Key Concept: Rectangle Diagonals, Square Diagonals

c) $147.48^\circ$
[Solution Description]
The diagonals of a rectangle bisect each other but are not perpendicular. To find the angle between them:
1. First, find the length of one diagonal using the Pythagorean theorem: $d = \sqrt{l^2 + w^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$ cm.
2. The diagonals in a rectangle are equal and bisect each other, forming four congruent triangles.
3. In any of these triangles, two sides are half the length of the diagonal ($5$ cm each), and the third side is the width or length of the rectangle.
4. Using the cosine rule for one triangle with sides $\frac{d}{2} = 5$ cm, $\frac{d}{2} = 5$ cm, and the included side $w = 6$ cm:
$\cos(\theta) = \frac{5^2 + 5^2 - 6^2}{2 \times 5 \times 5} = \frac{25 + 25 - 36}{50} = \frac{14}{50} = 0.28$
5. Therefore, $\theta = \cos^{-1}(0.28) \approx 73.74^\circ$. Since the angle between the diagonals is twice this angle (as the diagonals form two such triangles back-to-back), the angle between the diagonals is $\approx 147.48^\circ$.

Key Concept: Venn Diagram Representation

a) Squares are entirely within the set of rectangles.
[Solution Description]
Since every square is a rectangle but not every rectangle is a square, the set of squares is entirely contained within the set of rectangles. Therefore, option a) is correct.

Key Concept: Properties of a Rectangle

b) The diagonals are equal and bisect each other
[Solution Description]
The diagonals of a rectangle are equal and bisect each other, as stated in the properties of a rectangle.

Key Concept: Definition of a Rectangle

c) A quadrilateral with all angles equal to $90^\circ$ and opposite sides equal
[Solution Description]
A rectangle is a quadrilateral with all angles equal to $90^\circ$ and opposite sides equal. This matches the definition provided in the syllabus.

Key Concept: Construction of a Square

b) 50 cm$^2$
[Solution Description]
Given the diagonal $d = 10$ cm. Using the relationship $d = s\sqrt{2}$, we find the side length $s$ as $s = \frac{d}{\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2}$ cm. The area of the square is $s^2 = (5\sqrt{2})^2 = 25 \times 2 = 50$ cm$^2$.

Key Concept: Side Length from Diagonal

c) $40$ cm
[Solution Description]
Let the side of the square be $s$. The relationship between the side and the diagonal of a square is given by $d = s\sqrt{2}$. Given $d = 10\sqrt{2}$ cm, we can solve for $s$:
$10\sqrt{2} = s\sqrt{2} \implies s = 10 \, \text{cm}$
The perimeter of the square is $4s = 4 \times 10 = 40$ cm.

Key Concept: Diagonals, Opposite Sides

b) 5 cm
[Solution Description]
First, find the length of diagonal $AC$ using the Pythagorean theorem since all angles in a rectangle are $90^\circ$.
$AC = \sqrt{AB^2 + AD^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}$
Since the diagonals of a rectangle bisect each other, $AO$ is half of $AC$.
$AO = \frac{AC}{2} = \frac{10}{2} = 5 \text{ cm}$

Key Concept: Properties of Rectangles

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that in rectangle $ABCD$, diagonals $AC$ and $BD$ are equal in length. This is true because one of the properties of rectangles is that their diagonals are equal in length.
The reason states that the diagonals of a rectangle bisect each other. This is also true as it is another property of rectangles.
However, the reason does not explain why the diagonals are equal in length—it only states a different property of the diagonals. Thus, both statements are true, but the reason is not the correct explanation for the assertion.

Key Concept: Diagonals of a Rhombus

c) The diagonals bisect each other.
[Solution Description]
Both squares and rhombuses have diagonals that bisect each other. However, while squares also have equal-length diagonals, rhombuses do not necessarily have this property unless they are squares themselves.

Key Concept: Diagonals of a Rectangle

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because the diagonals of a rectangle are always equal in length, as proven by the properties of rectangles. The reason is also true because the diagonals of a rectangle bisect each other at their midpoints. Furthermore, the reason correctly explains why the assertion is true, as both properties (equality and bisecting each other) are fundamental to the definition and properties of a rectangle.

Key Concept: Diagonals of Rectangle and Square, Multi-step Solution

c) 26
[Solution Description]
Step 1: In a rectangle, diagonals are equal and bisect each other. So, AO = OC and BO = OD. Since AO = BO, we set $3x + 1 = 5x - 7$.
Step 2: Solve for x: subtract 3x from both sides to get $1 = 2x - 7$. Add 7 to both sides: $8 = 2x$. Divide by 2: $x = 4$.
Step 3: Substitute x back into AO: $AO = 3(4) + 1 = 13$.
Step 4: Diagonal AC is twice AO (since AO = OC): $AC = 2 \times 13 = 26$.

Key Concept: Diagonals of a Rectangle

d) The point $O$ divides both diagonals into two equal parts.
[Solution Description]
The diagonals of a rectangle bisect each other, so the point of intersection $O$ divides both diagonals into two equal parts. Each diagonal will be split into two segments of equal length.

Key Concept: Angles between diagonals of a rectangle

d) Assertion is false, but Reason is true.
[Solution Description]
In a rectangle, diagonals are equal in length and bisect each other, forming two pairs of isosceles triangles. For any angle $x$ between the diagonals, the base angles of these isosceles triangles can be calculated as $a = 90^\circ - \frac{x}{2}$ and $b = \frac{x}{2}$. The sum of the angles around the point of intersection must satisfy $a + b = 90^\circ$, leading to all four angles being $90^\circ$. However, while the diagonals of a rectangle are equal and bisect each other (Reason), this property alone does not necessarily imply that they intersect at $90^\circ$ (Assertion is false). Only in special cases like squares or rhombuses do diagonals intersect at $90^\circ$.

