Chapter 03 Understanding Quadrilaterals
3.1 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.
3.1.1 Convex and concave polygons
A simple closed curve made up of only line segments is called a polygon.


Curves that are polygons
Here are some convex polygons and some concave polygons. (Fig 3.1)


Convex 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.
3.1.2 Regular and irregular polygons
Aregular 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?


Regular polygons
[Note: Use of
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?
EXERCISE 3.1
1. Given here are some figures.




(1)




(5)
Classify each of them on the basis of the following.
(a) Simple curve
(d) Convex polygon
2. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides
3.2 Sum of the Measures of the Exterior Angles of a Polygon
On many occasions a knowledge of exterior angles may throw light on the nature of interior angles and sides.
DO THIS
Draw a polygon on the floor, using a piece of chalk. (In the figure, a pentagon
We want to know the total measure of angles, i.e,

Fig 3.2
Therefore,
This is true whatever be the number of sides of the polygon.
Therefore, the sum of the measures of the external angles of any polygon is
Example 1 : Find measure
Solution:
TRY THESE

Take a regular hexagon Fig 3.4.
1. What is the sum of the measures of its exterior angles
2. Is
3. What is the measure of each?
(i) exterior angle
(ii) interior angle
4. Repeat this activity for the cases of
(i) a regular octagon
(ii) a regular 20-gon

Fig 3.4
Example 2 : Find the number of sides of a regular polygon whose each exterior angle has a measure of
Solution Total measure of all exterior angles
Measure of each exterior angle
Therefore, the number of exterior angles
The polygon has 8 sides.
EXERCISE 3.2
1. Find

(a)

(b)
2. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides
3. How many sides does a regular polygon have if the measure of an exterior angle is
4. How many sides does a regular polygon have if each of its interior angles is
5. (a) Is it possible to have a regular polygon with measure of each exterior angle as
(b) Can it be an interior angle of a regular polygon? Why?
6. (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?
3.3 Kinds of Quadrilaterals
Based on the nature of the sides or angles of a quadrilateral, it gets special names.
3.3.1 Trapezium
Trapezium is a quadrilateral with a pair of parallel sides.


These are trapeziums
Study the above figures and discuss with your friends why some of them are trapeziums while some are not. (Note: The \to marks indicate parallel lines).
DO THIS
1. Take identical cut-outs of congruent triangles of sides

Fig 3.5
You get a trapezium. (Check it!) Which are the parallel sides here? Should the non-parallel sides be equal?
You can get two more trapeziums using the same set of triangles. Find them out and discuss their shapes.
2. Take four set-squares from your and your friend’s instrument boxes. Use different numbers of them to place side-by-side and obtain different trapeziums.
If the non-parallel sides of a trapezium are of equal length, we call it an isosceles trapezium. Did you get an isoceles trapezium in any of your investigations given above?
3.3.2 Kite
Kite is a special type of a quadrilateral. The sides with the same markings in each figure are equal. For example


These are kites
Study these figures and try to describe what a kite is. Observe that
(i) A kite has 4 sides (It is a quadrilateral).
(ii) There are exactly two distinct consecutive pairs of sides of equal length.
Check whether a square is a kite.
DO THIS
Take a thick white sheet.
Fold the paper once.
Draw two line segments of different lengths as shown in Fig 3.6.
Cut along the line segments and open up.
You have the shape of a kite (Fig 3.6).
Has the kite any line symmetry?

Fig 3.6
Fold both the diagonals of the kite. Use the set-square to check if they cut at right angles. Are the diagonals equal in length?
Verify (by paper-folding or measurement) if the diagonals bisect each other.
By folding an angle of the kite on its opposite, check for angles of equal measure.
Observe the diagonal folds; do they indicate any diagonal being an angle bisector?
Share your findings with others and list them. A summary of these results are given elsewhere in the chapter for your reference.
Show that

Fig 3.7
3.3.3 Parallelogram
A parallelogram is a quadrilateral. As the name suggests, it has something to do with parallel lines.



These are parallelograms
Study these figures and try to describe in your own words what we mean by a parallelogram. Share your observations with your friends.
Check whether a rectangle is also a parallelogram.
DO THIS
Take two different rectangular cardboard strips of different widths (Fig 3.8).


