﻿ How many diagonals does each of the following have?(a) A convex quadrilateral(b) A regular hexagon(c) A triangle from Mathematics Understanding Quadrilaterals Class 8 Manipur Board

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Find the angle measure x in the following figures.

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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’

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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.

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# How many diagonals does each of the following have?(a) A convex quadrilateral(b) A regular hexagon(c) A triangle

Solution:

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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°

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