# NCERT Class 8 Mathematics Solutions: Chapter 3 – Understanding Quadrilaterals Exercise 3.1 Part 3 (For CBSE, ICSE, IAS, NET, NRA 2023)

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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

Image:

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.