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(1) Find the Sum of the Interior Angles of a Polygon with:

(a) 5 sides

(b) 18 sides

(2) Find the Sum of the Interior Angles of a Polygon with:

(a) 15 sides

(b) 8 sides

(3) Find the Value of the Unknowns.

(a)

(4) Find the Value of the Unknowns.

(a)

(5) Find the Value of the Unknown Angle Shown in Figure of Regular Hexagon.

(a)

(6) Find the Value of the Unknowns.

(a)

(7) Do as Directed. Wherever Necessary, Write Out the Three Equalities and the Criterion Used to Show Congruence.

(a)

If and are right angle,

Then prove that which triangle is congruent to and why?

(8) Do as Directed. Wherever Necessary, Write Out the Three Equalities and the Criterion Used to Show Congruence.

(a)

For the given polygon and

Then prove that

(9) for Square ABCD Prove That Its Diagonal (AC and BD) Are Equal.

(10) Put a Tick (√) of the Given Statement is True. Put a Cross (X) , if Not.

(a) A diagonal of a rectangle divides its related interior angle exactly at half.

(b) In a kite, the diagonal intersects ant right angle.

(11) Put a Tick (√) of the Given Statement is True. Put a Cross (X) , if Not.

(a) A diagonal of parallelogram bisects each other.

(b) All angles of rhombus are congruent

(c) Square is also a kite

(12) Find the Value of the Unknowns.

(a) ABCD is rectangle

(13) Find the Value of Unknowns.

(a) If MNOP is parallelogram

(14) Do as Directed. Wherever Necessary, Write Out the Three Equalities and the Criterion Used to Show Congruence.

In the given parallelogram JKLM; their diagonal intersects at point x.

Prove that angle .

Hint: Which two triangles are congruent? Why?

(15) Do as Directed. Wherever Necessary, Write Out the Three Equalities and the Criterion Used to Show Congruence.

(a) In rhombus PQRS, QS is equal to ₹ . Find the measure of all interior angles of the rhombus.

Hint: What kinds of triangle are ? Why?

(16) Do as Directed. Wherever Necessary, Write Out the Three Equalities and the Criterion Used to Show Congruence.

(a) For Kite ABCD its two diagonals (AC and BD) intersect each other at point O, prove that O is midpoint of diagonal BD

Hint: What kinds of triangle are

We know that sum of interior angle of triangle is .

Now square/polygon having 4 side have 2 triangles (while we connect two opposite corner it become 2 triangle) hence for polygon having four side has sum of interior angle

Sum of interior angle of any polygon

Sum of interior angle of any polygon having 5 sides

Hence, sum of interior angle of any polygon having 5 sides

Sum of interior angle of any polygon having 18 sides

Therefore, sum of interior angle of any polygon having 18 sides

Sum of interior angle of any polygon having 15 sides

Hence, sum of interior angle of any polygon having 15 sides

Sum of interior angle of any polygon having 8 sides

Hence Sum of interior angle of any polygon having 18 sides

To find out angle first we must find rest of interior angle of polygon

So,

Guess the remaining angle as shown in figure.

Now from figure it՚s clear that Angle lies on same line hence sum of these two angles should be

Now from figure it՚s clear that Angle lies on same line hence sum of these two angles is

Polygon has 4 side hence sum of interior angle of polygon

Sum of interior angle of polygon

Sum of interior angle of polygon

Sum of interior angle of polygon

To find out angle first we must find rest of interior angle of polygon

So,

Guess the remaining angle as shown in figure.

Now from figure it՚s clear that Angle lies on same line hence sum of these two angles should be

From figure it՚s clear that Angle lies on same line hence sum of these two angles should be

From figure it՚s clear that Angle lies on same line hence sum of these two angles should be

From figure it՚s clear that Angle lies on same line hence sum of these two angles should be

Polygon has 6 side hence sum of interior angle of polygon

Put the value of Angle A, B, C and Angle d values in above equation,

Hexagon has 6 sides and for given hexagon; its regular hexagon hence it՚s all interior angle have same value.

Now, sum of all interior angle of hexagon.

Given hexagon is regular hence its all-interior angle has same value,

Interior angle of given hexagon

Now As shown in figure Angle lies on same line hence sum of these two angles should be

To find out angle first we must find rest of interior angle of polygon

So,

Guess the remaining angle as shown in figure.

