Understanding Quadrilaterals And Practical Geometry

Let the fourth angle be x

Since the sum of interior angles of a Quadrilateral is 360⁰

x + 3×75⁰ = 360⁰

⇒ x = 360⁰ – 225⁰

⇒ x = 135⁰

In a square the diagonals bisect each other because both the opposite side pairs of a square are parallel. But in a trapezium only one pair of opposite pair are parallel. In a kite no opposite pairs are parallel.

In a rectangle all the angles are at right angle and hence they are all equal. But in a kite trapezium or rhombus all the angles are not right angles hence not equal.

In a Kite one pair of opposite angles are at right angles and the adjacent sides are equal. So their diagonals are perpendicular. But in a trapezium, parallelogram or rectangle it doesn’t happen.

In a rectangle since all the four angles are equal hence the diagonals are equal. But in a trapezium, parallelogram or rhombus all the angles are not equal hence their diagonals are unequal.

R is a square. In a square all sides are equal, all angles are equal and at right angles and also opposite sides are parallel. P,Q,S are a trapezium, parallelogram and rectangle respectively whose all sides are not equal.

R is a kite whose adjacent sides are equal. P ,Q & S are a trapezium ,parallelogram and rectangle respectively whose adjacent sides are unequal.

P is a trapezium and has only one pair of parallel side. Q,R,S is a parallelogram ,square and rectangle respectively which has two pair of parallel side.

Q is a square. In a square all sides are equal, all angles are equal and at right angles and also opposite sides are parallel. P,S,R are a trapezium, parallelogram and rectangle respectively whose all sides are not equal.

A trapezium is a quadrilateral which has only one pair of sides which are parallel. The other properties are not true for trapezium.

In a parallelogram the opposite sides are parallel. The other properties are not true.

Sum of interior angles of a quadrilateral is 360⁰. An obtuse angle is an angle between 90⁰ and 180⁰ .So all the angles can’t be obtuse since then the sum will exceed 360⁰.So maximum it can have 3 obtuse angle.

There can be n-1 lines that can be drawn from one vertices. Out of these 2 sides are adjacent sides so n-3 lines can be drawn. When a n-gon is divided by n-3 lines the n-gon is divided into n-2 triangles.

Sum of interior angles of a polygon = (2n-4)×90⁰ where n represents the number of sides of a polygon

In this problem n = 5

⇒ Sum of interior angles = (10-4) ×90⁰

⇒ Sum of interior angles = 540⁰

Sum of interior angles of a polygon = (2n-4)×90⁰ where n represents the number of sides of a polygon

In this problem n = 6

⇒ Sum of interior angles = (12-4) ×90⁰

⇒ Sum of interior angles = 720⁰

Adjacent angles of a parallelogram are supplementary

5x-5 + 10x + 35 = 180

⇒ 15 x + 30 = 180

⇒ 15x = 150

⇒ x = 10

⇒ 5x-5 = 45⁰

⇒ 10x + 35 = 135⁰

⇒ Ratio = 1:3

All sides of a rhombus are equal and the diagonals bisect each other at right angles. It is also applicable for square but it is a superior form of rhombus whose all angles are equal.

A rectangle is a quadrilateral whose opposite sides and all angles are equal. It is also applicable for square but it is a superior form of rectangle whose all sides are equal.

A square is the only quadrilateral whose all sides, diagonals and angles are equal.

Number of diagonals of a polygon =

where n is the number of sides

Here n = 6

So the number of diagonals =

In parallelogram the opposite sides are equal. When the adjacent sides becomes equal all the sides of the parallelogram becomes equal and hence it becomes a rhombus.

A rectangle is a quadrilateral whose diagonals are equal and bisect each other. It is also applicable for square but it is a superior form of rectangle whose all sides are equal.

The sum of all exterior angles of all polygons is equal to 360⁰

A square is the only polygon whose all sides and all angles are equal.

In a rhombus the opposite sides are parallel, diagonals bisect at right angles and all the sides are equal. So it has all the properties of a kite and a parallelogram.

Let the angles be x ,2x ,3x ,4x

Sum of interior angles of a quadrilateral = 360⁰

x + 2x + 3x + 4x = 360⁰

⇒ 10x = 360⁰

⇒ x = 36⁰

D & A are supplementary

D + A = 180⁰

⇒ D = 180⁰-55⁰

⇒ D = 135⁰

Let the fourth angle be x

Sum of interior angles of a quadrilateral = 360⁰

x + 80× 3 = 360⁰

⇒ x = 360⁰-240⁰

⇒ x = 120⁰

Let the number of sides be n

sum of all exterior angles of all polygons is equal to 360⁰

n× 45⁰ = 360⁰

⇒ n = 8

Opposite angles are equal and adjacent angles are supplementary in a parallelogram.

