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
Solution:
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
2. How many diagonals does each of the following have?
a) A convex quadrilateral:
Solution: 2.
b) A regular hexagon:
Solution: 9.
c) A triangle:
Solution: 0
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!)
Solution:
Let us draw a convex quadrilateral ABCD.
From the figure, we understand that the quadrilateral ABCD is formed by two triangles,
i.e. ΔADC and ΔABC.
So, according to angle sum property of triangle we know that sum of interior angles of triangle is 180°,
The sum of the measures of the angles is 180° + 180° = 360°
Hence, the sum of all angles of a convex quadrilateral = 360°
Now,
Let us take another non-convex quadrilateral ABCD.
We can Join BC, Such that it divides ABCD into two triangles ΔABC and ΔBCD.
In ΔABC,
∠1 + ∠2 + ∠3 = 180° (angle sum property of triangle)
In ΔBCD,
∠4 + ∠5 + ∠6 = 180° (angle sum property of triangle)
∴, ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180°
⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°
⇒ ∠A + ∠B + ∠C + ∠D = 360°
Hence, this property also holds true for non-convex quadrilateral.
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
Solution:
According to above table we conclude that,
The angle sum of a polygon having side n = (n-2)×180°
Now,
a) 7
Here, n = 7
Hence, angle sum = (7-2)×180° = 5×180° = 900°
b) 8
Here, n = 8
Hence, angle sum = (8-2)×180° = 6×180° = 1080°
c) 10
Here, n = 10
Thus, angle sum = (10-2)×180° = 8×180° = 1440°
d) n
Here, n = n
Hence, angle sum = (n-2)×180°
5. What is a regular polygon? State the name of a regular polygon of
(i)3 sides
(ii) 4 sides
(iii) 6 sides
Solution:
Regular polygon: A polygon with equal sides and equal angles is called regular polygon. i.e., A regular polygon is both equilateral and equiangular.
(i) A regular polygon of 3 sides is called equilateral triangle.
(ii) A regular polygon of 4 sides is called square.
(iii) A regular polygon of 6 sides is called regular hexagon.
6. Find the angle measure x in the following figures.
(a)
Solution:
Given figure is a quadrilateral.
Sum of angles of the quadrilateral = 360°
Now,
⇒ 50° + 130° + 120° + x = 360°
⇒ 300° + x = 360°
⇒ x = 360° – 300° = 60°
b)
Given figure is a quadrilateral.
Sum of angles of the quadrilateral = 360°
Now,
⇒ 90° + 70° + 60° + x = 360°
⇒ 220° + x = 360°
⇒ x = 360° – 220° = 140°
c)
Given figure is having 5 sides. Hence, it is a pentagon.
So,
Sum of angles of the pentagon = 540°
Thus, 180° – 70° = 110°
180° – 60° = 120°
Now,
⇒ 30° + 110° + 120° + x + x = 540°
⇒ 260° + 2x = 540°
⇒ 2x = 540° – 260° = 280°
⇒ 2x = 280°
X = 140°
d)
Given figure is having 5 equal sides. Hence, it is a regular pentagon.
Now,
5x = 540° [All angles of a Regular Pentagon are equal]
⇒ x = 540°/5
⇒ x = 108°
7.
(a)Find x + y + z
Solution:
Sum of all angles of triangle = 180°
Now, y = 180° – a [linear pair]
Y = 120°
Also, z + 30° = 180° [linear pair]
Z = 180° – 30° = 150°
Also, x + 90° = 180° [linear pair]
X = 180° – 90° = 90°
Hence, x + y + z = 90° + 120° + 150° = 360°
b) x + y + z + w
Solution:
Sum of all angles of quadrilateral = 360°
∠r + 260° = 360°
∠r = 360° – 260° = 100°
Now,
X + 120° = 180° [linear pair]
X = 180° – 120° = 60°
Also, y + 80° = 180° [linear pair]
Y = 180° – 80° = 100°
Also, z + 60° = 180° [linear pair]
Z = 180° – 60° = 120°
Also, w = 180°- ∠r = 180° – 100° = 80° [linear pair]
Hence, x + y + z + w = 60° + 100° + 120° + 80° = 360°
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