Key Concept: Assertion-Reason

d) Assertion is false, but Reason is true.
[Solution Description]
In a rectangle, the diagonals are equal and bisect each other, but they do not necessarily intersect at $90^\circ$ unless it is a square. Thus, Assertion (A) is false. However, Reason (R) is true because diagonals of a rectangle are equal and bisect each other.

Key Concept: Diagonals, Congruence, Angle Properties

c) $36 \, \text{cm}^2$
[Solution Description]
The area of a rhombus is half the product of its diagonals. Given the shorter diagonal $d_1 = 6$ cm, the longer diagonal $d_2 = 2d_1 = 12$ cm.
The area is calculated as:
$\text{Area} = \frac{d_1 \times d_2}{2} = \frac{6 \times 12}{2} = 36 \, \text{cm}^2.$

Key Concept: Diagonals in a rhombus

c) They are all $90^\circ$
[Solution Description]
A rhombus has all sides equal, and its diagonals bisect each other at $90^\circ$. Therefore, the angles between the diagonals of a rhombus are all $90^\circ$.

Key Concept: Diagonals of a rectangle

b) 10 cm
[Solution Description]
The diagonals of a rectangle are equal in length. If one diagonal is 10 cm, the other diagonal must also be 10 cm.

Key Concept: Properties of Rectangles, Diagonals

b) 10 cm
[Solution Description]
In a rectangle, the diagonals are equal in length and bisect each other. Given AO = 5 cm, which is half of diagonal AC. Since diagonals are equal, BD = AC. Therefore, BD = 2 * AO = 10 cm.

Key Concept: Quadrilateral Properties, Congruence

a) Yes, because all sides and angles are equal.
[Solution Description]
A quadrilateral with four equal sides is a rhombus. If one angle is $90°$, then by properties of a rhombus, all angles must be $90°$ (since opposite angles are equal and consecutive angles are supplementary). Hence, it is a square.

Key Concept: Diagonals of Quadrilaterals

d) Square
[Solution Description]
If the diagonals of a quadrilateral are equal in length and bisect each other at $90^\circ$, then the quadrilateral is a square. This is because such properties ensure all angles are $90^\circ$ and all sides are equal.

Key Concept: Rectangle and Square Properties, Diagonal Congruence

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For a quadrilateral to be a square, its diagonals must satisfy three conditions: 1) They must be of equal length, 2) They must bisect each other, and 3) They must intersect at right angles ($90^\circ$). The Assertion states that joining equal-length strips at their midpoints with a $90^\circ$ angle forms a square, which is correct because this setup ensures all three conditions are met. The Reason correctly identifies these properties as sufficient for a quadrilateral to be a square. Therefore, both Assertion and Reason are true, and Reason explains Assertion.

Key Concept: Diagonals of Rectangle

d) 8 cm, joined at their midpoints
[Solution Description]
For a quadrilateral to be a rectangle, its diagonals must be equal in length and bisect each other. Given one diagonal is 8 cm, the other must also be 8 cm. The strips should be joined at their midpoints to ensure the diagonals bisect each other.

Key Concept: Definition of a Rectangle

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a quadrilateral with diagonals equal in length and bisecting each other is a rectangle, which aligns with the alternative definition of a rectangle mentioned in the syllabus. The reason correctly states that in a rectangle, the diagonals are equal and bisect each other. Moreover, the reason provides the justification for why such a quadrilateral must be a rectangle, making it the correct explanation of the assertion.

Key Concept: Rectangle Diagonals, Angle between Diagonals

c) $90^\circ$
[Solution Description]
In a quadrilateral with all angles equal to $90^\circ$ (rectangle), the diagonals are equal in length and bisect each other. The angle between the diagonals can be determined by considering two congruent right triangles formed by the diagonals. Let the point of intersection be O. Using properties of rectangles, $\angle AOB$ and $\angle BOC$ are equal and supplementary ($\angle AOB + \angle BOC = 180^\circ$). Hence, each angle must be $90^\circ$.

Key Concept: Quadrilateral Construction

c) Square
[Solution Description]
When the diagonals of a quadrilateral are equal in length and bisect each other at $90^\circ$, the quadrilateral must be a square. This is because such properties are specific to squares and not to other quadrilaterals like rectangles or rhombuses unless all sides are also equal.

Key Concept: Angle Sum Property of Quadrilateral

c) $115^\circ$
[Solution Description]
Sum of interior angles of a quadrilateral = $360^\circ$.
Let the fourth angle be $x$.
$$ 80^\circ + 95^\circ + 70^\circ + x = 360^\circ$$
$$ 245^\circ + x = 360^\circ$$
$$ x = 115^\circ$$

Key Concept: Properties of Squares

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the diagonals of a square bisect each other at $90^\circ$, which is true as per Property 4 of squares. The reason states that a square is a special type of rectangle with all sides equal, which is also true (Property 1 of squares). However, the reason does not explain why the diagonals bisect at $90^\circ$; it only describes another property of squares. Therefore, both statements are true, but the reason is not the correct explanation of the assertion.

