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: Squares, Diagonals

c) $40$ cm
[Solution Description]
In a square, the length of the diagonal $d$ is related to the side length $s$ by the formula $d = s \sqrt{2}$. Given $d = 10\sqrt{2}$ cm, we can solve for $s$:
$10\sqrt{2} = s \sqrt{2} \implies s = 10 \text{ cm}.$
The perimeter $P$ of a square is four times the side length:
$P = 4s = 4 \times 10 = 40 \text{ cm}.$
Thus, the perimeter is $40$ cm.

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: Area Calculation with Diagonals

c) $44\%$
[Solution Description]
Let the original diagonal be $d$ and the original side length be $s$. Then, $d = s\sqrt{2}$ and the area is $A = s^2 = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}$. If the diagonal increases by $20\%$, the new diagonal is $1.2d$. The new area becomes:
$A' = \frac{(1.2d)^2}{2} = \frac{1.44d^2}{2} = 1.44A$
The percentage increase in area is $(1.44 - 1) \times 100\% = 44\%$.

Key Concept: Diagonals and Angles

b) $45^\circ$
[Solution Description]
In a square, the diagonals bisect each other at right angles, forming four congruent right-angled triangles. In triangle $AOB$, $\angle AOB = 90^\circ$ and the diagonals also bisect the angles of the square. Thus, $\angle OAB$ is half of $\angle DAB$ since the diagonal $AC$ bisects $\angle DAB$. Since all angles in a square are $90^\circ$, $\angle DAB = 90^\circ$ and $\angle OAB = 45^\circ$.

Key Concept: Properties of a rectangle

d) Opposite sides are equal and parallel.
[Solution Description]
A rectangle has four right angles (each 90°), opposite sides equal and parallel, and diagonals that are equal in length and bisect each other.

Key Concept: Opposite Sides of a Rectangle

b) 8 cm
[Solution Description]
By the property of rectangles, opposite sides are equal in length. Here, $AB$ is opposite to $CD$. Since $AB = 8$ cm, then $CD$ must also be 8 cm.

Key Concept: Angles Between Diagonals

c) $90^\circ$
[Solution Description]
In a square, the diagonals bisect each other at right angles ($90^\circ$). This is a unique property of squares compared to other rectangles.

Key Concept: Diagonals of Squares and Rectangles

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that if the diagonals of a quadrilateral bisect each other at right angles and are equal in length, then the quadrilateral must be a square. This is not entirely accurate because a rhombus also has diagonals that bisect each other at right angles, but its diagonals are not necessarily equal in length. However, the reason given is that such a quadrilateral satisfies all properties of a square, which is true only for squares. Therefore, while both statements seem plausible, the assertion overgeneralizes by not considering other quadrilaterals (like rhombuses) that partially satisfy these conditions.
Thus:
- Assertion (A) is false because other quadrilaterals (e.g., rhombus) can have diagonals bisecting at right angles without being squares.
- Reason (R) is true because, for a square, the diagonals do indeed satisfy these properties, but this does not mean the quadrilateral must necessarily be a square.

Key Concept: Congruency of Triangles

c) Rectangle
[Solution Description]
A quadrilateral whose diagonals are equal and bisect each other must be a rectangle. While squares also satisfy this condition, the general case is a rectangle because squares have additional properties like all sides being equal and diagonals intersecting at $90^\circ$.

Key Concept: Rectangle and Rhombus Properties, Non-Straightforward Path

c) Rhombus
[Solution Description]
Step 1: Diagonals bisecting each other imply it could be a rectangle, rhombus, or square.
Step 2: Diagonals bisecting at right angles eliminate rectangles (their diagonals are equal but don’t bisect at right angles unless it's a square).
Step 3: Unequal diagonals further eliminate squares (their diagonals are equal).
Conclusion: It must be a rhombus (diagonals bisect at right angles and are unequal).

Key Concept: Diagonal Intersection, Angle Calculation

c) $100^\circ$
[Solution Description]
In a rectangle, the diagonals bisect each other and are equal in length, forming congruent isosceles triangles. For the given quadrilateral formed by the diagonals, the angles are $80^\circ$, $100^\circ$, $80^\circ$, and $100^\circ$.
Since the sum of angles around a point is $360^\circ$, the angle between the diagonals is:
$x + 180 - x = 180^\circ.$
Here, the given angles suggest two pairs of supplementary angles ($80^\circ + 100^\circ = 180^\circ$), which implies the angle between the diagonals alternates as $80^\circ$ and $100^\circ$. Thus, $x$ is either $80^\circ$ or $100^\circ$.
However, for a rectangle, the answer must align with the property that diagonals form congruent isosceles triangles, so $x = 100^\circ$ is correct (as the acute angle would be $80^\circ$).

Key Concept: Properties of diagonals in a square

b) $90^\circ$, $90^\circ$, $90^\circ$
[Solution Description]
In a square, the diagonals are equal, bisect each other, and intersect at $90^\circ$. Therefore, the angles at the intersection are all equal. If one angle is $x = 90^\circ$, the other three angles will also be $90^\circ$ due to vertically opposite angles and linear pairs.

