property of triangle, In ∆ABD, ∠1 + ∠2 + ∠3 = 180°.......(i) In ∆BDC, ∠4 + ∠5 + ∠6 = 180°.......(ii) Adding eq. (i) and (ii), ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360° ∠1 + ∠2 + ( ∠3 + ∠4) + ∠5 + ∠6 = 360° ∠A + ∠B + ∠C + ∠D = 360° Hence proved.

**Question-4 :-** Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

What can you say about the angle sum of a convex polygon with number of sides?

(a) 7 (b) 8 (c) 10 (d) n

(a) When n = 7, then Angle sum of a polygon = (n - 2) x 180° = (7 - 2) x 180° = 5 x 180° = 900° (b) When n = 8, then Angle sum of a polygon = (n - 2) x 180° = (8 - 2) x 180° = 6 x 180° = 1080° (c) When n = 10, then Angle sum of a polygon = (n - 2) x 180° = (10 - 2) x 180° = 8 x 180° = 1440° (d) When n = n, then Angle sum of a polygon = (n - 2) x 180°

**Question-5 :-** What is a regular polygon? State the name of a regular polygon of

(i) 3 sides (ii) 4 sides (iii) 6 sides

A regular polygon: A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon. (i) 3 sides : Polygon having three sides is called a Triangle. (ii) 4 sides : Polygon having four sides is called a Quadrilateral. (iii) 6 sides : Polygon having six sides is called a Hexagon.

**Question-6 :-** Find the angle measure x in the following figures.

(i) By the angle sum property of quadrilaeral 50° + 130° + 120° + x = 360° 300° + x = 360° x = 360° - 300° x = 60°

(ii) By the angle sum property of quadrilaeral 90° + 60° + 70° + x = 360° 220° + x = 360° x = 360° - 220° x = 140°

(iii) Linear pair angle = 180° First interior angle of base = 180° - 70° = 110° Second interior angle of base = 180° - 60° = 120° There are five sides, so n = 5 Angle sum of polygon = (n-2) x 180° = (5-2) x 180° = 3 x 180° = 540° Now, 30° + x + 110° + 120° + x = 540° 2x + 260° = 540° 2x = 540° - 260° 2x = 280° x = 280°/2 x = 140°

(iv) No. of angles = 5 Angle sum of polygon = (n-2) x 180° = (5-2) x 180° = 3 x 180° = 540° Now, x + x + x + x + x = 540° 5x = 540° x = 540°/5 x = 108°

**Question-7 :-** (a) Find x + y + z (b) Find x + y + z + w

(i) Sum of linear pair angle = 180° 90° + x = 180° x = 180° + 90° x = 90° and z + 30° = 180° z = 180° - 30° z = 150° Also, y = 90° + 30° = 120° [By Exeterior angle property] Hence, x + y + z = 90° + 120° + 150° = 360°

(ii) By using of angle sum property of quadrilateral, 60° + 80° + 120° + n = 360° 260° + n = 360° n = 360° - 260° n = 100° Now, sum of linear pair angles = 180° w + 100° = 180° w = 180° - 100° w = 80° x + 120° = 180° x = 180° - 120° x = 60° y + 80° = 180° y = 180° - 80° y = 100° z + 60° = 180° z = 180° - 60° z = 120° According to question, x + y + z + w = 60° + 100° + 120° + 80° = 360°

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