CBSE NCERT Solutions of Class 8 Mathematics
Chapter 3 Understanding Quadrilaterals
NCERT solutions of Understanding Quadrilaterals Exercise 3.1
Question 1: Given here are some figures.
Classify each of them on the basis of the following.
(a) Simple curve (b) Simple closed curve (c) Polygon
(d) Convex polygon (e) Concave polygon
(a) Simple Curve: 1, 2, 5, 6, 7
(b) Simple Closed Curve: 1, 2, 5, 6, 7
(c) Polygon: 1, 2, 4
(d) Convex Polygon: 2
(e) Concave Polygon: 1, 4
Question 2: How many diagonals does each of the following have?
(a) A convex quadrilateral (b) A regular hexagon (c) A triangle
(a) A convex quadrilateral has 2 diagonals.
(b) A regular Hexagon has 9 diagonals.
(c) A triangle has no diagonal.
Question 3: What is the sum of the measures of the angels of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)
The sum of the measures of the angles of a convex quadrilateral is 360° because a convex quadrilateral is made of two triangles. In the adjacent figure of convex quadrilateral ABCD, it is made of two triangles ΔABD and ΔBCD. Therefore, the sum of the measures of all the interior angles of this quadrilateral will be same as the sum of all the interior angles of these two triangles = 360°
Yes, this property also holds true even if the quadrilateral is not convex since any quadrilateral can be divided into two triangles. In the adjacent figure the concave quadrilateral ABCD, is made of two triangles ΔABD and ΔBCD. Hence, the sum of all the interior angles of this concave quadrilateral is also 360°.
Question 5: What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
Solution: A regular polygon is both ‘equiangular’ and ‘equilateral’. In other words, a polygon with equal sides and equal angles is called a regular polygon.
(i) Equilateral Triangle, (ii) Square, (iii) Regular Hexagon
Question 6: Find the angle measure x in the following figures.
(a) Sum of all interior angles of a quadrilateral is 360º.
50° + 130° + 120° + /x = 360°
/x = 60°
From the figure, it can be concluded that,
90º + y = 180º (Linear pair)
y = 180º − 90º = 90º
Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,
60° + 70° + x + 90° = 360°
220° + x = 360°
/x = 140°
70 + y = 180° (Linear pair)
y = 110°
60° + z = 180° (Linear pair)
z = 120°
Now, consider pentagon,
120° + 110° + 30° + x + x = 540°
(Sum of all interior angles of a pentagon is 540º)
⇒ 260° + 2x = 540°
⇒ 2x = 280°
⇒ x = 140°
Given figure is of a regular pentagon which is equilateral and equiangular.
Hence, sum of all interior angles of this pentagon is 540º.
5x = 540°
x = 108°
(a) find x + y + z (b) find x + y + z + w
(a) x + 90° = 180° (Linear pair)
x = 90°
Similarly, z = 150°
y = 90° + 30° (Exterior angle theorem)
y = 120°
∴ x + y + z = 90° + 120° + 150° = 360°
(b) Sum of the measures of all interior angles of a quadrilateral is 360O.
∴ a + 60° + 80° + 120° = 360°
⇒ a = 100°
And, x + 120° = 180° (Linear pair)
⇒ x = 60°
Similarly, y = 100°, z = 120° and w = 80°
∴ x + y + z + w = 60° + 100° + 120° + 80° = 360°