P = 60°

⇒ R = 60°

Q = ∠S = 180⁰- 60°

Q = ∠S = 120⁰

The three angles are 60°, 120°, 120°

Let the angles be 2x, 3x

The adjacent angles of a parallelogram are always supplementary and the opposite angles are always equal

2x + 3x = 180⁰

⇒ 5x = 180⁰

⇒ x = 36⁰

So, the angles are

2× 36 = 72⁰

3× 36 = 108⁰

The opposite angles of a parallelogram are always equal and so their difference is always 0⁰

The adjacent angles of a parallelogram are always supplementary so sum is 180⁰

BE & EF are perpendiculars from the same point

∠ EBF = 30⁰(given)

Sum of interior angles of a quadrilateral = 360⁰

∠ EBF + ∠ BED + ∠ EDF + ∠ DFB = 360⁰

∠ EDF = 360⁰-(90⁰ + 90⁰ + 30⁰)

∠ EDF = 150⁰ which is an obtuse angle.

∠ BAC = 30⁰ (Given)

∠ ABC = 150⁰(Supplementary angles)

∠ ABD = 150⁰-90⁰ = 60⁰

∠ BDC = 60⁰(Co interior angles)

∠ BEC = 60⁰(Opposite angles)

By using Pythagoras theorem:

Length of diagonal = √ (10² + 24²)

⇒ Length of diagonal = √676

⇒ Length of diagonal = 26cm

In parallelogram the opposite angles are equal. When the adjacent angles become equal all the angles become equal and so it becomes a rectangle.

Sum of interior angles of a quadrilateral = 360⁰

In option (a) & (d) this condition is true. But in option (d) one angle is 180⁰ which if considered correct the quadrilateral becomes a triangle.

Sum of interior angles of a quadrilateral = 360⁰ for both concave and convex quadrilateral.

Sum of exterior angles of a polygon = 360⁰

360⁰ divided by any one of the angles must be a whole number since it gives the number of sides.

But when 360⁰ is divided by option (a) it gives a fraction which can be the number of sides of a polygon.

BEST is a rhombus,
So, TS || BS
∠SBE = ∠BST = 40° (alternate interior angles)
As Diagonals of a rhombus bisect at right angles.
∠y = 90°

In Δ TSO,
∠STO + ∠OST = ∠EOS (Exterior angle property)
⇒ x + 40° = 90°
⇒ x = 90° - 40°
⇒ x = 50°

So, y - x = 90° -50°

= 40°

Only in (a) the line segments don’t intersect each other where as in the others it does.

The last option is wrong because each exterior angle

=

A square is a special type of rhombus whose all the angles are equal. We know that diagonals of a rhombus bisect at right angles so this is also true for a square.

Let the angles be x and 5x

The adjacent angles of a parallelogram are supplementary

x + 5x = 180⁰

6x = 180⁰

⇒ x = 30⁰

So the angles are 30°, 150°, 30°,150°

In a rectangle the adjacent sides are unequal and the angle between two sides is 90⁰.So the given parallelogram is a rectangle.

Given: Angles of a quadrilateral are in the ratio 1: 3: 7: 9

Formula Used: Sum of the angles of a quadrilateral = 360⁰

Let the angles be x, 3x, 7x, and 9x

Adding the angles we get,

x + 3x + 7x + 9x = 360⁰

⇒ 20x = 360⁰

⇒ x = 18⁰

Angles P, Q, R and S = 18⁰, 54⁰, 126⁰ & 162⁰

∠ P ,∠S & ∠Q, ∠R are supplementary

So, PQ ||RS and it’s a trapezium

In a trapezium the adjacent angles in the non-parallel sides are supplementary

∠ R + ∠ Q = 180⁰

⇒ ∠ R = 180⁰-110⁰

⇒ ∠ R = 70⁰

Each interior angle = 135⁰

Let the number of sides be n

Sum of interior angles = n×135⁰

Sum of interior angles is given by = (2n-4) × 90⁰

Equating we get

(2n-4)90 = 135n

⇒ 180n-360 = 135n

⇒ 45n = 360

⇒ n = 8

A parallelogram is a quadrilateral whose diagonal bisects both the angles. It is also true for rhombus and rectangle but in a rhombus all the sides are also equal and in a rectangle all the angles are also equal.

