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
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Hence, a convex quadrilateral has 2 diagonals AC and BD.
(b) A regular hexagon
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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.