Strip 1
Place one strip horizontally and draw lines along its edge as drawn in the figure (Fig 3.9).
Now place the other strip in a slant position over the lines drawn and use this to draw two more lines as shown (Fig 3.10).
Fig 3.9
These four lines enclose a quadrilateral. This is made up of two pairs of parallel lines (Fig 3.11).


Fig 3.10
It is a parallelogram.
A parallelogram is a quadrilateral whose opposite sides are parallel.
3.3.4 Elements of a parallelogram
There are four sides and four angles in a parallelogram. Some of these are equal. There are some terms associated with these elements that you need to remember.
Given a parallelogram

Fig 3.12
DO THIS
Take cut-outs of two identical parallelograms, say


Here
Place
Similarly examine the lengths
You may also arrive at this result by measuring
Property: The opposite sides of a parallelogram are of equal length.
TRY THESE
Take two identical set squares with angles
You can further strengthen this idea through a logical argument also.
Consider a parallelogram ABCD (Fig 3.15). Draw any one diagonal, say

Fig 3.15
Fig 3.14
Looking at the angles,
Since in triangles
and
This gives
Example 3 : Find the perimeter of the parallelogram PQRS (Fig 3.16).
Solution In a parallelogram, the opposite sides have same length.
Therefore,
So, Perimeter
3.3.5 Angles of a parallelogram

Fig 3.16
We studied a property of parallelograms concerning the (opposite) sides. What can we say about the angles?
DO THIS
Let

Fig 3.17
Does this tell you anything about the measures of the angles A and C? Examine the same for angles B and D. State your findings.
Property: The opposite angles of a parallelogram are of equal measure.
TRY THESE
Take two identical
You can further justify this idea through logical arguments.
If

Fig 3.18
Studying

Fig 3.19
This shows that
Alternatively,
Example 4 : In Fig 3.20, BEST is a parallelogram. Find the values
Solution
So,
We now turn our attention to adjacent angles of a parallelogram. In parallelogram

Fig 3.21
Identify two more pairs of supplementary angles from the figure.
Property: The adjacent angles in a parallelogram are supplementary.
Example 5 : In a parallelogram RING, (Fig 3.22) if
Solution Given
Then
because
Since

Fig 3.22
Also,
Thus,
THINK, DISCUSS AND WRITE
After showing
3.3.6 Diagonals of a parallelogram
The diagonals of a parallelogram, in general, are not of equal length. (Did you check this in your earlier activity?) However, the diagonals of a parallelogram have an interesting property.
DO THIS
Take a cut-out of a parallelogram, say,

Find the mid point of
Does this show that diagonal
Property: The diagonals of a parallelogram bisect each other (at the point of their intersection, of course!)
To argue and justify this property is not very difficult. From Fig 3.24, applying ASAcriterion, it is easy to see that

Fig 3.24
This gives
Example 6 : In Fig 3.25 HELP is a parallelogram. (Lengths are in cms). Given that
Solution : If
So
Therefore
Hence
(Why?)

Fig 3.25
EXERCISE 3.3
1. Given a parallelogram

2. Consider the following parallelograms. Find the values of the unknowns



(iii)

(iv)

(v)
3. Can a quadrilateral
4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
5. The measures of two adjacent angles of a parallelogram are in the ratio
6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

7. The adjacent figure HOPE is a parallelogram. Find the angle measures
8. The following figures GUNS and RUNS are parallelograms. Find

(ii)

9.

In the above figure both RISK and CLUE are parallelograms. Find the value of
10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.26)

Fig 3.26

Fig 3.27
11. Find
12. Find the measure of

3.4 Some Special Parallelograms
3.4.1 Rhombus
We obtain a Rhombus (which, you will see, is a parallelogram) as a special case of kite (which is not a a parallelogram).
DO THIS
Recall the paper-cut kite you made earlier.

Kite-cut

Rhombus-cut
When you cut along
Note that the sides of rhombus are all of same length; this is not the case with the kite.
A rhombus is a quadrilateral with sides of equal length.
Since the opposite sides of a rhombus have the same length, it is also a parallelogram. So, a rhombus has all the properties of a parallelogram and also that of a kite. Try to list them out. You can then verify your list with the check list summarised in the book elsewhere.