In a parallelogram the opposite sides and opposite angles are equal. So to construct a parallelogram we need the measurements of the two adjacent sides of the parallelogram and the angle between them.

We only need the measurement of the length and the breadth of a rectangle to construct it. So only two measurements are required.

HO ,PE & HE ,OP

HO ,PE & HE ,OP are the opposite sides because they have no common vertices.

∠R , ∠O; ∠O, ∠P; ∠P, ∠E; ∠E, ∠R

∠R , ∠O; ∠O, ∠P; ∠P, ∠E; ∠E, ∠R are the pair of adjacent angles because they have a common vertices.

∠W , ∠Y ; ∠X , ∠Z

∠W , ∠Y ; ∠X , ∠Z are the pair of opposite angles because they have no common sides.

DF & EG

DF & EG are the opposite vertices of the quadrilateral and hence they create two diagonals.

Angles

Sum of both interior as well as exterior angles of a quadrilateral is 360°.

72⁰

Each exterior angle =

where n is the number of sides

For this problem n = 5 and so each exterior angle = 72⁰

720⁰

Sum of interior angle = (2n-4)×90⁰

where n is the number of sides

In this problem n = 6

(12-4) × 90⁰

= 8×90⁰ = 720⁰

20⁰

Each exterior angle =

where n is the number of sides

For this problem n = 18

and so each exterior angle = 20⁰

n× Each exterior angle = 360⁰

where n is the number of sides

For this problem each exterior angle = 36⁰

n =

⇒ n = 10

Concave Polygon

The polygon has more than one reflex angle .So it is a concave polygon.

Kite

One pair of opposite angles are equal but the other pair of angle is not equal in a kite.

108⁰

Sum of interior angle = (2n-4) × 90⁰

where n is the number of sides

In this problem n = 5

(10-4) × 90⁰

= 6×90⁰ = 540⁰

Each interior angle =

⇒ Each interior angle = 108⁰

Equilateral triangle

Equilateral triangle is a three-sided regular polygon.

9

Number of diagonals of a polygon =

where n is the number of sides

Here n = 6

So the number of diagonals =

Line segments

A polygon is a simplest closed curve made up of only line segments which don’t intersect with each other.

angles

All the sides as well as angles are equal for regular polygon.

2n-4

Sum of interior angle = (2n-4)×90⁰

So Sum of interior angles is 2n-4 right angles

360⁰

The sum of all exterior angles of any polygon is always 360⁰

Square

All the angles and sides of square are equal. So it is a regular quadrilateral

Trapezium

Trapezium is a in which a pair of opposite sides is parallel .

Rhombus

Rhombus is a quadrilateral whose all sides are equal.

Right

Since all sides are equal in a rhombus so they intersect at right angles.

Five

We need 5 measurements to determine a quadrilateral uniquely. It can be four sides one angle or 3 sides and 2 included angle.

Two included

We need 5 measurements to determine a quadrilateral uniquely. It can be drawn uniquely if its three sides and two included angles are given.

All

rhombus is a parallelogram in which all the sides are equal and the diagonals bisect at right angle.

One

A concave polygon has one reflex angle which is greater than 180⁰

Opposite

Diagonals are drawn by joining two opposite vertices of a quadrilateral.

5

Number of sides × each exterior angle = 360⁰

So number of sides =

⇒ Number of sides = 5

Parallelogram

The diagonals of a parallelogram always bisects each other.

28cm

Perimeter of parallelogram = 2× (Sum of adjacent sides)

⇒ 2×14 = 28cm

9

Nonagon is a polygon having 9 sides.

Equal

Diagonals of a rectangle are always equal in magnitude.

Decagon

Decagon is a polygon having 10 sides.

Square

If the adjacent sides of rectangle becomes equal then all the sides of the rectangle becomes equal and hence it becomes a square.

6cm

The diagonals of a rectangle are always equal in magnitude.

Supplementary

Adjacent angles of a parallelogram always add up to 180⁰

Kite

Only one diagonal of a kite bisects each other and the diagonals meet at right angle.

80⁰

The adjacent angles of non-parallel sides are supplementary

∠ A + ∠D = 180⁰

⇒ ∠D = 180⁰-100⁰

⇒ ∠D = 80⁰

Quadrilateral

Quadrilateral is a polygon whose sum of all exterior angles is equal to the sum of interior angles.

