NCERT Class 8 Mathematics Solutions: Chapter 3 –Understanding Quadrilaterals Exercise 3.1 Part 3

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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 convex quadrilateral

Image:

Hence, a convex quadrilateral has 2 diagonals AC and BD.

(b) A regular hexagon

Image:

Hence, a regular hexagon has 9 diagonals AE, AD, AC, BF, BE, BD, CE, CF and DF.

(c) A triangle

Triangle is a polygon with three edges and three vertices. A triangle has no any 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:

Case-1

Let ABCD is a convex quadrilateral, and then we draw a diagonal AC which divides the quadrilateral in two triangles. and.

In

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Since, we know that sum of interior angles of triangle is Thus, the sum of the measures of the angles is .

Yes, if quadrilateral is not convex then, this property will also be applied.

Case-2

Let ABCD is a non-convex quadrilateral and join BD, which also divides the quadrilateral in two triangles.

Image:

Using angle sum of property of triangle,

In , ………….(eq.1)

…………… (eq.2)

Adding eq.1 and eq.2

Thus, this property holds if the quadrilateral is not convex.