CHAPTER 3: UNDERSTANDING QUADRILATERALS
Simple curve: If a curve does not cross itself, then it is called a simple curve.
Polygons: A simple closed curve made up of only line segments is called a polygon.
Diagonals: A diagonal is a line segment connecting two non-consecutive vertices of a polygon.
Convex Polygons: Polygons that are convex have no portions of their diagonals in their exteriors.
Concave Polygons: Polygons that are concave have some portions of their diagonals in their exteriors.
Regular polygons: A regular polygon is both ‘equiangular’ and ‘equilateral’.
Trapezium: Trapezium is a quadrilateral with a pair of parallel sides.
Isosceles trapezium: If the non-parallel sides of a trapezium are of equal length, we call it an isosceles trapezium.
Kite: A kite is a quadrilateral with exactly two distinct consecutive pairs of sides of equal length.
Parallelogram: A parallelogram is a quadrilateral whose opposite sides are parallel.
The opposite sides of a parallelogram are of equal length.
The opposite angles of a parallelogram are of equal measure.
The adjacent angles in a parallelogram are supplementary.
The diagonals of a parallelogram bisect each other at the point of their intersection.
Rhombus: A rhombus is a quadrilateral with sides of equal length.
The opposite sides of a rhombus are of equal length.
The opposite angles of a rhombus are of equal measure.
The adjacent angles in a rhombus are supplementary.
The diagonals of a rhombus are perpendicular bisectors of one another.
Rectangle: A rectangle is a parallelogram with equal angles
The opposite sides of a rectangle are of equal length.
The opposite angles of a rectangle are of equal measure.
The adjacent angles in a rectangle are supplementary.
The diagonals of a rectangle are of equal length.
The diagonals of a rectangle bisect each other at the point of their intersection.
Square: A square is a rectangle with equal sides.
All angles of a square are of equal measure.
The diagonals of a square are of equal length.
The diagonals of a square are perpendicular bisectors of one another.
Measure of interior angle of a regular polygon =
The number of diagonals in a polygon of n sides is
The sum of interior angles of a polygon of n sides is (n – 2) straight angles, i.e.
The sum of the measures of the external angles of any polygon is 360°.
Each exterior angle of a regular polygon of n sides =
Exercise 3.1
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:
(a) Simple curve: 1, 2, 5, 6, 7
(b) Simple closed curve: 1, 2, 5, 6, 7
(c) Polygon: 1, 2
(d) Convex polygon: 2
(e) Concave polygon: 1
2. How many diagonals does each of the following have?
(a) A convex quadrilateral
Solution: We know that, the number of diagonals in a polygon of n sides is
Here, n = 4.
Number of diagonals =
(b) A regular hexagon (c) A triangle
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 non-convex quadrilateral and try!)
Solution:
ABCD is a convex quadrilateral and AC diagonal.
From ∆ABC we get,
∠ABC+∠BAC+∠ACB = 180 ………. (1) [By Angle sum property of triangle]
Also, from ∆ADC we get,
∠ADC+∠DAC+∠ACD = 180 ………. (2) [By Angle sum property of triangle]
(1) + (2) ⇒
∠ABC+∠BAC+∠ACB+∠ADC+∠DAC+∠ACD = 180 +180
⇒∠BAC+∠DAC+∠ABC+∠ACB+∠ACD +∠ADC = 360
⇒∠A+∠B+∠C+∠D = 360
Hence, the sum of measures of the angles of a convex quadrilateral is 360°.
Yes, if quadrilateral is not convex then, this property will also be applied.
ABCD is a non-convex quadrilateral and AC diagonal.