Key Concept: Properties of Rectangle

c) The quadrilateral must be a rectangle but not necessarily a square
[Solution Description]
When diagonals of a quadrilateral are equal and bisect each other, it must be a rectangle. However, for rectangles, diagonals should intersect at $90°$ if it's a square, otherwise they form angles not equal to $90°$ unless specified. Here the angle is $60°$, which means it's a rectangle but not necessarily a square.
Step-by-step reasoning:
1) Diagonals being equal and bisecting each other confirms it's a rectangle (Property 4 of rectangles).
2) The angle between diagonals being $60°$ shows it's not a square (where diagonals intersect at $90°$).
3) Thus, the quadrilateral is a rectangle with unequal adjacent sides.

Key Concept: Special Rectangle and Square Properties

c) Opposite sides are equal
[Solution Description]
We need to identify a property that is common to both squares and rectangles.
Property 1: All angles are $90^\circ$. This is true for both squares and rectangles.
Property 2: Opposite sides are equal. This is also true for both.
Property 3: Diagonals bisect each other. This is true for both.
However, only in a square are all sides equal and diagonals perpendicular. So, the correct statement must be one that applies to both but not exclusively to either.
The most general correct statement common to both is that "Opposite sides are equal."

Key Concept: Properties of Rectangles and Squares

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that every square is a rectangle. This is true because a square satisfies all the properties of a rectangle: all angles are $90^\circ$, opposite sides are equal and parallel, and diagonals bisect each other.
The reason given is that in a square, all angles are $90^\circ$ and opposite sides are equal and parallel. This is also true as it aligns with the properties of both squares and rectangles. However, while the reason correctly describes properties of a square, it does not explicitly link these properties to why a square is a rectangle. The key observation from the syllabus mentions this relationship directly ("Every square is a rectangle"), but the reason does not provide an explanation for the assertion; it merely lists properties. Hence, both the assertion and reason are true, but the reason is NOT the correct explanation of the assertion.

Key Concept: Venn Diagram Representation

c) Square $\subset$ Rectangle
[Solution Description]
The set of all squares is a subset of the set of all rectangles because every square is a rectangle (all angles $90^\circ$ and opposite sides equal), but not all rectangles are squares. Thus, the correct representation is Square $\subset$ Rectangle.

Key Concept: Angles in a Rectangle

c) $90^\circ$
[Solution Description]
By definition, all angles in a rectangle are right angles. A right angle measures $90^\circ$. Therefore, each angle in a rectangle is $90^\circ$.

Key Concept: Quadrilateral Angle Sum, Triangle Angle Property

b) $90^\circ$
[Solution Description]
We know that the sum of all angles in a quadrilateral is $360^\circ$. Let angles R and S be $x$ each. So, we can write the equation:
$60^\circ + 120^\circ + x + x = 360^\circ$.
Simplifying this: $180^\circ + 2x = 360^\circ$.
Subtract $180^\circ$ from both sides: $2x = 180^\circ$.
Divide by 2: $x = 90^\circ$.
Therefore, angle R is $90^\circ$.

Key Concept: Quadrilateral Angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A quadrilateral can indeed be divided into two triangles by drawing one diagonal. The sum of angles in each triangle is $180^\circ$, so the total sum of angles for both triangles is $360^\circ$. Since the angles of the triangles collectively form the angles of the quadrilateral, the sum of the quadrilateral's angles must also be $360^\circ$. Thus, both Assertion (A) and Reason (R) are true, and R correctly explains why A is true.

Key Concept: Opposite angles in a parallelogram

c) $20^\circ$
[Solution Description]
In a parallelogram, opposite angles are equal. Hence, $\angle A = \angle C$.
So, $3x + 10^\circ = 5x - 30^\circ$.
Subtract $3x$ from both sides: $10^\circ = 2x - 30^\circ$.
Add $30^\circ$ to both sides: $40^\circ = 2x$.
Divide by 2: $x = 20^\circ$.

Key Concept: Quadrilateral Angle Sum, Algebraic Equation

c) $76^\circ$
[Solution Description]
Let angle B be $x$. Then:
- Angle A = $2x$,
- Angle C = $x + 20^\circ$,
- Angle D = $x - 40^\circ$.
The sum of all angles is $360^\circ$, so:
$2x + x + (x + 20^\circ) + (x - 40^\circ) = 360^\circ$.
Combine like terms: $5x - 20^\circ = 360^\circ$.
Add $20^\circ$ to both sides: $5x = 380^\circ$.
Divide by 5: $x = 76^\circ$.
Therefore, angle B is $76^\circ$.

Key Concept: Sum of angles in a quadrilateral

c) 110°
[Solution Description]
The sum of all angles in any quadrilateral is $360°$. Given three angles as $80°$, $100°$, and $70°$, let the fourth angle be $x$.
Therefore, $80° + 100° + 70° + x = 360°$.
Adding the known angles: $250° + x = 360°$.
Subtracting $250°$ from both sides: $x = 110°$.

Key Concept: Angles formed by diagonals in a quadrilateral

c) 360°
[Solution Description]
The sum of angles in the first triangle is $40° + 60° + 80° = 180°$.
The sum of angles in the second triangle is $50° + 70° + 60° = 180°$.
Adding these gives the total sum of angles in the quadrilateral: $180° + 180° = 360°$.

Key Concept: Sum of angles in a quadrilateral

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sum of angles in any quadrilateral is 360°, which is true based on the syllabus. The reason explains that when a quadrilateral is split into two triangles via a diagonal, the sum of angles in each triangle is 180°, leading to a total of 360° for the quadrilateral. Therefore, both the assertion and reason are correct, and the reason justifies the assertion.