FALSE

In a trapezium all the angles are not equal , rather , Adjacent angles (next to each other) along the sides are supplementary. This means that their measures add up to 180 degrees.

x° + y° = 180°

TRUE

Every square is a rectangle because it is a quadrilateral with all four angles right angles(90°).

FALSE

All kites are not squares.

A kite is a quadrilateral (four sided shape) where the four sides can be grouped into two pairs of adjacent (next to/connected) sides that are equal length.

So, if all sides are equal, and all angles of the quadrilateral are equal, then only we have a square.

TRUE

A parallelogram is defined as a quadrilateral (4-sided polygon) with both pairs of opposite sides parallel. A rectangle certainly fits that description. In addition, a rectangle must have 4 right angles.

So, you can say that all rectangles are parallelograms

FALSE

A rhombus is a quadrilateral with all sides equal in length.

A square is a quadrilateral with all sides equal in length and all interior angles right angles.

Thus a rhombus is not a square unless the angles are all right angles.

FALSE

Sum of all the angles of a quadrilateral is 360°.

Because ,the quadrilateral can be divided into 2 triangles and sum of angles of each of the triangles is 180°.

⸫ when we join these 2 triangles to get back the original quadrilateral then the sum of all the angles of the quadrilateral is sum of angles of both the triangles ie. 180°(sum of angles of triangle1) + 180° (sum of angles of triangle2) = 360° (sum of angles of quadrilateral)

TRUE

A diagonal is a line segment connecting 2 non consecutive vertices of a polygon , ie. A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.

So, as we can see , we can draw only 2 such line segments.

⸫ A quadrilateral can have only two diagonals.

TRUE

We know that,

Sum of Interior angles of Polygon = 180(n-2)
Sum of Exterior angles of Polygon = 360
Here, for triangle , n=3,ie. triangle

So, Sum of Interior angles of triangle = 180(n-2)° = 180(3-2)° = 180°

And ,Sum of Exterior angles of triangle = 360°

⸫ we can say that ,

2 × Sum of Interior angles of triangle = Sum of Interior angles of triangle

2 × 180° = 360°

TRUE

FALSE

A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set.

⸫ Kite is a convex polygon.

*KITE*

TRUE

We know that,

Sum of Interior angles of Polygon = 180(n-2)
Sum of Exterior angles of Polygon = 360

Here, we have n=4

⸫ Sum of Interior angles of Polygon = 180(4-2)=360°

And Sum of Exterior angles of Polygon = 360

⸫ The sum of interior angles and the sum of extreme angles taken in an order are equal in case of quadrilaterals only.

TRUE

In a hexagon number of sides = 6.

∴ Sum of the interior angles of a hexagon = (2n - 4) x 90°

= (2 x 6) - 4) x 90° = 8 x 90° = 720°. -------(1)

Sum of the exterior angles of a hexagon = 360°.

Given that 2 times the sum of the exterior angles of a hexagon ie. 2 x 360°= 720° -----(2)

from (1) and (2) we get

∴ If the sum of interior angles is double the sum of exterior angles taken in an order of a polygon, then it is hexagon.

TRUE

A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

FALSE

As we know , A regular polygon is a shape whose sides are all the same length, and whose interior angles are all the same measure. A quadrilateral is a polygon with four sides. So a regular quadrilateral is a shape that has four equal sides, with all the interior angles equal., and in a rectangle , only the angles are equal , all the side are not equal , only opposite sides are equal.

⸫ Rectangle is not a regular quadrilateral.

FALSE

It is not required that the quadrilateral must be a rectangle , if its diagonals are equal.

In the above picture , we can see that in both the figures , both diagonals are each 5 units long, and the quadrilateral is an isosceles trapezium.

If the diagonals of a quadrilateral are equal the figure could be a rectangle, a square or an isosceles trapezium.

TRUE

Let ABCD be a parallelogram, with A = α and B = β.

We have to Prove that C = α and D = β.

α + β = 180° (co-interior angles, AD || BC),

C = α (co-interior angles, AB || DC)

D = β (co-interior angles, AB || DC).

False,

Given:

Ratio of interior angles = 1:2:3

Let the interior angles be x, 2x and 3x

Now, the sum of the interior angles of the triangle are 180⁰

⇒ x+ 2x + 3x = 180⁰

⇒ 6x = 180⁰

⇒ x =

⇒ x = 30⁰

Therefore, the interior angles are: 30⁰, 60⁰ and 90⁰.

Now, the exterior angles will be found by subtracting from 180⁰ because they form a linear pair.

The exterior angles are: 150⁰, 120⁰ and 90^0.