From ∆ABC we get,
∠ABC+∠BAC+∠ACB = 180 ………. (3) [By Angle sum property of triangle]
Also, from ∆ADC we get,
∠ADC+∠DAC+∠ACD = 180 ………. (4) [By Angle sum property of triangle]
(3) + (4) ⇒
∠ABC+∠BAC+∠ACB+∠ADC+∠DAC+∠ACD = 180 +180
⇒∠BAC+∠DAC+∠ABC+∠ACB+∠ACD +∠ADC = 360
⇒∠A+∠B+∠C +∠D = 360°
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
Solution:
Angle sum = (7 – 2) × 180° = 5×180° = 900°
(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
Solution:
A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon.
(i) Regular polygon having 3 sides is called an equilateral triangle.
(ii) Regular polygon having 4 sides is called a square.
(iii) Regular polygon having 6 sides is called a Regular hexagon.
6. Find the angle measure x in the following figures.
a)
Solution:
We know that, the sum of the angles of a quadrilateral is 360°.
⸫x+50+130+120 = 360
⇒x+300 = 360
⇒x = 60°
b)
Solution:
We know that, the sum of the angles of a quadrilateral is 360°.
⸫x+70+60+90 = 360
⇒x+220 = 360
⇒x = 360 – 220
⇒x = 140°
c)
Solution:
We know that, the sum of the angles of a pentagon is 540°.
⸫30 + x + x + (180 – 70) + (180 – 60) = 540
30 + 2x + 110 + 120 = 540
⇒2x + 260 = 540
⇒2x = 540 – 260
⇒x =
⇒x = 140
d)
Solution:
We know that, the sum of the angles of a pentagon is 540°.
5x = 540
⇒x =
⇒x = 108
7.
(a) Find x + y + z
Solution:
Sum of all angles of triangle= 180°
⇒30 + 90 + (180 – y) = 180
⇒120 + 180 – y = 180
⇒– y = –120
⇒y = 120
x + 90° = 180° ⇒x = 180° – 90° ⇒x = 90°
z + 30° = 180° ⇒ z = 180° – 30° ⇒ z = 150°
⸫x + y + z = 90° + 120° + 150° = 360°
(b) Find x + y + z + w
Solution:
The sum of the angles of a quadrilateral = 360°
⇒60 + 80 + 120 + (180 – w) = 360
⇒260 + 180 – w = 360
⇒440 – w = 360
⇒– w = 360 – 440
⇒– w = – 80
⇒w = 80
⸫x + y + z + w
= (180 – 120) + (180 – 80) + (180 – 60) + 80
= 60 + 100 + 120 + 80
= 360
Exercise 3.2
1. Find x in the following figures.
a)
Solution:
The sum of the measures of the external angles of any polygon = 360°
⇒ x + 125 + 125 = 360
⇒ x + 250 = 360
⇒ x = 360 – 250
⇒ x = 110
b)
Solution:
The sum of the measures of the external angles of any polygon = 360°
⇒ x + 70 + 90 + 60 + (180 – 90) = 360
⇒ x + 310 = 360
⇒ x = 360 – 310
⇒ x = 50
2. Find the measure of each exterior angle of a regular polygon of
(i) 9 sides (ii) 15 sides
Solution:
(i) We know that, each exterior angle of a regular polygon of n sides =
Here, n = 9.
⸫Measure of each exterior angle =
(ii) We know that, each exterior angle of a regular polygon of n sides =
Here, n = 15.
⸫Measure of each exterior angle =
3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
Solution: We know that, number of sides of a regular polygon with exterior angle θ° =
Here, θ = 30°.
⸫Number of sides =
4. How many sides does a regular polygon have if each of its interior angles is 165°?
Solution:
We know that, interior angles + exterior angle = 180°.
Here, interior angles = 165°.
⸫Exterior angle = 180 – 165 = 15°.
Again, we know that, number of sides of a regular polygon with exterior angle θ° =
Here, θ = 15°.
⸫Number of sides =
5. (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?
Solution:
(a) We know that, number of sides of a regular polygon with exterior angle θ° =
Here, θ = 22°.
⸫Number of sides =
But, number of sides cannot be fraction.
