Understanding Quadrilaterals

Simple curved figures : These figures are the figures which contain curved side in it.

Simple closed curves : All the simple curved figures that are closed are called simple closed curves

Polygons : Polygons are closed figures that does not contain any curves. For example - triangle, rectangle, square, hexagon etc.

Convex polygons : A convex polygon is defined as a polygon with all its interior angles less than 180°. This means that all the vertices of the polygon will point outwards, away from the interior of the shape.

Concave Polygons : A concave polygon is defined as a polygon with one or more interior angles greater than 180°. It looks sort of like a vertex has been 'pushed in' towards the inside of the polygon.



A. Simple curved figures are: 3, 5, 6, 7, 8

B. Simple closed curved figures are: 3, 5, 6, 7

C. Polygons are: 1, 2

D. Convex polygons are: 2

E. Concave polygon are: 1

(a) There are 2 diagonals in a convex quadrilateral.(See in figure below)

(b) There are 9 diagonals in a regular hexagon

(Represented in figure)

(c) A triangle does not have any diagonal in it

(Represented in figure)

The angle sum of a convex quadrilateral is 360° as a convex quadrilateral is made of two triangles:

So, 2 × 180° = 360°

Here,

ABCD is a convex quadrilateral, made of two triangles ΔABD and ΔBCD.

Therefore,

The sum of all the interior angles of this quadrilateral will be same as the sum of all the interior angles of these two triangles i.e.,

180° + 180° = 360°

Yes, this property also holds true for a quadrilateral which is not convex. This is because any quadrilateral can be divided into two triangles.

Again, ABCD is a concave quadrilateral, made of two triangles ΔABD and ΔBCD.

Therefore,

Sum of all the interior angles of this quadrilateral will also be:

= 180° + 180°

= 360°

From the table, it can be concluded that the angle sum of a convex polygon of n sides

is (n −2) × 180°.

Hence, the angle sum of the convex polygons having number of sides as above will be as follows:

A. Given number of sides =7

Therefore,

Sum of angles = (7 − 2) × 180^o = 900°

B. Given number of sides = 8

Therefore,

Sum of angles = (8 − 2) × 180^o = 1080°

C. Given number of sides = 10

Therefore,

Sum of angles = (10 − 2) × 180^o = 1440°

D. Given number of sides =n

Therefore,

Sum of angles = (n − 2) × 180^o

A polygon with regular sides and equal angles is called regular polygon.

Now,

(i) Equilateral Triangle

(ii) Square

(iii) Regular Hexagon

A. We know that,

Sum of the measures of all interior angles of a quadrilateral is 360^o.

Therefore, in the given quadrilateral,

50° + 130° + 120° + x = 360°

300° + x = 360°

x = 60°

B.

We know that, angle sum of quadrilateral = 360⁰

90^o + a = 180^o (Linear pair)

a = 180^o – 90^o

= 90^o

Sum of the measures of all interior angles of a quadrilateral is 360^o.

Therefore, in the given quadrilateral,

60° + 70° + x + 90° = 360°

220° + x = 360°

x = 140°

C. From the figure,

70 + a = 180° (Linear pair)

a = 110°

60° + b = 180° (Linear pair)

b = 120°

Sum of the measures of all interior angles of a pentagon is 540^o

Therefore, in the given pentagon,

120° + 110° + 30° + x + x = 540°

260° + 2x = 540°

2x = 280°

x = 140°

D.
As, every angle of a regular pentagon is equal, hence every angle of given pentagon is x
Also, Sum of the measures of all interior angles of a pentagon is 540^o

Hence, we have
5x = 540°

x = 108°

A.
We know that,

x + 90° = 180° (Linear pair)
x = 180° - 90°

x = 90°

z + 30° = 180° (Linear pair)

z = 150°

y = 90° + 30° (Exterior angle theorem)

y = 120°

x + y + z = 90° + 120° + 150° [ all exterior angles make angles around point]

= 360°

B. We know that angle sum of quadrilateral 360^o. Therefore, in the given quadrilateral,

x + 120° = 180° (Linear pair)

x = 60°

y + 80° = 180° (Linear pair)

y = 100°

z + 60° = 180° (Linear pair)

z = 120°

w + 100° = 180° (Linear pair)

w = 80°

Sum of the measures of all exterior angles = x + y + z + w

= 60° + 100° + 120° + 80°

= 360°

Sum of all exterior angles of polygon = 360^o. ..... (1)

A.

125° + 125° + x = 360°

⇒250° + x = 360°

⇒x = 110°

B.



The figure is having five sides and five angles.

This figure is a pentagon.

As, the sum of all the exterior angles of a polygon is 360°.

