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: The sum of measures of angles of a convex quadrilateral = 360° Yes, this property holds, even if the quadrilateral is not convex.
Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that).
|
Figure |
|
|
|
|
|
Side |
3 |
4 |
5 |
6 |
|
Angle sum |
180° |
2 x 180° = (4 - 2) x 180° |
3 x 180° = (5 - 2) x 180° |
4 x 180° = (6 - 2) x 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
Solution: From the above table, we conclude that sum of the interior angles of polygon with n-sides = (n - 2) x 180°
(a) When n = 7
Substituting n = 7 in the above formula, we have Sum of interior angles of a polygon of 7 sides (i.e. when n = 7)
= (n - 2) x 180° = (7 - 2) x 180°
= 5 x 180°
= 900°
(b) When n = 8
Substituting n = 8 in the above formula, we have
Sum of interior angles of a polygon having 8 sides
= (n - 2) x 180° = (8 - 2) x 180°
= 6 x 180°
= 1080°
(c) When n = 10
Substituting n = 10 in the above formula, we have
Sum of interior angles of a polygon having 10 sides
= (n - 2) x 180° = (10 - 2) x 180°
= 8 x 180°
= 1440°
(d) When n = n
The sum of interior angles of a polygon having n-sides = (n - 2) x 180°
How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
(c) A triangle
Solution:
What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
Solution: A polygon is said to be a regularpolygon if
(a) The measures of its interior angels are equal, and
(b) The lengths of its sides are equal.
The name of a regular polygon having
(i) 3 sides is ‘equilateral-triangle’
(ii) 4 sides is ‘square’
(iii) 6 sides is ‘regular hexagon’
Switch