⸫Measure of exterior angle cannot be 22°.
(b) We know that, interior angles + exterior angle = 180°.
Here, interior angle = 22°.
⸫Exterior angle = 180 – 22 = 158°.
Again, we know that, number of sides of a regular polygon with exterior angle θ° =
Here, θ = 158°.
⸫Number of sides =
But, number of sides cannot be fraction.
⸫Measure of interior angle cannot be 22°.
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?
Answer:
(a) The equilateral triangle being a regular polygon of 3 sides has the least measure of an interior angle = 60°.
(b) By (a), we can see that the greatest exterior angle is 120°.
Exercise 3.3
1. 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 = ......
Solution:
(i) AD = BC [⸪The opposite sides of a parallelogram are of equal length]
(ii) ∠DCB = ∠DAB [⸪The opposite angles of a parallelogram are of equal measure]
(iii) OC = OA [⸪The diagonals of a parallelogram bisect each other]
(iv) m∠DAB + m ∠CDA = 180 [⸪The adjacent angles in a parallelogram are supplementary]
2. Consider the following parallelograms. Find the values of the unknowns x, y, z.
(i)
Solution:
We know that, the adjacent angles in a parallelogram are supplementary.
⸫∠DCB + ∠CBA = 180
⇒ x + 100 = 180
⇒ x = 180 – 100
⇒ x = 80
∠ADC = ∠ABC [⸪The opposite angles of a parallelogram are of equal measure]
⇒ y = 100
∠DAB = ∠DCB [⸪The opposite angles of a parallelogram are of equal measure]
⇒ z = 80
(ii)
Solution:
x + 50 = 180 [⸪The adjacent angles in a parallelogram are supplementary]
⇒ x = 130
y = x [⸪The opposite angles of a parallelogram are of equal measure]
⇒ y = 130
z = x [Corresponding angles]
⇒ z = 130
(iii)
Solution:
x = 90 [Vertically opposite angles]
x + y + 30 = 180 [Angle sum property of a triangle]
⇒ 90 + y + 30 = 180
⇒ 120 + y = 180
⇒ y = 180 – 120
⇒ y = 60
z = y [Alternate angles]
⇒ z = 60
(iv)
Solution:
x + 80 = 180 [⸪The adjacent angles in a parallelogram are supplementary]
⇒ x = 180 – 80
⇒ x = 100
y = 80 [Opposite angles of a parallelogram are equal]
z = 80 [Corresponding angles]
(v)
Solution:
y = 112 [Opposite angles of a parallelogram are equal]
x + y + 40 = 180 [Angle sum property of a triangle]
⇒ x + 112 + 40 = 180
⇒ x + 152 = 180
⇒ x = 180 – 152
⇒ x = 28
z = x [Alternate angles]
⇒ z = 28
3. 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°?
Answer:
(i) Can be, but need not be.
(ii) No; (in a parallelogram, opposite sides are equal; but here, AD ≠BC).
(iii) No; (in a parallelogram, opposite angles are equal; but here, ∠A ≠∠C).
4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
Solution:
5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.
Solution:
Let, the measures of two adjacent angles be 3x and 2x
We know that, the adjacent angles in a parallelogram are supplementary.
A/Q,
⸫The angles of the parallelogram are 72°, 108°, 72° and 108°.
6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Solution:
Let, the measures of two adjacent angles be x and x
We know that, the adjacent angles in a parallelogram are supplementary.
A/Q,
x + x = 180
⇒ 2x = 180
⇒ x = 90
⸫The angles of the parallelogram are 90°, 90°, 90° and 90°.