Key Concept: Sum of angles in quadrilateral, properties of triangles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason:
1. Assertion (A): Any quadrilateral can indeed be divided into two triangles by drawing one diagonal. This is true as a quadrilateral has four sides and connecting two non-adjacent vertices with a diagonal splits it into two distinct triangles.
2. Reason (R): The sum of the interior angles of any quadrilateral is $360^\circ$. This is true because when you divide the quadrilateral into two triangles, the sum of angles in each triangle is $180^\circ$. Adding them gives $180^\circ + 180^\circ = 360^\circ$. Thus, the reason correctly explains why the sum of angles in a quadrilateral is $360^\circ$.
Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Sum of angles in a quadrilateral

c) $360^\circ$
[Solution Description]
The sum of all angles in any quadrilateral is always $360^\circ$. This can be derived by dividing the quadrilateral into two triangles and adding their angle sums ($180^\circ + 180^\circ = 360^\circ$).

Key Concept: Sum of angles in a quadrilateral

b) $80^\circ$
[Solution Description]
The sum of all four angles in a quadrilateral is $360^\circ$. Given three angles as $90^\circ$, $90^\circ$, and $100^\circ$, let the fourth angle be $x$. Then:
$90^\circ + 90^\circ + 100^\circ + x = 360^\circ$
$280^\circ + x = 360^\circ$
$x = 80^\circ$
So, the fourth angle measures $80^\circ$.

Key Concept: Construction of quadrilaterals

c) The fourth angle must be a right angle
[Solution Description]
The sum of angles in any quadrilateral is $360^\circ$. If a quadrilateral has three right angles ($90^\circ$ each), their sum would be $270^\circ$. Therefore, the fourth angle must be:
$360^\circ - 270^\circ = 90^\circ$
Hence, if a quadrilateral has three right angles, the fourth angle must also be a right angle.

Key Concept: Quadrilateral angles, diagonal properties

c) The fourth angle must be $90^\circ$
[Solution Description]
Let's analyze the given quadrilateral PQRS with three right angles (let's say $\angle P = \angle Q = \angle R = 90^\circ$) and diagonals intersecting at O forming angle POQ = $60^\circ$.
Since sum of angles in any quadrilateral is $360^\circ$, we have:
$\angle P + \angle Q + \angle R + \angle S = 360^\circ$
Substituting known values:
$90^\circ + 90^\circ + 90^\circ + \angle S = 360^\circ$
$270^\circ + \angle S = 360^\circ$
Therefore, $\angle S = 90^\circ$.
The information about diagonal intersection angle doesn't affect the angle sum property. Hence, the fourth angle must also be $90^\circ$.

Key Concept: Properties of Parallelograms and Rhombus

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the diagonals of a rhombus bisect each other at $90^\circ$, which is a known property of a rhombus. The reason states that a rhombus is a special type of parallelogram with all sides equal, which is also true. However, the reason does not directly explain why the diagonals bisect at $90^\circ$; it merely defines a rhombus. Therefore, both statements are true, but the reason is not the correct explanation of the assertion.

Key Concept: Properties of Parallelograms

c) $115^\circ$
[Solution Description]
In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Given $\angle A = 65^\circ$, $\angle B$ will be $180^\circ - 65^\circ = 115^\circ$. Since $\angle D$ is opposite to $\angle B$, $\angle D = 115^\circ$.

Key Concept: Adjacent Angles in Parallelogram

b) $130^\circ$
[Solution Description]
Adjacent angles in a parallelogram are supplementary, meaning they add up to $180^\circ$. If one angle is $50^\circ$, the adjacent angle will be $180^\circ - 50^\circ = 130^\circ$.

Key Concept: Diagonals of Parallelogram, Bisection Property

c) $10$ cm
[Solution Description] The diagonals of a parallelogram bisect each other. This means $PO = OR$ and $QO = OS$. Here, $PO = 5$ cm and $OR = 3$ cm implies the diagonals are not bisecting each other equally, which contradicts the property. Therefore, the correct values must be equal. Assuming $PO = OR = x$, then $PR = PO + OR = x + x = 2x$. However, the problem states $PO = 5$ cm and $OR = 3$ cm, which is impossible for a parallelogram. But interpreting it as a misprint, if $PO = 5$ cm and $OR = 5$ cm, then $PR = 10$ cm.

Key Concept: Adjacent Angles in Parallelogram

c) $120°$
[Solution Description]
Property 3 of parallelograms states that adjacent angles add up to $180°$. So, if one angle is $60°$, the adjacent angle will be $180° - 60° = 120°$.

Key Concept: Adjacent Angles in Parallelogram

b) $92^{\circ}$
[Solution Description]
In a parallelogram, adjacent angles are supplementary, meaning they add up to $180^{\circ}$. Therefore, $\angle A + \angle D = 180^{\circ}$.
Given:
$\angle A = (3x + 10)^{\circ}$
$\angle D = (2x + 40)^{\circ}$
Set up the equation:
$(3x + 10) + (2x + 40) = 180$
Combine like terms:
$5x + 50 = 180$
Subtract $50$ from both sides:
$5x = 130$
Divide by $5$ on both sides:
$x = 26$
Now find $\angle A$:
$\angle A = 3(26) + 10 = 88^{\circ}$
In a parallelogram, opposite angles are equal, so $\angle B = \angle D$.
Thus, $\angle B = 2(26) + 40 = 92^{\circ}$
The measure of $\angle B$ is $92^{\circ}$.