FALSE

The given figure is not a pentagon because it has 6 sides.

FALSE

Consider rhombus ABCD,

You know that , AB=BC=CD=AD

Now , in ΔAED and ΔCED,

EA = EC (diagonals of a parallelogram bisect each other)

ED= ED (common)

AD=CD

⸫ ΔAED is congruent to ΔCED (SSS)

This gives that AED = CED (CPCT)

But, AED + CED = 180° (linear pair)

So , 2 × AED = 180°

⸫AED = 90°

So , the diagonals of rhombus are only perpendicular to each other.

⸫ the diagonals of rhombus are perpendicular but not equal.

TRUE

Given ABCD is a rectangle

then AC and BD are diagonals

then in ΔABC and ΔBCD,

b is common angle

BC is common side

AB = CD

so by SAS congruency,

Δ ABC is congruent to ΔBCD

so by cpct,

AC = BD

so ,diagonals are equal.

TRUE

In rectangle ABCD, diagonals bisect the angles.
Consider ΔAOD and ΔBOC
AD = BC (ABCD is a rectangle)
∠AOD = ∠BOC (Vertically opposite angles)
∠OAD = ∠OCB = 45° (Diagonals bisect the angles)
ΔAOD ≅ ΔBOC (AAS congruence criterion)
Therefore, OA = OC and OB = OD
Thus the diagonals bisect each other in a rectangle.

FALSE

Kite is a quadrilateral that has 2 adjacent equal sides, that is upper and lower side , whereas in parallelogram , opposite side should be equal.

⸫every kite is not a parallelogram.

FALSE

In a parallelogram, both the pairs of sides should be parallel. So , in a trapezium , only one pair of sides is parallel that is it doesn’t fulfill the conditions for parallelogram.

FALSE

In a rectangle, all the angles should be equal to 90°. So , all the parallelograms may not have all the angles as 90°.

FALSE

In a rectangle , all the angles should be equal to 90°. So , all the trapeziums may not have all the angles as 90° , even if they have a pair of parallel sides.

FALSE

A trapezium is defined to be a quadrilateral that has exactly one pair of parallel sides. Whereas rectangle has 2 pairs of parallel sides. So every rectangle is not a trapezium.

TRUE

In rhombus, all the sides should be equal and the diagonals should be perpendicular to each other.

This condition is fulfilled by the square.

TRUE

In a parallelogram, opposite pairs of sides should be equal and opposite angles should also be equal. These conditions are fulfilled by square.

FALSE

A trapezium is defined to be a quadrilateral that has exactly one pair of parallel sides. Whereas square has 2 pairs of parallel sides. So every square is not a trapezium.

FALSE

A trapezium is defined to be a quadrilateral that has exactly one pair of parallel sides. Whereas rhombus has 2 pairs of parallel sides. So every rhombus is not a trapezium.

FALSE

To draw a quadrilateral, we need at least 2 angles as well.

FALSE

As we know that the sum of all the interior angles of the quadrilateral = 360°.

And , if all the 4 angles are obtuse then the above property is violated.

So, A quadrilateral cannot have all four angles as obtuse.

TRUE

Let AB,BC,CD,AD(sides) and AC(diagonal) be given.

Draw ΔABC using SSS construction.

Now, we have to locate the 4^th point D. this ‘D’ would be on the side opposite to B with reference to AC.

So, with A as centre draw an arc of radius of length equal to AD.

Now, with C as centre draw an arc of radius of length equal to CD.

The intersection of these 2 arcs is D. Mark D and complete ABCD

FALSE

To construct a quadrilateral, we need at least 3 sides.

FALSE

To construct a quadrilateral, we need at least 2 angles.

TRUE

Here, let BC,AD,CD(sides) are given and AC , BD (diagonals)are also given.

Draw ΔACD by using SSS construction.

With D as centre, draw an arc of radius of length equal to BD.

With C as centre, draw an arc of radius of length equal to BC.

The intersection of these 2 arcs is B. Mark B and complete ABCD.

FALSE

It is not required that the quadrilateral must be a parallelogram , if its diagonals are equal.

In the above picture , we can see that in both the figures, both diagonals are each 5 units long, and the quadrilateral is an isosceles trapezium.

TRUE

Let AB, BC(sides) and A,B,C are given.

Draw AB and then B.

From B, draw BC.

At C, draw C.

At M, draw A and mark the point of intersection of these 2 angles as D.

So, we get the required quadrilateral ABCD.

TRUE

Mark a point A and draw a line through A.