7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
Solution:
x = 180 – 70 [Opposite angles of a parallelogram are equal]
⇒ x = 110
y = 40 [Alternate angles]
y + z + (180 – 70) = 180 [Angle sum property of a triangle]
⇒ 40 + z + 110 = 180
⇒ 150 + z = 180
⇒ z = 180 – 150
⇒ z = 30
8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)
(i)
Solution:
We know that, the opposite sides of a parallelogram are of equal
GU = SN
⇒ 3y – 1 = 26
⇒ 3y = 27
⇒ y = 9
GS = UN
⇒ 3x = 18
⇒ x = 6
⸫ x = 6 and y = 9
(ii)
Solution:
We know that, the diagonals of a parallelogram bisect each other.
y + 7 = 20
⇒ y = 20 – 7
⇒ y = 13
x + y = 16
⇒ x + 13 = 16
⇒ x = 16 – 13
⇒ x = 3
⸫ x = 3 and y = 13
9. In the above figure both RISK and CLUE are parallelograms. Find the value of x.
Solution:
In parallelogram RISK,
∠RKS + ∠KSI = 180 [The adjacent angles in a parallelogram are supplementary]
⇒ 120 + ∠KSI = 180
⇒ ∠KSI = 180 – 120
⇒ ∠KSI = 60
In parallelogram CLUE,
CEU = ULC [Opposite angles of a parallelogram are equal]
⇒ ∠CEU = 70
Again,
x + 60 + 70 = 180 [Angle sum property of a triangle]
⇒ x = 180 – 130
⇒ x = 50
10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig)
Answer: NM|| KL [⸪Sum of interior opposite angles is 180°.]
⸫KLMN is a trapezium.
11. Find m∠C in Fig if AB || DC.
Solution:
We know that, sum of interior opposite angles is 180°.
∠C + ∠B = 180
⇒ ∠C + 120 = 180
⇒ ∠C = 180 – 120
⇒ ∠C = 60
12. Find the measure of ∠P and ∠S if SP || RQ in Fig. (If you find m∠R, is there more than one method to find m∠P?)
Solution:
We know that, sum of interior opposite angles is 180°.
∠P + ∠Q = 180
⇒ ∠P +130 = 180
⇒ ∠P = 180 – 130
⇒ ∠P = 50
∠S + ∠R = 180
⇒ ∠S + 90 = 180
⇒ ∠S = 180 – 90
⇒ ∠S = 90.
Exercise 3.4
1. State whether True or False.
(a) All rectangles are squares Answer: False
(b) All rhombuses are parallelograms Answer: True
(c) All squares are rhombuses and also rectangles Answer: True
(d) All squares are not parallelograms. Answer: False
(e) All kites are rhombuses. Answer: False
(f) All rhombuses are kites. Answer: True
(g) All parallelograms are trapeziums. Answer: True
(h) All squares are trapeziums. Answer: True
2. Identify all the quadrilaterals that have.
(a) four sides of equal length Answer: Rhombus and square
(b) four right angles Answer: Rectangle and square
3. Explain how a square is.
(i) a quadrilateral
Answer: A square has four sides, therefore it is a quadrilateral.
(ii) a parallelogram
Answer: Opposite sides of a square are parallel, therefore it is a parallelogram.
(iii) a rhombus
Answer: A square has four equal sides, therefore it is a rhombus.
(iv) a rectangle
Answer: Opposite sides of a square are parallel and all angles are 90°, therefore it is a rectangle.
4. Name the quadrilaterals whose diagonals.
(i) bisect each other
Answer: Parallelogram, rhombus, square and rectangle.
(ii) are perpendicular bisectors of each other
Answer: Rhombus and square
(iii) are equal
Answer: Square and rectangle
5. Explain why a rectangle is a convex quadrilateral.
Answer: A rectangle is a convex quadrilateral, since both the diagonals lie inside the rectangle.
6. ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).
Solution:
AD and DC are drawn so that AD || BC and AB || DC
AD = BC and AB = DC
ABCD is a rectangle as opposite sides are equal and parallel to each other and all the interior angles are of 90°.
In a rectangle, diagonals are of equal length and also bisects each other.
Hence, AO = OC = BO = OD
Thus, O is equidistant from A, B and C.
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