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Understanding Quadrilaterals
Main Concepts and Results
• A simple closed curve made up of only line segments is called a polygon.
• A diagonal of a polygon is a line segment connecting two nonconsecutive vertices.
• A convex polygon is a polygon in which no portion of its any diagonal is in its exterior.
• A quadrilateral is a polygon having only four sides.
•A regular polygon is a polygon whose all sides are equal and also all angles are equal.
• The sum of interior angles of a polygon of n sides is (n-2) straight angles.
• The sum of interior angles of a quadrilateral is 360°.
• The sum of exterior angles, taken in an order, of a polygon is 360°.
• Trapezium is a quadrilateral in which a pair of opposite sides is parallel.
• Kite is a quadrilateral which has two pairs of equal consecutive sides.
• A parallelogram is a quadrilateral in which each pair of opposite sides is parallel.
• A rhombus is a parallelogram in which adjacent sides are equal.
• A rectangle is a parallelogram in which one angle is of 900.
• A square is a parallelogram in which adjacent sides are equal and one angle is of 900.
• In a parallelogram, opposite sides are equal, opposite angles are equal and diagonals bisect each
other.
• In a rhombus diagonals intersect at right angles
. • In a rectangle diagonals are equal.
• Five measurements can determine a quadrilateral uniquely.
• A quadrilateral can be constructed uniquely if the lengths of its four sides and a diagonal are
given.
Exercise:
1) What is a polygon? How polygons are classified?
2) What do you mean by a diagonal?
3) What do you mean by interior and exterior of a closed curve? Does the exterior of a closed
curve have a boundary?
4)
5)
6)
7)
What are convex and concave polygons?
What are regular and irregular polygons?
What is the sum of three angles of a triangle?
Draw each of the following and also mention the number of sides or vertices they have.
(i)
Triangle (ii) Quadrilateral (iii) Pentagon (iv) Hexagon (v) Heptagon
(vi)
Octagon (vii) Octagon
(viii) Nonagon (ix) Decagon
8) An “n-gon” has how many sides or vertices?
9) What is a diagonal? Give examples.
10) The sum of measures of the four angles of a quadrilateral is?
11) If three angles of a quadrilateral are each equal to 75°, the fourth angle is?
12) The sum of all interior angles of a polygon is called the angle sum. True or false?
13) Angle sum of a polygon = (n - 2) x 180°, where n = number of sides of the polygon. True or
false?
14)
15) How many diagonals does each of the following have?
(a) A convex quadrilateral
(b) A regular hexagon (c) A triangle
16) What is the sum of measures of the angles of a convex quadrilateral? Will this property hold
if the quadrilateral is not convex?
17) 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
18) What is a regular polygon? State the name of a regular polygon of
(i)
3 sides
(ii) 4 sides
(iii) 6 sides
19) Find the angle measure x in the following figures.
20) Two adjacent angles of a parallelogram are in the ratio 4:5. Find their measures.
21) The four angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find the angles.
22) ABCD is a quadrilateral in which AB = 5 cm, CD = 8 cm and the sum of angle A and angle D is
180°. What is the name of this quadrilateral?
23) The angles of a quadrilateral ABCD taken in an order are in the ratio 3 : 7 : 6 : 4. Then ABCD
is a_________________.
24)
25) What is the sum of the measures of the exterior angles of a polygon?
26) Find measure x in the fig below?
27)
28) Find the number of sides of a regular polygon whose each exterior angle has a measure of
45°.
29) Find the measure of each exterior angle of a regular polygon of
(i)
9 sides (ii) 15 sides (iii) 11 sides
30) How many sides does a regular polygon have if the measure of an exterior angle is 24°?
31) How many sides does a regular polygon have if each of its interior angles is 165°?
32) (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°.
(b) Can it be an interior angle of a regular polygon?Why?
33) (a) What is the minimum interior angle possible for a regular polygon? Why?
(c) What is the maximum exterior angle possible for a regular polygon?
34) Give the definition and draw the diagram of (i) Trapezium (ii) Parrellelogram (iii) Rhombus
(iv) Rectangle (v) Square
35) Study the below mentioned figures and try to describe what a kite is.
36) Draw a rough figure of a quadrilateral that is not a parrellelogram but has exactly two
opposite angles of equal measure.