From A, draw an arc of radius of length equal to one diagonal to cut the above line at C.

Bisect AC at O.

From A and C, draw arcs of radius more than half the length of AC on both sides, Let they cut at X and Y. XY is the perpendicular bisector of AC. It cuts AC at O.

From O, draw a line at the given angle to AC on both sides.

From O, draw an arc of radius of half the length of another diagonal on both sides of AC to cut the above line at B and D.

Join AB,BC,CD,AD.

TRUE

Here only 2 measurements of the rhombus are given ie. The diagonals.

However , since it is a rhombus , we can find more help from its properties.

The diagonals of a rhombus are perpendicular bisectors of one another.

So, first draw AC (any one diagonal) and then construct its perpendicular bisector. Let them meet at E. Cut off half the length of second diagonal on either side of the drawn bisector.

You now get B and D.

Similarly, get A and C, draw the rhombus now.

Given, the diagram below:

diagonals of a rhombus, d₁ = 8cm

d₂ = 15cm

Divide d1 and d2 in two equal halves.

We will get four right angled triangles

Now, the base of each triangle would be, d1/2 = 8/2 = 4cm

And perpendicular would be, d2/2 = 15/2 = 7.5cm

Now, we will find the hypotenuse of each right angled

triangle.

Hence, hypotenuse = side of rhombus

Given, ratio of the adjacent angles of a parallelogram is 1:3

⇒ By the property of parallelogram, the sum of adjacent angles of a parallelogram is 180°.

⇒ Let the adjacent angles are ∠A and ∠B.

∠A = x

∠B = 3x

∠A + ∠B = x + 3x {∠A + ∠B = 180

4x = 180 ̇

x = 45

⇒ ∠A = 45

⇒∠B = 45×3 = 135

According to question, we have four quadrilaterals- square,

Rectangle, rhombus and trapezium.

By the property of parallelogram, a parallelogram is a quadrilateral whose opposite sides are always parallel.

But in case of trapezium, only one pair of opposite sides are parallel. Hence, it is not a parallelogram.

it is clear from the above points that trapezium is different from square, rectangle and rhombus.

The figure is given below:

Given, a rectangle ABCD

AB = 25cm

BC = 15cm

Bisector of ∠C divides AB i.e. ∠BCE = 45

⟹ By angle sum property of triangle.

∠BEC = 45

⟹ BC = BE, because sides having equal angles are equal.

BE = 15cm

And AE = AB-BE = 25-15

AE = 10cm

AE:BE = 10:15

AE:BE = 2:3

The figure is given below:

PQRS is a rectangle. The perpendicular ST from S on PR divides angle S in the ratio 2:3.

∠PSR = 90

⟹ It is given that, the perpendicular ST from S on PR divides angle S in the ratio 2:3.

⟹ divide the 90° angle into the ratio 2:3.

2x + 3x = 90

5x = 90

x = 18

⟹ therefore, 2x = 2×18 = 36 and 3x = 3×18 = 54

now, ∠PST = 36 and

∠TSR = 54 = ∠TSQ + ∠QSR

∠PST = 36

∠TSR = 54 = ∠TPS

Hence, ∠TPQ = ∠SPQ-∠TPS = 90-54 = 36

A photo frame is in the shape of quadrilateral with one diagonal longer than the other. So, That quadrilateral-shaped frame is not in shape of rectangle. As we know rectangle have equal diagonals.

Given, adjacent angles are (2x-4) and (3x-1)

⟹ (2x-4) + (3x-1) = 180

⟹ (5x-5) = 180

⟹ 5(x-1) = 180

⟹ x-1 = 36

⟹ x = 37

⟹ First angle = 2x-4 = (2×37)-4 = 70

⟹ Second angle = 3x-1 = (3×37)-1 = 110

No, it cannot be a parallelogram because the diagonal of a parallelogram are always bisect each other i.e. in the ratio 1:1.

Given, the ratio between exterior angle and interior angle is

1:2.

Let, exterior angle = x

And, interior angle = 5x

Sum of exterior and interior angle is equal to 180.

Hence, x + 5x = 180

6x = 180

x = 30

⟹Total measure of all the exterior angle = 360

⟹Measure of each exterior angle = 30

Therefore, the number of all exterior angles

Hence, the polygon has 12 sides.

Given, two sticks each of length 5cm i.e. equal in length.

If the diagonal of a quadrilateral are equal and bisects each

Other then the quadrilateral is a rectangle. Hence, the shape

Formed by joining given two sticks will be a rectangle.