Key Concept: Congruent Triangles in Parallelogram

c) The opposite sides are equal and parallel.
[Solution Description]
The congruence of triangles $WXY$ and $YZW$ implies certain properties of the parallelogram $WXYZ$.
For triangles $WXY$ and $YZW$ to be congruent, their corresponding sides and angles must be equal. Specifically:
1. Opposite sides of a parallelogram are equal and parallel, so $WX = YZ$ and $XY = WZ$.
2. Diagonally opposite angles in a parallelogram are equal, so $\angle WXY = \angle YZW$ and $\angle XYW = \angle ZWY$.
This means the diagonal $WY$ divides the parallelogram into two congruent triangles. Additionally, the diagonals of a parallelogram bisect each other but do not necessarily have to be equal in length.
Therefore, the correct property deduced from the given information is that the opposite sides of $WXYZ$ are equal and parallel.

Key Concept: Parallelogram Properties

d) Assertion is false, but Reason is true.
[Solution Description]
Assertion (A) claims that the diagonals of a parallelogram are equal in length. However, according to the syllabus, while the diagonals of a parallelogram bisect each other (Property 4), they need not be equal in length (as stated under "Note"). Therefore, the assertion is false.
Reason (R) states that the diagonals of a parallelogram bisect each other, which is true as per Property 4 in the syllabus.
Thus, Assertion is false while Reason is true.

Key Concept: Rectangles as a type of parallelogram

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A rectangle is a quadrilateral with all angles equal to $90^\circ$ and opposite sides parallel and equal, which satisfies the definition of a parallelogram. The reason correctly states that a rectangle has opposite sides parallel and equal. Since both statements are true, and the Reason correctly explains why the Assertion is true, the correct answer is option a).

Key Concept: Special Cases of Parallelograms

d) Square
[Solution Description]
A quadrilateral with all sides equal and all angles equal to $90^\circ$ is called a square. This is because a square is a special type of rectangle where all sides are equal.

Key Concept: Opposite Sides, Diagonals

b) Yes, because the sides satisfy Pythagoras' theorem
[Solution Description]
Step 1: Check if the given sides satisfy the Pythagorean theorem for a right angle. Here, $6^2 + 8^2 = 36 + 64 = 100 = 10^2$.
Step 2: This means the angle between sides $6$ cm and $8$ cm is $90°$.
Step 3: For a parallelogram to be a rectangle, all its angles must be $90°$. Since one angle is already $90°$ and opposite angles in a parallelogram are equal, all angles will be $90°$.
Step 4: Hence, the given conditions make it a rectangle.

Key Concept: Rectangles as a type of parallelogram, Diagonals in rectangles and parallelograms

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the diagonals of a rectangle are perpendicular to each other. However, this is not a property of rectangles. The correct property is that the diagonals of a rectangle are equal in length and bisect each other but are not necessarily perpendicular. They are perpendicular only in the case of a square, which is a special type of rectangle. Therefore, the Assertion is false. The reason states that the diagonals of a rectangle bisect each other at $90^{\circ}$, which is also false because while they do bisect each other, it is not necessary that they do so at $90^{\circ}$. Hence, both the Assertion and Reason are false. Thus, none of the given options is correct, but among the provided options, option d) best matches since the Reason is partially true (diagonals bisect each other, though not always at $90^{\circ}$). Note: There might be an issue with the question design as all options involve truth values that don't perfectly match.

Key Concept: Venn Diagram, Classification

c) Every square is a rhombus.
[Solution Description]
A square is a special type of rhombus where all angles are $90^\circ$. However, not all rhombuses are squares because a rhombus does not require all angles to be $90^\circ$. Therefore, every square is a rhombus, but not every rhombus is a square.

Key Concept: Special Cases of Parallelograms

c) All its angles are $90^\circ$
[Solution Description]
A rectangle is a special parallelogram where all angles are $90^\circ$. While it has all properties of a parallelogram, the defining feature is its right angles. Thus, the correct answer is "All its angles are $90^\circ$."

Key Concept: Properties of Quadrilaterals

c) Rhombus
[Solution Description]
A rhombus has diagonals that are perpendicular bisectors of each other. This is one of its defining properties. Hence, the correct answer is Rhombus.

Key Concept: Properties of Quadrilaterals

c) Either a Square or a Rhombus.
[Solution Description]
Diagonals of a rhombus bisect each other at right angles and also bisect its angles. A square also has this property since it is a special type of rhombus. However, not all quadrilaterals with perpendicular bisecting diagonals are squares—they could also be rhombuses. Rectangles have diagonals that bisect each other but not necessarily at right angles unless they are squares.

Key Concept: Properties of Rhombus

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that every rhombus is a square, which is false because a rhombus only requires all sides to be equal but does not require all angles to be $90^\circ$. The Reason correctly describes a square as having all sides equal and all angles equal to $90^\circ$, which is true but does not explain the Assertion since the Assertion is incorrect.
Therefore, the Assertion is false, but the Reason is true.

Key Concept: Quadrilaterals with Equal Side Lengths, Properties of Rhombus and Square

d) Assertion is false, but Reason is true.
[Solution Description]
To analyze this assertion-reason pair, let's examine the properties of quadrilaterals with equal side lengths:
1. Assertion states that a quadrilateral with all sides equal must be a square. However, from the syllabus, we know that a rhombus also has all sides equal but does not necessarily have all angles as 90\textdegree. Hence, the assertion is false because there are other quadrilaterals (like rhombuses) with equal side lengths that are not squares.
2. Reason states that the diagonals of a rhombus bisect each other at 90\textdegree. This is true as per Property 4 of a rhombus mentioned in the syllabus.
3. Since the assertion is false and the reason is true, the correct answer corresponds to option d).