37) Study the below mentioned figures and try to describe what a parallelogram is.
38) A parallelogram is a quadrilateral whose opposite sides are parallel. True or false?
39) Prove taht the opposite sides of a parallelogram are of equal length.
40) Find the perimeter of the parallelogram PQRS.
41) Prove that the opposite angles of a parallelogram are of equal measure.
42) In Fig given below, BEST is a parallelogram. Find the values x, y and z.
43) In a parallelogram RING, (Fig 3.28) if m˂R = 70° , find all the other angles.
44) Prove that the diagonals of a parallelogram bisect each other (at the point of their
intersection)
45) In Fig given below HELP is a parallelogram.( Lengths are in cms). Given that OE=4 and HL is 5
more than PE? Find OH?
46) Given a parallelogram ABCD. Complete each statement along with the definition or property
used.
(i)
AD = ......
(II) ˂DCB = ......
(III) OC= .....
(iv)
m<DAB + m <CDA = ......
47) Consider the following parallelograms. Find the values of the unknown x ,y , z.
48) Can a quadrilateral ABCD be a parallelogram if
(i)
<D + <B = 180°
(II) AB= DC= 8 cm , AD = 4 cm and BC= 4.4 cm (iii) <A = 70° and
<C= 65° ?
49) The measure of two adjacent angles of a parallelogram are in the ratio 3:2. Find the measure
of each of the angles of the parallelogram.
50) Two adjacent angles of a parallelogram have equal measure. Find the measure of each of
the angles of the parallelogram.
51) The below mentioned figure HOPE is a parallelogram. Find the angle measures x , y and z.
State the properties you use to find them.
52) The following figures GUNS and RUNS are parallelograms. Find x and y.( Lengths are in cm)
53) In the below mentioned figure both RISK and CLUE are parallelograms. Find the value of x.
54) Explain how this fighure is a trapezium. Which of its two sides are parallel?
55) Find m<C in Fig. Below if AB II DC.
56) Find the measure of <P and <S if SP II RQ in fig. Below:
57) Prove that the diagonals of a rhombus are perpendicular bisectors of one another.
58) RICE is a rhombus(fig. below). Find x , y , z. Justify your findings.
59) A rectangle is a parallelogram with equal angles.True or false?
60) Prove that the diagonals of a rectangle are of equal length.
61) RENT is a rectangle (Fig below) . Its diagonals meet at O. Find x, if OR = 2x+4 and OT = 3x+1.
62) A square is a rectangle with equal sides. True or false?
63) The squares , like the rectangle, has diagonals of equal length.
64) Prove that the diagonals of a square are perpendicular bisectors of each other.
65) State whether True or False.
(i)
All rectangles are squares
(ii)
All rhombuses are parallelograms
(iii)
All squares are rhombuses and also rectangles
(iv)
All squares are not parallelograms
(v)
All kites are rhombuses
(vi)
All rhombuses are kites
(vii)
All parallelograms are trapeziums
66) Identify all the quadrilaterals that have.
(i)
Four sides of equal length
(ii)
Four right angles
67) Explain how a square is.
(i)
A quadrilateral
(ii) a parallelogram (iii) a rhombus (iv) a rectangle
68) Name the quadrilaterals whose diagonals
(i)
Bisect each other
(ii)
Are perpendicular bisectors of each other
(iii)
Are equal
69) Expalin why a rectangle is a convex quadrilateral.
70) ABC is aright-angled triangle and O is the mid point of the side oppsite to the right angle.
Explain why O is equidistant from A, B and C.( The dotted lines are drawn additionally to help
you).
Multiple choice questions:
71) ABCD is a rhombus in which the altitude from D to side AB bisects AB. Then < A and <B
respectively, are___
(a) 60°, 120°
(b) 120° , 60°
(c) 80°, 100°
(d) 100°, 80°
72) The exterior angle of an regular polygon is one-third of its interior angle. How many sides
does the polygon has?
