please please please mark me as BRAINLIEST and thanks if it helped you good day Step-by-step explanation:3.1 Class 8 Maths Question 1. Q1.1Classify each of the above figure on the basis of the following:(a) Simple curve(b) Simple closed curve(c) Polygon(d) Convex polygon(E) Concave polygonSolution:(a) Simple curve: (1), (2), (5), (6) and (7)(b) Simple closed curve: (1), (2), (5), (6) and (7)(c) Polygon: (1) and (2)(d) Convex polygon: (2)(e) Concave polygon: (1)Ex 3.1 Class 8 Maths Question 2.How many diagonals does each of the following have?(a) A convex quadrilateral(b) A regular hexagon(c) A triangleSolution:(a) In Fig. (i) ABCD is a convex quadrilateral which has two diagonals AC and BD.NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Q2(b) In Fig. (ii) ABCDEF is a regular hexagon which has nine diagonals AE, AD, AC, BF, BE, BD, CF, CE and DF.Q2.1(c) In Fig. (iii) ABC is a triangle which has no diagonal. Q2.2Ex 3.1 Class 8 Maths 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 verify)Solution:In the given figure, we have a quadrilateral ABCD. Join AC diagonal which divides the quadrilateral into two triangles ABC and ADC.Q3In ∆ABC, ∠3 + ∠4 + ∠6 = 180°…(i) (ANGLE sum property)In ∆ADC, ∠1 + ∠2 + ∠5 = 180° …(ii) (angle sum property)Adding, (i) and (ii)∠1 + ∠3 + ∠2 + ∠4 + ∠5 + ∠6 = 180° + 180°⇒ ∠A + ∠C + ∠D + ∠B = 360°Hence, the sum of all the angles of a convex quadrilateral = 360°.Let us draw a non-convex quadrilateral.Yes, this property also holds true for a non-convex quadrilateral. Q3.1Ex 3.1 Class 8 Maths Question 4.Examine the table. (Each figure is divided into triangles and the sum of the angles reduced from that).Q4What can you say about the angle sum of a convex polygon with number of sides?(a) 7(b) 8(c) 10(d) nSolution:From the above table, we conclude that the sum of all the angles of a polygon of side ‘n’= (n – 2) × 180°(a) Number of sides = 7Angles sum = (7 – 2) × 180° = 5 × 180° = 900°(b) Number of sides = 8Angle sum = (8 – 2) × 180° = 6 × 180° = 1080°(c) Number of sides = 10 Angle sum = (10 – 2) × 180° = 8 × 180° = 1440°(d) Number of sides = nAngle sum = (n – 2) × 180°Ex 3.1 Class 8 Maths Question 5.What is a regular polygon? State the name of a regular polygon of(i) 3 sides(ii) 4 sides(iii) 6 sidesSolution:A polygon with equal sides and equal angles is called a regular polygon.(i) Equilateral triangle Q5.1(iii) Regular HexagonNCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Q5.2Ex 3.1 Class 8 Maths Question 6.Find the angle measure x in the following figures:Q6Solution:(a) Angle sum of a quadrilateral = 360°⇒ 50° + 130° + 120° + x = 360°⇒ 300° + x = 360°⇒ x = 360° – 300° = 60°(b) Angle sum of a quadrilateral = 360°⇒ x + 70° + 60° + 90° = 360° [∵ 180° – 90° = 90°]⇒ x + 220° = 360°⇒ x = 360° – 220° = 140°(c) Angle sum of a pentagon = 540°⇒ 30° + x + 110° + 120° + x = 540° [∵ 180° – 70° = 110°; 180° – 60° = 120°]⇒ 2x + 260° = 540°⇒ 2x = 540° – 260°⇒ 2x = 280°⇒ x = 140°(d) Angle sum of a regular pentagon = 540°⇒ x + x + x + x + x = 540° [All angles of a regular pentagon are equal]⇒ 5x = 540°⇒ x = 108°Ex 3.1 Question 7.(a) Find x + y + zQ7(b) Find x + y + z + wQ7.1Solution:(a) ∠a + 30° + 90° = 180° [Angle sum property]⇒ ∠a + 120° = 180°Q7.2⇒ ∠a = 180° – 120° = 60°Now, y = 180° – a (Linear pair)⇒ y = 180° – 60°⇒ y = 120°and, z + 30° = 180° [Linear pair]⇒ z = 180° – 30° = 150°also, x + 90° = 180° [Linear pair]⇒ x = 180° – 90° = 90°Thus x + y + z = 90° + 120° + 150° = 360°(b) ∠r + 120° + 80° + 60° = 360° [Angle sum property of a quadrilateral ]Q7.3∠r + 260° = 360°∠r = 360° – 260° = 100°Now x + 120° = 180° (Linear pair)x = 180° – 120° = 60°y + 80° = 180° (Linear pair)⇒ y = 180° – 80° = 100°z + 60° = 180° (Linear pair)⇒ z = 180° – 60° = 120°w = 180° – ∠r = 180° – 100° = 80° (Linear pair)x + y + z + w = 60° + 100° + 120° + 80° = 360°.