So for the given angles of the pentagon from (1),


60° + 90° + 70° + x + 90° = 360°

⇒ 310° + x = 360°

⇒ x = 50°

(i) Since, 9 sides of polygon has 9 angles

And we know that:

Sum of all exterior angles of a polygon = 360^o

Each exterior angle of a regular polygon has the same measure

Therefore,

Measure of each exterior angle of a regular polygon of 9 sides =

= 40^o

(ii) Sum of all exterior angles of the given polygon = 360^o

Each exterior angle of a regular polygon has the equal measure

Thus, measure of each exterior angle of a regular polygon of 15 sides =

= 24^o

We know that, sum of all exterior angles of the given polygon = 360^o

Measure of each exterior angle = 24^o


Measure of each exterior angle of a regular polygon = 360º/n

where, n = number of sides of polygon

From the given,

24° = 360° / n


n = 15

So there are 15 sides of the polygon with 24° as each exterior angle.

As per question,

Measure of each interior angle = 165°

Hence, Measure of each exterior angle = 180° − 165°

= 15°

Sum of all exterior angles of polygon = 360^o

Thus, number of sides of the polygon =360^o/15°

= 24

We know that the sum of all exterior angles of all polygons is 360^o.

And, in a regular polygon, each exterior angle is of the same measure.

Therefore, if 360^o is a perfect multiple of the given exterior angle, then the given polygon will be possible

Exterior angle = 22°

360^o is not a perfect multiple of 22^o.

Hence, such polygon is not possible

Interior angle = 22°

Exterior angle = 180° − 22°

= 158°

Such a polygon is not possible as 360° is not a perfect multiple of 158°

To find the maximum exterior angle of a regular polygon, we will have to take the lowest possible number of sides (i.e., an equilateral triangle).

Now,

The exterior angle of this triangle will be the maximum exterior angle possible for any regular polygon

Exterior angle of an equilateral triangle = 360^o/3

= 120^o

Hence, the maximum possible measure of exterior angle for any polygon is 120^o.

And, we know that an exterior angle and an interior angle are always in a linear pair

Hence, minimum interior angle = 180^o − 120°

= 60^o

To find the maximum exterior angle of a regular polygon, we will have to take the lowest possible number of sides (i.e., an equilateral triangle).

Now,

The exterior angle of this triangle will be the maximum exterior angle possible for any regular polygon

Exterior angle of an equilateral triangle = 360^o/3

= 120^o

Hence, the maximum possible measure of exterior angle for any polygon is 120^o.

(i) In a parallelogram, opposite sides are equal in length

AD = BC

(ii) In a parallelogram, opposite angles are equal in measure

∠DCB = ∠DAB

(iii) In a parallelogram, diagonals bisect each other

Hence, OC = OA

(iv) In a parallelogram, adjacent angles are supplementary to each other

Hence,

m ∠DAB + m ∠CDA = 180°

(i) x + 100° = 180° (Adjacent angles are supplementary)

x = 80°

z = x = 80^o (Opposite angles are equal)

y = 100° (Opposite angles are equal)

(ii) x, y and z will be complimentary to 50⁰

Hence, required angle = 180 -50

= 130⁰

x = y = 130° (Opposite angles are equal)

z = x = 130^o (Corresponding angles)

(iii) x = 90° (Vertically opposite angles)

x + y + 30° = 180° (Angle sum property of triangles)

120° + y = 180°

y = 60°

z = y = 60° (Alternate interior angles)

(iv) z = 80° (Corresponding angles)

y = 80° (Opposite angles are equal)

x+ y = 180° (Adjacent angles are supplementary)

x = 180° − 80°

= 100°

(v) y = 112° (Opposite angles are equal in a parallelogram)

x+ y + 40° = 180° (Angle sum property of triangles)

x + 112° + 40° = 180°

x + 152° = 180°

x = 28°

z = x = 28° (Alternate interior angles)

(i) Considering, ∠D + ∠B = 180°, quadrilateral ABCD may or may not be a parallelogram

Along with this, the following criteria should also be fulfilled

Opposite angles should also be of same measures

The sum of the measures of adjacent angles should be 180°

(ii) No because opposite sides AD and BC are of different lengths

(iii) No as opposite angles A and C have different measures

As per the question,

Quadrilateral ABCD (kite) has two of its interior angles, ∠B and ∠D, of same measures.

But, still the it is not a parallelogram as the measures of the remaining pair of opposite angles, ∠A and ∠C, are unequal.