(a) 10
(b) 8
(c) 9
(d) 13
73) If the diagonals of a quadrilateral bisect each other at right angle, then it is a ____
(a) Kite
(b) Parallelogram
(c) Rhombus
(d) Rectangle
74) The number of sides of a regular polygon whose eavh exterior angle has a measure of 45° is
___
(a) 4
(b) 6
(d) 8
(d) 10
75) If ABCD is an isosceles trapezium, < C is equal to ___
(a) < B
(b) < A
(c) < D
(d) 90°
76) If each pair of opposite sides of a quadrilateral are equal and parallel, then it is a ___
(a) Kite
(b) Trapezium
(c) Parallelogram (d) None of these
77) In a parallelogram ABCD, if < A = 45°, then the other angles are ___
(a) 45° , 120°, 120°
(b) 45°, 135 °, 135°
(c) 60°, 135°, 135° (d) 60°, 120°, 120°
78) ABCD is a rhombus. If < ACB = 40°, then < ADB is ___
(a) 40°
(b) 45°
(c) 50°
(d) 60°
79) Four angles of quadrilateral are in the ratio 3 : 5 : 7 : 9 . The greates angle is __
(a) 125°
(b) 75°
(c) 135°
(d) 120°
80) It is possible to construct a quadrilateral with the susufficient dat ( other than five simple
cases), where less than _____ parts but some other relatons between them are given.
(a) Four
(b) Five
(c) Three
(d) Two
81) If a quadrilateral has exactly two pairs of equal adjacent sides and the unequal opposite
sides, then it is called____.
(a) Parallelogram
(b) Square
(c) Rectangle
(d) Kite
82) In the given figure, line RT is drawn parallel to SQ.If < QPS = 100°, <PQS = 40°, < PSR = 85°and
<QRS = 70°, then <QRT is ___
(a) 45°
(b) 65°
83) Match the following:
(c) 85°
(d) 90°
84) Choose the correct statement.
(a) The diagonals of a paralelogram are equal
(b) The diagonals of a rectangle are perpendicular to each other.
(c) If the diagonals of a quadrilateral intersect at right angles, it is not necessarily a
rhombus.
(d) Every quadrilateral is either a trapezium or a parallelogram or a ite.
85) Two adjacent angles of a parallelogram are in the ratio 1:5. Then all the angles of the
parallelogram are
(a) 30°, 150°, 30°, 150°
(b) 85°, 95°, 85°, 95°
(c) 45°, 135°, 45°, 135°
(d) 30°, 180°, 30°, 180°
86) A parallelogram PQRS is constructed with sides QR = 6 cm, PQ = 4 cm and ∠PQR = 90°. Then
PQRS is a
(a) square
(b) rectangle
(c) rhombus
(d) trapezium
87) If three angles of a quadrilateral are each equal to 75°, the fourth angle is
a) 150°
(b) 135°
(c) 45°
(d) 75°
88) Which of the following figures satisfy the following properties?
- All sides are congruent.
- All angles are right angles.
- Opposite sides are parallel.
89) Which of the following figures satisfy the following property? - Has two pairs of congruent
adjacent sides.
90) Which of the following figures satisfy the following property? - Only one pair of sides are
parallel.
91) Which of the following figures do not satisfy any of the following properties?
- All sides are equal.
- All angles are right angles.
- Opposite sides are parallel.
92) Which of the following properties describe a trapezium?
(a) A pair of opposite sides is parallel.
(b) The diagonals bisect each other.
(c) The diagonals are perpendicular to each other.
(d) The diagonals are equal.
93) Which of the following is a property of a parallelogram?
(a) Opposite sides are parallel.
(b) The diagonals bisect each other at right angles.
(c) The diagonals are perpendicular to each other.
(d) All angles are equal.
94) What is the maximum number of obtuse angles that a quadrilateral can have?
(a) 1
(b) 2
(c) 3
(d) 4
95) How many non-overlapping triangles can we make in an n-gon (polygon having n sides), by
joining the vertices?
(a) n –1
(b) n –2
(c) n –3
(d) n –4
96) What is the sum of all the angles of a pentagon?
(a) 180°
(b) 360°
(c) 540°
(d) 720°
97) What is the sum of all angles of a hexagon?
(a) 180°
(b) 360°
(c) 540°
(d) 720°
98) If two adjacent angles of a parallelogram are (5x – 5)° and (10x + 35)°, then the ratio of these
angles is
(a) 1 : 3
(b) 2 : 3
(c) 1 : 4
(d) 1 : 2
99) How many diagonals does a hexagon have?
(a) 9
(b) 8
(c) 2
(d) 6