Key Concept: Properties of a Rhombus

a) All sides are equal
[Solution Description]
A rhombus has all sides equal, opposite sides parallel, adjacent angles summing to $180^\circ$, and diagonals that bisect each other. From Property 1, we know all sides are equal.

Key Concept: Diagonals of Rhombus

b) They bisect each other at right angles.
[Solution Description]
The diagonals of a rhombus bisect each other at right angles and also bisect the angles of the rhombus. However, they are not necessarily equal in length.

Key Concept: Diagonals of Rhombus, Right Triangles

b) 40 cm
[Solution Description]
All sides of a rhombus are equal, so the perimeter is simply 4 times the side length. Given the side length is 10 cm:
$\text{Perimeter} = 4 \times 10\,\text{cm} = 40\,\text{cm}$
Note that the information about the diagonal is irrelevant for calculating the perimeter.

Key Concept: Adjacent Angles of a Rhombus

c) $180^\circ$
[Solution Description]
Adjacent angles in a rhombus add up to $180^\circ$.

Key Concept: Properties of Rhombus, Angle Relationships

c) $180^\circ$
[Solution Description]
Opposite angles in a rhombus are equal, so $\angle PQR = \angle PSR = 110^\circ$. Adjacent angles in a rhombus are supplementary, so $\angle QPS = 180^\circ - \angle PQR = 70^\circ$. Therefore, the sum is:
$\angle QPS + \angle RSP = 70^\circ + 110^\circ = 180^\circ$

Key Concept: Properties of Rhombus, Angles in Rhombus

c) $60^\circ$
[Solution Description]
Since diagonals of a rhombus bisect each other at right angles, we know $\angle AOD = 90^\circ$. Given $\angle OAD = 30^\circ$, we can find $\angle ADO$ using the sum of angles in triangle $AOD$:
$\angle OAD + \angle ADO + \angle AOD = 180^\circ \implies 30^\circ + \angle ADO + 90^\circ = 180^\circ \implies \angle ADO = 60^\circ$
The diagonal also bisects the angle, so $\angle ADC = 2 \times \angle ADO = 120^\circ$. Then, since adjacent angles in a rhombus are supplementary:
$\angle ABC = 180^\circ - \angle ADC = 180^\circ - 120^\circ = 60^\circ$

Key Concept: Side length, Diagonal relationship

c) 6 cm
[Solution Description]
In a rhombus, the diagonals bisect each other at right angles and divide the rhombus into four right-angled triangles. Let's denote half of diagonal 1 as $d_1/2 = 4$ cm and half of diagonal 2 as $d_2/2 = x$ cm. Using Pythagoras theorem: $(d_1/2)^2 + (d_2/2)^2 = \text{side}^2$ → $4^2 + x^2 = 5^2$ → $16 + x^2 = 25$ → $x^2 = 9$ → $x = 3$ cm. Thus the full diagonal length is $6$ cm.

Key Concept: Diagonals of Rhombus

c) $90^\circ$
[Solution Description]
Property 4 states that the diagonals of a rhombus bisect each other at right angles ($90^\circ$). Therefore, the angle between the diagonals is $90^\circ$.

Key Concept: Properties of Rhombus

c) They are equal.
[Solution Description]
According to Property 3 of a rhombus, the opposite angles are equal. For example, if one angle is $50^\circ$, its opposite angle will also be $50^\circ$.

Key Concept: Properties of Rhombus

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A rhombus has the property that its diagonals bisect each other at $90^{\circ}$. This means the diagonals intersect at right angles and divide each other into two equal parts. Additionally, the diagonals divide the rhombus into four congruent right-angled triangles because all sides of a rhombus are equal, and the diagonals bisect the angles. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Extending Diagonals of Quadrilaterals

b) Square
[Solution Description]
Initially, the square has diagonals that are equal in length and perpendicular bisectors. Extending both diagonals equally by 2 cm on both sides means:
- The new endpoints are symmetrically placed around the original square.
- The distance between any two adjacent new vertices will be equal due to symmetry.
- The angles at the vertices remain 90 degrees because the extensions preserve the perpendicularity of the diagonals.
Thus, the resulting quadrilateral formed by joining the new endpoints is another square, as all sides and angles remain equal and perpendicular.

Key Concept: Construction of Parallelograms

b) At the midpoints of both strips
[Solution Description]
To form a parallelogram, the strips must be joined at their midpoints. The lengths of the strips determine the sides of the parallelogram, but the key property is that opposite sides must be parallel and equal in length, which is ensured when the strips intersect at their midpoints.

Key Concept: Properties of Quadrilaterals, Constructing Quadrilaterals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that in a quadrilateral with two pairs of parallel sides, the diagonals bisect each other. This is true because such a quadrilateral is a parallelogram, and one of the properties of a parallelogram is that its diagonals bisect each other. The reason correctly identifies the quadrilateral as a parallelogram and states the property of diagonals bisecting each other, which explains the assertion. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Properties of Quadrilaterals

d) Square
[Solution Description]
When the diagonals of a quadrilateral are perpendicular and equal in length, the quadrilateral must be a square. In a square, the diagonals bisect each other at $90^{\circ}$ and are equal in length.

