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Given here are some figures.
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| (1) | (2) | (3) |
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| (4) | (5) | (6) |
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| (7) | (8) |
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
(d)convex polygon-2
(e)concave polygon-1
How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
(c) A triangle
In figure dotted lines represent diagonals.
(a)2
(b)9
(c)0
What is the sum of the measures of the angels of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)
A quadrilateral can be split into 2 triangles. sum of internal angles of a quadrilateral is the sum of internal angles of 2 triangles, which is= 180+180=360°. this property holds true even if the quadrilateral is not convex. An example for non convex quadrilateral is shown in the figure below.
we can split quadrilateral ABCD in to two triangles ABC and ACD, hence sum of angles =180+180=360
Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)
| Figure |
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| Side | 3 | 4 | 5 | 6 |
| Angle sum | 180° | 2 × 180° = (4 − 2) × 180° | 3 × 180° = (5 − 2) × 180° | 4 × 180° = (6 − 2) × 180° |
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)angle sum=(7-2)X180°=5X180°=900º
(b)angle sum=(8-2)X180°=6X180°=1080º
(c)angle sum=(10-2)X180°=8X180°=1440º
(d)angle sum=(n-2)X180°
What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides
(ii) 4 sides
(iii) 6 sides
A polygon with equal side lengths and equal exterior angles called regular polygon.
(i)equilateral triangle
(ii)square
(iii)regular hexagon
Find the angle measure x in the following figures.
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| (a) | (b) | ||
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| (c) | (d) (a) Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral, 50° + 130° + 120° + x = 360° 300° + x = 360° x = 60° (b)
From the figure, it can be concluded that, 90º + a = 180º (Linear pair) a = 180º − 90º = 90º Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral, 60° + 70° + x + 90° = 360° 220° + x = 360° x = 140° (c)
From the figure, it can be concluded that, 70 + a = 180° (Linear pair) a = 110° 60° + b = 180° (Linear pair) b = 120° Sum of the measures of all interior angles of a pentagon is 540º. Therefore, in the given pentagon, 120° + 110° + 30° + x + x = 540° 260° + 2x = 540° 2x = 280° x = 140° (d) Sum of the measures of all interior angles of a pentagon is 540º. 5x = 540° x = 108°
(a) find x + y + z (b) find x + y + z + w |
(a) x + 90° = 180° (Linear pair)
x = 90°
z + 30° = 180° (Linear pair)
z = 150°
y = 90° + 30° (Exterior angle theorem)
y = 120°
x + y + z = 90° + 120° + 150° = 360°
(b)
Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,
a + 60° + 80° + 120° = 360°
a + 260° = 360°
a = 100°
x + 120° = 180° (Linear pair)
x = 60°
y + 80° = 180° (Linear pair)
y = 100°
z + 60° = 180° (Linear pair)
z = 120°
w + 100° = 180° (Linear pair)
w = 80°
Sum of the measures of all interior angles = x + y + z + w
= 60° + 100° + 120° + 80°
= 360°
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