## Chapter 3 Understanding Quadrilaterals Exercise 3.1

**Question 1: Given here are some figures.**

**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

Answer:

a) Simple curve

b) Simple closed curve

c) Polygon

d) Convex polygon

e) Concave polygon

Answer:

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

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

a) A convex quadrilateral

b) A regular hexagon

c) A triangle

Answer:

a)

a) A convex quadrilateral

b) A regular hexagon

c) A triangle

Answer:

Number of sides a convex quadrilateral has = 4

Formula for finding diagonal = n × (n - 3)

= 4 × (4 - 3)/2

= 4 × 1/2

= 4/2

= 2 diagonals

b)

Number of sides a regular hexagon has = 6

Formula for finding diagonal = n × (n - 3)

= 6 × (6 - 3)/2

= 6 × 3/2

= 18/2

= 9 diagonals

c)

Number of sides a triangle has = 3

Formula for finding diagonal = n × (n - 3)

= 3 × (3 - 3)/2

= 3 × 0/2

= 0/2

= 0 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!)**

Answer:The sum of measures of the angles of a convex quadrilateral is 360°. Yes, this property will hold if the quadrilateral is not convex.

Answer:

Sum of all angles in a triangle = 180°

= △ABC + △ACD = ?

= 180° + 180° = 360°

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

**a)**

a) 7

b) 8

c) 10

d) n

Answer:

**What can you say about the angle sum of a convex polygon with number of sides?**a) 7

b) 8

c) 10

d) n

Answer:

Number of sides = 7

Formula for finding angle sum property = (n - 2) × 180°

= (7 - 2) × 180°

= 5 × 180°

= 900°

b)

Number of sides = 8

Formula for finding angle sum property = (n - 2) × 180°

= (8 - 2) × 180°

= 6 × 180°

= 1080°

c)

Number of sides = 10

Formula for finding angle sum property = (n - 2) × 180°

= (10 - 2) × 180°

= 8 × 180°

= 1440°

d)

Number of sides = n

Formula for finding angle sum property = (n - 2) × 180°

= (n - 2) × 180°

= (n - 2) × 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

Answer:

State the name of a regular polygon of

i) 3 sides

ii) 4 sides

iii) 6 sides

Answer:

Polygons which are equiangular and equilateral are called regular polygon.

i) A regular polygon of 3 sides is equilateral triangle.

ii) A regular polygon of 4 sides is square.

iii) A regular polygon of 6 sides is regular hexagon.

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

**Answer:**a) Angle sum property of quadrilateral = 360°

= ∠50° + ∠130° + ∠120° + ∠x = 360°

= 300° + x = 360°

= x = 360 - 300

= x = 60°

b) Angle sum property of quadrilateral = 360°

= ∠60° + ∠70° + ∠x + ∠90° = 360°

= 220° + x = 360°

= x = 360 - 220

= x = 140°

c) Angle sum property of pentagon = 540°

= ∠30° + ∠110° + ∠120° + ∠x + ∠x = 540°

= 260° + 2x = 540°

= 2x = 540 - 260

= 2x = 280

= x = 280/2

= x = 140°

d) Angle sum property of regular pentagon = 540°

= ∠x + ∠x + ∠x + ∠x + ∠x = 540°

= 5x = 540°

= x = 540/5

= x = 108°

**Question 7:**

**Answer:**a)

Angle sum property of triangle = 180°

= ∠30° + ∠90° + ∠a = 180°

= 120° + a = 180°

= a = 180 - 120

= a = 60°

Therefore,

z = 180° - 30° = 150° (supplementary angles equal to 180°)

x = 180° - 90° = 90° (supplementary angles equal to 180°)

y = 180° - 60° = 120° (supplementary angles equal to 180°)

b)

Angle sum property of quadrilateral = 360°

= ∠120° + ∠80° + ∠60°+ ∠a = 360°

= 260° + a = 360°

= a = 360 - 260

= a = 100°

Therefore,

w = 180° - 100° = 30° (supplementary angles equal to 180°)

x = 180° - 120° = 60° (supplementary angles equal to 180°)

y = 180° - 80° = 100° (supplementary angles equal to 180°)

z = 180° - 60° = 120° (supplementary angles equal to 180°)

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