Key Concept: Geoboard Activity: Properties of Quadrilaterals

c) Square
[Solution Description]
When two diagonals are equal in length and perpendicular to each other, joining their endpoints forms a square. This is because all sides become equal due to the properties of symmetry and the right angles between the sides.

Key Concept: Diagonals of Quadrilaterals, Rectangle Formation

d) Square
[Solution Description]
When the diagonals of a quadrilateral are equal in length and bisect each other at $90^\circ$, all angles formed are $90^\circ$, and opposite sides are equal. This satisfies the properties of a square. However, since the sides are not necessarily equal, it could also be a rectangle. But given the diagonals intersect at $90^\circ$, it must be a square.

Key Concept: Quadrilateral Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When two identical equilateral triangles are joined side-by-side, the resulting figure will have four sides of equal length since all sides of an equilateral triangle are equal. Additionally, opposite sides will be parallel because the angles at each vertex will be 60°, making the sum of adjacent angles 180°, which ensures parallelism. Therefore, the quadrilateral meets all the properties of a rhombus: all sides equal and opposite sides parallel. Hence, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Key Concept: Quadrilateral Transformation

b) Kite
[Solution Description]
When one diagonal of a square is extended equally on both sides, the angles between the sides change, but the adjacent sides remain equal. This results in a kite, as two pairs of adjacent sides are equal, but not all four sides.

Key Concept: Joining Triangles, Kite Properties

d) All of the above
[Solution Description]
When two equilateral triangles of side 8 cm are joined along one common side, the resulting quadrilateral is a rhombus. A rhombus has all sides equal and opposite angles equal. The diagonals bisect each other at 90 degrees and also bisect the vertex angles. Hence, all the given properties apply.

Key Concept: Constructing quadrilaterals using isosceles triangles

b) Kite
[Solution Description]
When two identical isosceles triangles (with sides 8 cm, 8 cm, 6 cm) are joined along their unequal sides (6 cm), the resulting quadrilateral will have two pairs of adjacent sides equal. This forms a kite because $AB = BC$ (8 cm each) and $CD = DA$ (8 cm each).

Key Concept: Constructing quadrilaterals using equilateral triangles

c) Rhombus
[Solution Description]
When two identical equilateral triangles are joined along one common side, the resulting figure has four sides. Since all sides of the original triangles are equal and they are joined along a common side, the new quadrilateral will have opposite sides equal in length. This makes it a rhombus.

Key Concept: Constructing quadrilaterals by joining triangles of equal sides

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion (A) is true because two equilateral triangles of side length 8 cm can indeed be joined along one of their sides to form a quadrilateral (specifically, a rhombus). However, the Reason (R) is false because joining two congruent triangles along their equal sides does not necessarily result in a rectangle; it could also form other types of quadrilaterals such as a rhombus or parallelogram.

Key Concept: Trapezium Angles

c) $110^\circ$
[Solution Description]
For a trapezium with parallel sides, the sum of adjacent angles is $180^\circ$. Given $\angle P = 70^\circ$, then $\angle S = 180^\circ - 70^\circ = 110^\circ$.

Key Concept: Trapezium Angles

d) $120^\circ$
[Solution Description]
One property of a trapezium states that the sum of angles on the same side of a leg is $180^\circ$. Here, $\angle P + \angle S = 180^\circ$. Given $\angle P = 60^\circ$, we calculate $\angle S = 180^\circ - 60^\circ = 120^\circ$.

Key Concept: Isosceles Trapezium Properties

b) $\angle U = \angle V$
[Solution Description]
In an isosceles trapezium, the non-parallel sides are equal, and the angles opposite these sides are equal. Thus, $\angle U = \angle V$.

Key Concept: Isosceles Trapezium

d) Both $\angle U = \angle V$ and $\angle W = \angle X$
[Solution Description]
An isosceles trapezium has equal non-parallel sides and equal base angles. Therefore, angles opposite to the equal sides are equal.

Key Concept: Kite Properties

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the diagonals of a kite bisect each other at right angles. According to the properties of a kite, one diagonal bisects the other and is perpendicular to it, but only one diagonal gets bisected (not both). Hence, the assertion is false.
The reason states that a kite has two pairs of adjacent sides equal, which is true as per the definition of a kite. Therefore, the assertion is false, but the reason is true.

Key Concept: Classification of Quadrilaterals

a) Every rhombus is a kite, but not every kite is a rhombus.
[Solution Description]
A rhombus is a special case of a kite where all four sides are equal. However, not every kite is a rhombus because a kite only requires two distinct pairs of adjacent sides to be equal.

Key Concept: Kite construction, side lengths

c) $30$ cm
[Solution Description]
A kite must satisfy the triangle inequality for each pair of triangles formed by the diagonals. For sides 6 cm and 9 cm, let the shorter sides be repeated.
Then, the sides are 6, 6, 9, 9, giving a perimeter of $6 + 6 + 9 + 9 = 30$ cm. This satisfies all geometric constraints.

Key Concept: Property of Diagonals in Kite

b) Bisects $\angle ABC$ and $\angle ADC$
[Solution Description]
According to the properties of a kite, diagonal BD bisects $\angle ABC$ and $\angle ADC$. Additionally, it bisects the other diagonal AC, making $AO = OC$, and is perpendicular to AC.

