Скачать или смотреть Q.2 - EXERCISE 3.1 - Chapter 3 Understanding Quadrilaterals - NCERT CLASS 8 Math Solutions

2. How many diagonals does each of the following have?
(a) A convex quadrilateral (b) A regular hexagon (c) A triangle
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

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 anon-convex quadrilateral and try!)
4. Examine the table. (Each figure is divided into triangles and the sum of the angles
deduced from that.)
What can you say about the angle sum of a convex polygon with number of sides?
(a) 7 (b) 8 (c) 10 (d) n
5. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides (ii) 4 sides (iii) 6 sides
6. Find the angle measure x in the following figures.
Introduction
You know that the paper is a model for a plane surface. When you join a number of
points without lifting a pencil from the paper (and without retracing any portion of the
drawing other than single points), you get a plane curve.
Try to recall different varieties of curves you have seen in the earlier classes.
Match the following: (Caution! A figure may match to more than one type).
Polygons
A simple closed curve made up of only line segments is called a polygon.
Classification of polygons
We classify polygons according to the number of sides (or vertices) they have.
Diagonals
A diagonal is a line segment connecting two non-consecutive vertices of a polygon.
Can you name the diagonals in each of the above figures? (Fig 3.1)
Is PQ a diagonal? What about LN ?
You already know what we mean by interior and exterior of a closed curve
Convex and concave polygons
Can you find how these types of polygons differ from one another? Polygons that are
convex have no portions of their diagonals in their exteriors or any line segment joining any
two different points, in the interior of the polygon, lies wholly in the interior of it . Is this true
with concave polygons? Study the figures given. Then try to describe in your own words
what we mean by a convex polygon and what we mean by a concave polygon. Give two
rough sketches of each kind.
In our work in this class, we will be dealing with convex polygons only.
Regular and irregular polygons
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides of
equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is
equiangular but not equilateral. Is a rectangle a regular polygon? Is an equilateral triangle a
regular polygon? Why?
In the previous classes, have you come across any quadrilateral that is equilateral but not
equiangular? Recall the quadrilateral shapes you saw in earlier classes – Rectangle, Square,
Rhombus etc.
Is there a triangle that is equilateral but not equiangular?
Angle sum property
Do you remember the angle-sum property of a triangle? The sum of the measures of the
three angles of a triangle is 180°. Recall the methods by which we tried to visualise this
fact. We now extend these ideas to a quadrilateral.
Take any quadrilateral, say ABCD (Fig 3.4). Divide
it into two triangles, by drawing a diagonal. You get
six angles 1, 2, 3, 4, 5 and 6.
Use the angle-sum property of a triangle and argue
how the sum of the measures of ∠A, ∠B, ∠C and
∠D amounts to 180° + 180° = 360°.
Take four congruent card-board copies of any quadrilateral ABCD, with angles
as shown [Fig 3.5 (i)]. Arrange the copies as shown in the figure, where angles
∠1, ∠2, ∠3, ∠4 meet at a point [Fig 3.5 (ii)].
As before consider quadrilateral ABCD (Fig 3.6). Let P be any
point in its interior. Join P to vertices A, B, C and D. In the figure,
consider ∆PAB. From this we see x = 180° – m∠2 – m∠3;
similarly from ∆PBC, y = 180° – m∠4 – m∠5, from ∆PCD,
z = 180º – m∠6 – m∠7 and from ∆PDA, w = 180º – m∠8
– m∠1. Use this to find the total measure m∠1 + m∠2 + ...
m∠8, does it help you to arrive at the result? Remember
∠x + ∠y + ∠z + ∠w = 360°.
These quadrilaterals were convex. What would happen if the
quadrilateral is not convex? Consider quadrilateral ABCD. Split it
into two triangles and find the sum of the interior angles.