Given, two sticks each of length 7cm i.e. equal in length.

If the diagonal of a quadrilateral are equal and bisects each

Other at right angle then the quadrilateral is a rhombus. Hence, the shape formed by joining given two sticks will be a rhombus.

The kite has four sides. There are exactly two distinct consecutive pairs of sides of equal length.

Let two different length be A and B.

Then the perimeter = 2A + 2B _______(i)

Given, perimeter = 106m

And one side i.e = 23m

According to Eq.(i), (2 × 23) + 2B = 106

2B = 60

B = 30

Hence, all sides are 23, 23, 30, 30.

Given, ∠ROE = 60

According to given figure, ∠AOE = 120, ∠AOD = 60 and ∠ROD = 120.

⟹In triangle AOE, consider ∠EAR = ∠DEA = x

∠AOE + ∠EAR + ∠DEA = 180

x + x + 120 = 180

2x + 120 = 180

2x = 60

x = 30, i.e. ∠EAR = 30

⟹In triangle AOD, consider ∠RAD, = ∠EDA = y

∠RAD + ∠EAR + ∠AOD = 180

y + y + 60 = 180

2y = 120

y = 60, i.e.∠RAD = 60.

Given, ∠RAI = 35

⟹In ∆ARI,

∠A + ∠R + ∠I = 180

35 + ∠ARI + 90 = 180

∠ARI = 55

⟹In ∆RMI,

∠R + ∠M + ∠I = 180

∠ARI + ∠RMI + ∠ARI = 180 {∠ARI = ∠MIR = 55}

(2×55) + ∠RMI = 180

∠RMI = 70

And, ∠PMA = 180-∠RMI

∠PMA = 180-70

∠PMA = 110

Given, ∠A = 80

⟹The sum of the adjacent angles of a parallelogram is 180

Hence, ∠A + ∠B = 180

∠B = 180-80

∠B = 100

⟹The opposite angles of a parallelogram are equal.

Hence, ∠C = ∠A = 80

And, ∠D = ∠B = 100

Given:

∠RQY = 60°

∵ ∠RQY and ∠RQP form a linear pair

⇒ ∠RQP = 180° - ∠RQY = 120°

∴ ∠RQP =120°

Now, in a parallelogram, the opposite sides are equal,

⇒ ∠RQP = ∠S = 120°

Similarly, ∠P = ∠R

Now, from the angle sum property of a quadrilateral,

∠P + ∠R + ∠Q + ∠S = 360°

⇒ ∠P + ∠R + 120° + 120° = 360°

⇒ 2∠P = 120°

⇒ ∠P = 60°

⇒ ∠R = 60°

Now,

Opposite sides of the parallelogram are equal.

⇒ SR = PQ = 15 cm

Also, PS = QR = 11 cm

And, the diagonals of the parallelogram bisect each other.

Hence, PR = 2×PQ = 2×6 = 12 cm.

In the rhombus,

∠BAM = 70⁰

Also, we know that the diagonals of a rhombus bisect at 90⁰

∴ ∠AOM = 90

Now, in ΔAOM,

∠AOM + ∠OMA + ∠MOA = 180⁰

⇒ 90 + ∠OMA + 70 = 180⁰

⇒ ∠AME = ∠OMA = 20⁰

Also,

The sides in the rhombus are equal.

Ans, we know, equal sides make equal angles.

⇒ ∠AEM = 20⁰

Given, ∠TOF,∠TIS and ∠TIF

⟹In ∆TOF,

∠T + ∠O + ∠F = 180

25 + 110 + ∠SFT = 180

∠SFT = 180-135

∠SFT = 45

⟹In ∆TOS,

∠T + ∠O + ∠S = 180

35 + (180-110) + ∠OST = 180

35 + 70 + ∠OST = 180

∠OST = 180-105

∠OST = 75

∠STO = ∠T(in triangle SOT) = 35

Given, ∠RUO = 120 and ∠SRY = 50

⟹The opposite angles of a parallelogram are equal.

Hence, ∠RYO = 120

⟹In ∆YSR,

∠Y + ∠R + S = 180

(180-120) + 50 + ∠YSR = 180

110 + ∠YSR = 180

∠YSR = 70

In ∆ARW,

∠R + ∠A + ∠W = 180

⟹Let ∠A = ∠W = x

80 + 2x = 180

2x = 180-80

x = 50

⟹In ∆WEA,

∠W + ∠E + ∠A = 180

⟹Let ∠W = ∠A = y

2y + 70 = 180

2y + 180-70

y = 55

⟹Hence, ∠RWE = ∠RAE = (x + y)

= (50 + 55)

= 105

(i) Yes,

∵ MORE is a rectangle.