Now from figure it՚s clear that Angle lies on same line hence sum of these two angles should be

From figure it՚s clear that Angle lies on same line hence sum of these two angles should be

Polygon has 5 side hence sum of interior angle of polygon

Now, put the value of Angle A and Angle B in above equation than,

From figure lies on same line hence sum of these two lines is

 Triangle △ABC Triangle △DEF ∠ BAC = 90 given ∠ EDF = 90 given AF = CD given AF = CD given AB = DE given AB = DE given

Now, for triangle

Hence

Hence for both Triangle we can write that

Hence As per Congruent rule SAS this two triangle Congruent

Both triangle Congruent so we can say that,

Hence proved that

Given polygon have two triangles as shown in figure

Now For triangle it is given that

Means Angle

And for Triangle it is given that

Means Angle

Hence for both Triangle we can write that

Hence As per Congruent triangle rule SAS this two triangle Congruent

Both triangle Congruent so we can say that,

Hence proved that

Polygon ABCD is square given,

From figure we can say that rectangle have two triangles

BDC

As ABCD polygon is square; so, it՚s all interior angle is And all side are equal.

Now for Triangle

Hence As per Congruent triangle rule SAS this two triangle

Congruent

Hence, we can say that,

AC and BD is Diagonal of rectangle ABCD and; above we prove that both are equal.

Hence, it՚s prove that for square ABCD its diagonal (AC and BD) is equal.

A Diagonal of a Rectangle divides its related interior angle exactly at half.

As shown in figure Rectangle ABCD have two diagonal AC and BD.

For Rectangle Triangle are congruent.

Hence, we can say that

For Rectangle Angle is right angle and AC is its related diagonal

But Angle are same as derived above

Hence, we can say that; A Diagonal of a Rectangle divide its related interior angle exactly at half

A Diagonal of a Rectangle divide its related interior angle exactly at half

Given Statement: In a kite, the diagonal intersects ant right angle.

It՚s a property of kite quadrilaterals that its diagonal always intersects at right angle,

Hence, in a kite, the diagonal intersects ant right angle.

Given Statement: A diagonal of parallelogram bisect each other

The properties of parallelogram are;

Opposite sides are parallel and congruent.

Opposite angles are congruent.

Diagonals bisect each other and each diagonal divides the parallelogram into two congruent triangles.

If one of the angles of a parallelogram is a right angle then all other angles are right and it becomes a rectangle.

It՚s a property of parallelogram quadrilaterals that its diagonal always bisects each other,

Hence, a diagonal of parallelogram bisects each other

Given Statement: All angles of rhombus are congruent

The properties of Rhombus are as below;

All sides are congruent.

Opposite angles are congruent.

The diagonals are perpendicular to and bisect each other.

Adjacent angles are supplementary (For e. g. , ∠ A + ∠ B = 180°) .

A rhombus is a parallelogram whose diagonals are perpendicular to each other.

Hence, it՚s clear that only opposite side Angle of rhombus are congruent not all

Hence, all angles of rhombus are congruent

Given statement: Square is also a kite.

The properties of Kite are:

Two distinct pairs of adjacent sides are congruent.

Diagonals of a kite intersect at right angles.

One of the diagonals is the perpendicular bisector of another.

Angles between unequal sides are equal.

And this all properties are having a square hence we can say that squarer is also a kite

Hence, Square is also a kite

And in given rectangle

And for rectangle, it՚s a property of rectangle that its opposite side are congruent.

Hence,

For rectangle it՚s a property of rectangle that it՚s all angle is right angle

Therefore

Now for triangle Angel is Right angle; so, it՚s a Right triangle.

And for right triangle , AC is hypotenuse

Hence,

And for given parallelogram

And for parallelogram, it՚s a property of parallelogram that its opposite side are congruent.

Hence

Now, given that,

Hence,

Now for triangle ,

It՚s a property of parallelogram that its Adjacent angles are supplementary

Means,

Now, from figure,

Hence,

Given quadrilateral is parallelogram hence as per its properties its two opposite side are parallel and congruent,

Now, it՚s parallelogram՚s properties that its two diagonals bisect each other,

Hence, we can write that

Now from given figure, for two triangle

Hence As per Congruent triangle rule SSS this two triangle

Congruent

Hence, we can say that,

And for rhombus as per its properties it՚s opposite side are congruent

Hence,

Now for Triangle

Hence As per Congruent triangle rule SSS this two triangle

Congruent

Hence, we can say that

Now PQRS is Rhombus hence it՚s all side are congruent

Hence

So, for Triangle

Hence all side of triangle

So as per theorem it՚s all opposite angle are also congruent.

Now for Triangle

Same we can derive for triangle

Now, for angle

Now, for angle

Hence, angle of rhombus PQRS are,

As per it՚s properties its two adjacent side are congruent

Hence,

And Kites diagonal intersect each other at right angle hence,

Now for triangle

Here Triangle are right angle and its hypotenuse is congruent and one led is also congruent.

Hence this two triangle are congruent;

So, we can say that

Now,

So, we can say that “O” is midpoint of Diagonal BD.

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