Let us consider that the measures of two adjacent angles, ∠A and ∠B, of parallelogram ABCD are in the ratio of 3: 2 are 3 x and 2 x

We know that the sum of the measures of adjacent angles is 180^o for a parallelogram

∠A + ∠B = 180^o

3 x + 2 x = 180^o

5 x = 180^o

∠A = ∠C = 3 x = 3 x 36° = 108^o (Opposite angles)

∠B = ∠D = 2 x = 2 x 36° = 72^o (Opposite angles)

Hence, the measure of the angles of the parallelogram are 108^o, 72^o, 108^o, and 72^o

We know,

Sum of adjacent angles = 180°

∠A + ∠B = 180^o

2∠A = 180^o (∠A = ∠B)

∠A = 90^o

∠B = ∠A = 90^o

∠C = ∠A = 90^o (Opposite angles of parallelogram)

∠D = ∠B = 90^o (Opposite angles of parallelogram)

Hence, each angle of the parallelogram measures 90^o

As per the question,

y = 40° (Alternate interior angles)

70° = z + 40^o (Corresponding angles)

70° − 40° = z

z = 30°

x + (z + 40^o) = 180° (Adjacent pair of angles)

x + 70°= 180°

x = 110°

(i) We know that the lengths of opposite sides of a parallelogram are equal to each other

GU = SN

3y − 1 = 26

3y = 27

y = 9

SG = NU

3x = 18

x = 6

Therefore, the measures of x and y are 6 cm and 9 cm respectively

(ii) We know that the diagonals of a parallelogram bisect each other

y + 7 = 20

y = 13

x + y = 16

x + 13 = 16

x = 3

Hence, the measures of x and y are 3 cm and 13 cm respectively

Adjacent angles of a parallelogram are supplementary

In parallelogram RISK,

∠RKS + ∠ISK = 180°

120° + ∠ISK = 180°

∠ISK = 60°

And, opposite angles of a parallelogram are equal

In parallelogram CLUE,

∠ULC = ∠CEU = 70°

The sum of the measures of all the interior angles of a triangle is 180^o

x + 60° + 70° = 180°

x = 50°

We know that if a transversal line intersects two given lines in such a way that the sum of the measures of the angles on the same side of transversal is 180^o, then the given two lines are parallel to each other.

As from the figure,

∠NML + ∠MLK = 180°

Hence, NM||LK

A quadrilateral with one pair of sides parallel is called trapezium. As for the given figure two sides are parallel, the given figure is a trapezium.

It is mentioned in the question that:

∠B + ∠C = 180° (Angles on the same side of transversal)

120^o + ∠C = 180°

∠C = 60°

According to the figure,

∠P + ∠Q = 180° (Angles on the same side of transversal)

∠P + 130° = 180°

∠P = 50°

∠R + ∠S = 180° (Angles on the same side of transversal)

90° + ∠R = 180°

∠S = 90°

Yes, there is one more method to find the measure of m ∠P, m ∠R and m ∠Q are given. After finding m ∠S, the angle sum property of a quadrilateral can be applied to find m ∠P

(a) False. All squares are rectangles but all rectangles are not squares

(b) True. Opposite sides of a rhombus are equal and parallel to each other

(c) As all sides of square are equal and the diagonals intersect at right angles, all squares are rhombus. And also opposite sides of square are equal and all angles are right angles, therefore all squares are also rectangles.

(d) False.

All squares are parallelograms as opposite sides are equal and parallel

(e) False.

A kite does not have all sides of the same length

Therefore, all kites aren’t rhombus

(f) True.

A rhombus also has two distinct consecutive pairs of sides of equal length

(g) True. All parallelograms have a pair of parallel sides

(h) True. All squares have a pair of parallel sides

(a) Rhombus and Square are the quadrilaterals that have 4 sides of equal length

(b) Square and rectangle are the quadrilaterals that have 4 right angles

Properties of Square :

1. Each Interior Angle = 90º.

2. Each Side is equal.

3. Diagonals bisect each other.

4. Opposite sides are parallel.




(i) A square is a quadrilateral since it has four sides

(ii) A square is a parallelogram since its opposite sides are parallel to each other

(iii) A square is a rhombus because its four sides are of the same length

(iv) A square is a rectangle because its each interior angle measures 90°

(i) The diagonals of a parallelogram, rhombus, square, and rectangle bisect each other

(ii) The diagonals of a rhombus and square act as perpendicular bisectors

(iii) The diagonals of a rectangle and square are equal

In a rectangle, there are two diagonals, both lying in the interior of the rectangle

Hence, it is a convex quadrilateral

Draw lines AD and DC such that AD||BC, AB||DC

AD = BC, AB = DC

ABCD is a rectangle as opposite sides are equal and parallel to each other and all the interior angles are of 90^o

In a rectangle, diagonals are of equal length and also these bisect each other

Hence, AO = OC = BO = OD

Thus, O is equidistant from A, B, and C