And, the opposite sides of the rectangle are equal.

∴ RE = OM

(ii) Yes,

In the MORE rectangle,

MY and RX are perpendicular to OE

⇒ ∠RXO = ∠RXE = ∠MYE = ∠MYO = 90⁰

∴ ∠MYO = ∠RXE

(iii) Yes,

∵ in rectangle MORE, RE||OM and EO is transveral.

From the property of alternate interior angles being equal.

∴ ∠MOE = ∠OER

⇒ ∠MOY = ∠REX

(iv) Yes,

In, ΔMYO and Δ RXE,

MO = RE

∠MOY = ∠REX

∠MYO = ∠RXE

∴ ΔMYO ≅ ΔRXE

(v) Yes.

Corresponding parts of congruent triangles are equal.

⟹In ∆STM,

∠S + ∠T + ∠M = 180

40 + ∠STM + 90 = 180

∠STM = 180-130 = 50

∠STM = ∠SON (these are opposite angles).

∠SON = 50

∠SON and ∠OST are adjacent angles.

⟹Hence, ∠SON + ∠NSO + ∠NSM + ∠MST = 180

50 + (180-(90 + 50)) + ∠NSM + 40 = 180

∠NSM = 140-90

∠NSM = 50

Adjacent angles along a side of a trapezium are supplementary.

Hence, ∠R + ∠A = 180 ________(i)

Similarly, ∠E + ∠H = 180 ________(ii)

According to given data in question,

The complete ∠R = 2×30 = 60

And ∠E = 2×25 = 50

Because, EP and RP are bisectors of E and R respectively.

By Eq (i) and (ii),

∠A = 180-60 = 120

∠H = 180-50 = 130

Let MODE be the parallelogram and Q be the point of intersection of the bisector of ∠M and ∠O

The figure is attached below:

Now, in a parallelogram, the adjacent angles are supplementary.

∴ ∠EMO + ∠DOM = 180°

⇒ 1/2 (∠EMO + ∠DOM) = 1/2 180°

⇒ 1/2∠EMO + 1/2∠DOM = 1/2 180°

∠QMO + ∠QOM = 90°

Now, in ΔMOQ,

∠QMO + ∠QOM + ∠MQO = 180⁰

⇒ 90⁰ + ∠MQO = 180⁰

⇒ ∠MQO = 90⁰

EBF formed a isosceles triangle.

⟹Hence, ∠EBF = ∠EFB = 45

⟹According to question,

∠EBF + ∠x = 180

∠x = 180-45 = 135

⟹And ∠EFB + y = 90

∠y = 90-45 = 45

⟹In llgm ABDH,

∠D = ∠BAH = 180-130(adjacent angles)

⟹In llgm CEFG,

∠C = ∠GCE = ∠GFE = 30(opposite angles)

⟹Hence, in ∆CDX,

∠C + ∠D + ∠x = 180

30 + 130 + ∠x = 180

∠x = 180-160 = 20

In Rhombus ABCD,

AO = OP + PA = 2+2 = 4 units.

And, OB = OQ + QB = 2 + 1 = 2 units.

Now,

We know that the diagonals of the rhombus bisect at 90⁰

∴ in ΔOAB,

AB² = OA² + OB²

⇒ AB² = 4² + 3²

⇒ AB = √25

⇒ AB = 5 units.

Now, AB is also the diameter of the semi-circle.

∴ radius of the circle will be 2.5 units.

The figure is attached below:

I have circumscribed the regular Pentagon with a circle. Since, the 360° of the circle is divided equally by the pentagon to five parts.

⇒ ∠COD =

Using the properties of circle,

∠CAD = 1/2 ∠COD

⇒ ∠CAD = 1/2 72⁰

∠CAD = 36⁰

Now, in ∆ACD.

36° + ∠ACD + ∠ADC = 180°

Since, ∆ACD is an isosceles ∆,

36° + 2×∠ACD = 180°

∠ACD = 72°

Now, in ∆AMC.

∠MAC = 1/2 ∠CAD = 18°

Now,

∠MAC + ∠AMC + ∠ACM = 180°

18° + ∠AMC + 72° = 180°

∠AMC = 90°

By the property of rectangle, the length of both the diagonals are equal.

Hence, FH = EG

2(JF) = EG

2(8x + 4) = (24x-8)

16x + 8 = 24x-8

8x = 16

x = 2