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
Answer 1:
(a) Simple curve
(b) Simple closed curve
(c) Polygons
(d) Convex polygons
(e) Concave polygon
Question 2:
How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon
(c) A triangle
Answer 2:
(a) A convex quadrilateral has two diagonals.
Here, AC and BD are two diagonals.
(b) A regular hexagon has 9 diagonals.
Here, diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD.
(c) A triangle has no diagonal.
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 nonconvex quadrilateral and try)
Answer 3:
Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles.
= 180^{o }+180^{o} [By Angle sum property of triangle]
= 360^{o}
Hence, the sum of measures of the triangles of a convex quadrilateral is 360^{o}.
Yes, if quadrilateral is not convex then, this property will also be applied.
Let ABCD is a nonconvex quadrilateral and join BD, which also divides the quadrilateral in two triangles.
Using angle sum property of triangle,
In ΔABD, 1 + 2 + 3 = 180^{o} ……….(i)
In ΔBDC, 4 + 5 + 6 = 180^{o} ……….(i)
Adding eq. (i) and (ii),
Hence proved.
Question 4:
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  1 x 180^{0} = (3 2) x 180^{0}  2x180^{0} = (42)xl80^{0}  3x180^{0} = (52)xl80^{0}  4x180^{0} = (62)xl80^{0} 
What can you say about the angle sum of a convex polygon with number of sides?
Answer 4:
(a) When n = 7, then
Angle sum of a polygon = (n  2)x 180^{o} = (7  2)x180^{o} = 5x180^{o} = 900^{o}
(b) When n = 8, then
Angle sum of a polygon = (n  2)x 180^{o} = (8  2)x180^{o} = 6x180^{o} =1080^{o}
(c) When n = 10, then
Angle sum of a polygon = (n  2)x 180^{o} = (10  2)x180^{o} = 8x180^{o} =1440^{o}
(d) When n = n, then
Angle sum of a polygon = (n  2)x 180^{o}
Question 5:
What is a regular polygon? State the name of a regular polygon of:
(a) 3 sides
(b) 4 sides
(c) 6 sides
Answer 5:
A regular polygon: A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon.
(i) 3 sides  Polygon having three sides is called a triangle.
(ii) 4 sides  Polygon having four sides is called a quadrilateral.
(iii) 6 sides  Polygon having six sides is called a hexagon.
Question 6:
Find the angle measures x in the following figures:
Answer 6:
(a) Using angle sum property of a quadrilateral,
50^{o}+130^{o}+120^{o}+ x = 360^{o}
300^{o}+ x = 360^{o}
x = 360^{o } 300^{o}
x = 60^{o}
(b) Using angle sum property of a quadrilateral,
90^{o}+60^{o}+70^{o}+ x = 360^{o}
220^{o}+ x = 360^{o}
x = 360^{o}220^{o}
x =140^{o}
(c) First base interior angle = 180^{o} 70^{o} =110^{o}
Second base interior angle = 180^{o} 60^{o} = 120^{o}
There are 5 sides, n = 5
Angle sum of a polygon = (n  2)x180^{o}
= (52)x 180^{o }= 3x180^{o }= 540^{o}
(d) Angle sum of a polygon = (n  2)x180^{o}
Hence each interior angle is 108^{o}
Question 7:
(a) Find x + y + z
(b) Find x+y+z+w
Answer 7:
(a) Since sum of linear pair angles is 180^{o}.
[Exterior angle property]
x+y+x=90^{o}+120^{o}+150^{o}=360^{o}
(b) Using angle sum property of a quadrilateral,
Since sum of linear pair angles is 180.
w+100 +180^{o} ……….(i)
x+120^{o} +180^{o} ……….(ii)
y +80^{o} +180^{o} ……….(iii)
z +60^{o} +180^{o} ……….(iv)
Adding eq. (i), (ii), (iii) and (iv),
x + y + z + w+ 100^{o} +120^{o} + 80^{o} + 60^{o} +180^{o} +180^{o} +180^{o} +180^{o}
x + y + z + w+ 360^{o} = 720^{o}
x + y + z + w = 720^{o}  360^{o}
x + y + z + w = 360^{o}
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