Kite

Rhombus
The most useful property of a rhombus is that of its diagonals.
Property: The diagonals of a rhombus are perpendicular bisectors of one another.
DO THIS
Take a copy of rhombus. By paper-folding verify if the point of intersection is the mid-point of each diagonal. You may also check if they intersect at right angles, using the corner of a set-square.
Here is an outline justifying this property using logical steps.
Since diagonals bisect each other,
We have to show that
It can be seen that by SSS congruency criterion

Fig 3.29
Therefore,
Since
Since
Example 7 :
RICE is a rhombus (Fig 3.30). Find
Solution:

Fig 3.30
3.4.2 A rectangle
A rectangle is a parallelogram with equal angles (Fig 3.31).
What is the full meaning of this definition? Discuss with your friends.
If the rectangle is to be equiangular, what could be the measure of each angle?

Fig 3.31
Let the measure of each angle be
Then
Thus each angle of a rectangle is a right angle.
So, a rectangle is a parallelogram in which every angle is a right angle.
Being a parallelogram, the rectangle has opposite sides of equal length and its diagonals bisect each other.
In a parallelogram, the diagonals can be of different lengths. (Check this); but surprisingly the rectangle (being a special case) has diagonals of equal length.
Property: The diagonals of a rectangle are of equal length.

Fig 3.32

Fig 3.33

Fig 3.34
This is easy to justify. If ABCD is a rectangle (Fig 3.38), then looking at triangles
The congruency follows by SAS criterion.
Thus
and in a rectangle the diagonals, besides being equal in length bisect each other (Why?)
Example 8 : RENT is a rectangle (Fig 3.35). Its diagonals meet at O. Find
Solution
Diagonals are equal here. (Why?)
So, their halves are also equal.
Therefore
or

Fig 3.35 A square is a rectangle with equal sides.
This means a square has all the properties of a rectangle with an additional requirement that all the sides have equal length.
The square, like the rectangle, has diagonals of equal length.
In a rectangle, there is no requirement for the diagonals to be perpendicular to one another, (Check this).

In a square the diagonals.
(i) bisect one another (square being a parallelogram)
(ii) are of equal length (square being a rectangle) and
(iii) are perpendicular to one another.
Hence, we get the following property.
Property: The diagonals of a square are perpendicular bisectors of each other.
DO THIS
Take a square sheet, say PQRS (Fig 3.37).
Fold along both the diagonals. Are their mid-points the same? Check if the angle at
This verifies the property stated above.
We can justify this also by arguing logically:

Fig 3.36
By SSS congruency condition, we now see that
Therefore,
These angles being a linear pair, each is right angle.
EXERCISE 3.4
1. State whether True or False.
(a) All rectangles are squares
(b) All rhombuses are parallelograms
(c) All squares are rhombuses and also rectangles
(d) All squares are not parallelograms. (e) All kites are rhombuses.
(f) All rhombuses are kites.
(g) All parallelograms are trapeziums.
(h) All squares are trapeziums.

Fig 3.37
2. Identify all the quadrilaterals that have. (a) four sides of equal length (b) four right angles
3. Explain how a square is. (i) a quadrilateral (ii) a parallelogram (iii) a rhombus (iv) a rectangle
4. Name the quadrilaterals whose diagonals. (i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal
5. Explain why a rectangle is a convex quadrilateral.
6.

THINK, DISCUSS AND WRITE
1. A mason has made a concrete slab. He needs it to be rectangular. In what different ways can he make sure that it is rectangular?
2. A square was defined as a rectangle with all sides equal. Can we define it as rhombus with equal angles? Explore this idea.
3. Can a trapezium have all angles equal? Can it have all sides equal? Explain.
WHAT HAVE WE DISCUSSED?
Quadrilateral | Properties |
---|---|
Parallelogram: A quadrilateral with each pair of opposite sides parallel. ![]() |
(1) Opposite sides are equal. (2) Opposite angles are equal. (3) Diagonals bisect one another. |
Rhombus: A parallelogram with sides of equal length. ![]() |
(1) All the properties of a parallelogram. (2) Diagonals are perpendicular to each other. |
Rectangle: A parallelogram with a right angle. ![]() |
(1) All the properties of a parallelogram. (2) Each of the angles is a right angle. (3) Diagonals are equal. |
Square: A rectangle with sides of equal length. ![]() |
All the properties of a parallelogram, rhombus and a rectangle. |
Kite: A quadrilateral with exactly two pairs of equal consecutive sides ![]() |
(1) The diagonals are perpendicular to one another (2) One of the diagonals bisects the other. (3) In the figure |