Key Concept: Basic property of trapezium angles

c) $120^\circ$
[Solution Description]
Since $PQ || SR$, by the property of trapezium, the sum of angles on the same side of the non-parallel side is $180^\circ$. Therefore, $\angle P + \angle S = 180^\circ$. Given $\angle P = 60^\circ$, we can find $\angle S$ as follows:
$\angle S = 180^\circ - 60^\circ = 120^\circ$

Key Concept: Kite: Diagonal Properties

b) Only one diagonal bisects the other and they are perpendicular.
[Solution Description]
In a kite, one diagonal bisects the other. Specifically, the diagonal connecting the vertices where the adjacent sides are equal (e.g., $BD$) bisects the other diagonal ($AC$) perpendicularly. Thus, $AO = OC$ and $BD \perp AC$.

Key Concept: Sum of angles in a quadrilateral

c) $20^\circ$
[Solution Description]
The sum of the interior angles of any quadrilateral is $360^\circ$. Let the fourth angle be $x$. Then:
$105^\circ + 135^\circ + 100^\circ + x = 360^\circ$
Simplifying:
$105 + 135 = 240^\circ$
$240 + 100 = 340^\circ$
$x = 360^\circ - 340^\circ = 20^\circ$
So, the fourth angle is $20^\circ$.

Key Concept: Trapezium: Angle Calculation

b) $\angle B = 75^\circ$, $\angle C = 105^\circ$
[Solution Description]
Since $AB \parallel CD$ and $AD$ is the transversal, the sum of the angles on the same side of the transversal is $180^\circ$. Therefore, $\angle A + \angle D = 105^\circ + 75^\circ = 180^\circ$ which is correct. Now, the sum of all angles in a quadrilateral is $360^\circ$. So, $\angle B + \angle C = 360^\circ - (\angle A + \angle D) = 360^\circ - 180^\circ = 180^\circ$. Also, since $AB \parallel CD$ and $BC$ is the transversal, $\angle B + \angle C = 180^\circ$. Hence, both $\angle B$ and $\angle C$ must be supplementary but not necessarily equal unless it's an isosceles trapezium.

Key Concept: Kite Diagonal Properties, Trapezium Angle Relationships

c) Assertion is true, but Reason is false.
[Solution Description]
For assertion (A), in a kite, one diagonal (main diagonal) is known to be perpendicular to the other and bisects it. This property makes $\angle AOB = 90^\circ$ true because diagonals of a kite are perpendicular by definition. Therefore, Assertion (A) is true.
For reason (R), the diagonals of a trapezium do not necessarily bisect each other unless it is an isosceles trapezium. Even in an isosceles trapezium, diagonals are equal but they only bisect each other if additional conditions are met. Hence, Reason (R) is false.
Thus, Assertion (A) is true, but Reason (R) is false.

Key Concept: Properties of Kite: Diagonals

c) Assertion is true, but Reason is false.
[Solution Description]
According to the property of a kite, one of the diagonals (the one connecting the vertices with equal adjacent sides) bisects the other diagonal at $90^\circ$ and also bisects the angles at those vertices. However, the diagonals of a kite do not necessarily bisect each other completely; only one diagonal is bisected by the other. Therefore, the Assertion is true, but the Reason is false.

Key Concept: Trapezium properties, Angle relationships

b) $23.75$
[Solution Description]
Recall that in an isosceles trapezium, the angles adjacent to each non-parallel side are equal.
So, $\angle P = \angle Q$ and $\angle S = \angle R$.
Also, since $PQ \parallel SR$, the sum of consecutive angles on the same side of the legs is $180^\circ$.
So, $\angle P + \angle S = 180^\circ$.
Given $\angle P = (3x + 10)^\circ$ and $\angle S = (5x - 20)^\circ$.
Therefore, $(3x + 10) + (5x - 20) = 180$.
Simplifying, $8x - 10 = 180$.
Then, $8x = 190$.
So, $x = \frac{190}{8} = 23.75$.

Key Concept: Kite properties, Angle relationships

b) $100^\circ$
[Solution Description]
First, recall that in a kite, one diagonal bisects the angles at the vertices it connects. So, diagonal $BD$ bisects $\angle ABC$ and $\angle ADC$.
Given $\angle ABC = 120^\circ$, so $\angle ABO = \angle CBO = \frac{120^\circ}{2} = 60^\circ$.
Similarly, $\angle ADC = 80^\circ$, so $\angle ADO = \angle CDO = \frac{80^\circ}{2} = 40^\circ$.
Now, consider triangle $AOD$. The sum of angles in a triangle is $180^\circ$.
We know $\angle OAD$ can be found using the property that diagonals of a kite are perpendicular. Thus, $\angle AOD = 90^\circ$.
However, let's verify this by calculating $\angle OAD$:
In quadrilateral $ABCD$, the sum of all angles is $360^\circ$.
So, $\angle BAD + \angle ABC + \angle BCD + \angle CDA = 360^\circ$.
Since $AB = BC$ and $CD = DA$, we can find $\angle BAD = \angle BCD$ as follows:
Let $\angle BAD = x$, then $\angle BCD = x$ (because $AB = BC$ and $CD = DA$).
So, $x + 120^\circ + x + 80^\circ = 360^\circ$.
Which gives $2x + 200^\circ = 360^\circ$.
So, $2x = 160^\circ$ and $x = 80^\circ$.
Now, $\angle OAD = \frac{\angle BAD}{2} = \frac{80^\circ}{2} = 40^\circ$.
In triangle $AOD$, $\angle AOD + \angle OAD + \angle ODA = 180^\circ$.
Substituting known values: $\angle AOD + 40^\circ + 40^\circ = 180^\circ$.
So, $\angle AOD = 100^\circ$.