Key Concept: Diagonals of a square

c) $10\ \text{cm}$
[Solution Description]
For a square with diagonal $d$, the side length $s$ is related by $d = s\sqrt{2}$. Given $d = 10\sqrt{2}\ \text{cm}$, solving for $s$ gives:
$s = \frac{d}{\sqrt{2}} = \frac{10\sqrt{2}}{\sqrt{2}} = 10\ \text{cm}$

Key Concept: Angles between diagonals of a rhombus

d) $120^\circ$
[Solution Description]
The diagonals of a rhombus bisect the vertex angles and intersect at $90^\circ$ in a standard case. However, given an intersection angle of $60^\circ$, we calculate:
For triangle formed by half-diagonals, let the base angles be $a$.
Then, $a + a + 60^\circ = 180^\circ$ implies $2a = 120^\circ$, so $a = 60^\circ$.
Thus, each interior angle adjacent to the $60^\circ$ angle is $60^\circ + 60^\circ = 120^\circ$.

Key Concept: Properties of Rectangle

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

Key Concept: Properties of Rectangles and Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that if a quadrilateral has equal-length diagonals that bisect each other at 90°, it must be a square. This is true because such properties are unique to squares among all quadrilaterals. The Reason correctly explains the Assertion as it lists the defining properties of the diagonals of a square. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Quadrilateral diagonals

c) Square
[Solution Description]
A quadrilateral with equal diagonals that bisect each other at $90^\circ$ is a square because all angles will be $90^\circ$ and all sides will be equal.

Key Concept: Properties of Rectangles

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
In a rectangle, the diagonals are always equal in length. This is a proven property of rectangles. The diagonals also bisect each other, but this property alone does not explain why they are equal in length. Both statements are true, but the Reason does not correctly explain the Assertion.

Key Concept: Quadrilateral Angles, Side Lengths

b) Opposite sides are equal.
[Solution Description]
A quadrilateral with all angles equal to $90^\circ$ is a rectangle, where opposite sides are equal in length. This can be proved by dividing the quadrilateral into two congruent right triangles using a diagonal and applying the properties of congruent triangles. Thus, opposite sides must be equal.

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: 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: 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: Square Properties

c) They bisect each other at $90^\circ$
[Solution Description]
Both squares and rectangles have diagonals that are equal in length and bisect each other. However, the diagonals of a square additionally bisect each other at right angles ($90^\circ$), which is not necessarily true for all rectangles.

Key Concept: Square Properties

a) $3\sqrt{2}\ \text{cm}$
[Solution Description]
For any square, diagonals are equal and bisect each other perpendicularly. Given AB = 6 cm (side length), we find diagonal AC first using Pythagorean theorem:
$AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2}\ \text{cm}$.
Since diagonals bisect each other, AO = half of AC = $\frac{6\sqrt{2}}{2} = 3\sqrt{2}\ \text{cm}$.
Calculation steps:
1) Calculate diagonal AC using Pythagoras: $AC = \sqrt{6^2 + 6^2} = 6\sqrt{2}\ \text{cm}$.
2) Find AO as half of AC: $AO = 3\sqrt{2}\ \text{cm}$.

Key Concept: Properties of Squares and Rectangles, Geometric Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A square has diagonals that bisect each other at $90^\circ$ and are equal in length (Property 4 of squares). Since all sides of a square are equal (Property 1), the four triangles formed by the intersection of diagonals have two sides equal (half the diagonal) and the included angle as $90^\circ$. By SAS congruency, these triangles are congruent. Hence, both Assertion and Reason are true, and the Reason explains why the Assertion is true.

Key Concept: Construction of Parallelograms and Relation to Rectangles

a) 6.24 cm
[Solution Description]
To find the shorter diagonal of the parallelogram, we use the Law of Cosines.
Step 1: Let the sides be $a = 5$ cm and $b = 7$ cm, with the included angle $\theta = 60^\circ$.
Step 2: The shorter diagonal $d_1$ can be calculated using:
$d_1 = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}$
Step 3: Substituting the values:
$d_1 = \sqrt{5^2 + 7^2 - 2 \times 5 \times 7 \cos(60^\circ)} = \sqrt{25 + 49 - 70 \times 0.5} = \sqrt{74 - 35} = \sqrt{39} \approx 6.24 \text{ cm}$
Therefore, the length of the shorter diagonal is approximately $6.24$ cm.

Key Concept: Square and Rectangle Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if a quadrilateral has all sides equal and all angles equal to $90^{\circ}$, it must be a square. This is true because by definition, a square is a rectangle with all sides equal.
The reason states that every square is a rectangle (since it meets the rectangle's condition of having all angles equal to $90^{\circ}$ and opposite sides parallel and equal), but not every rectangle is a square (as rectangles can have unequal adjacent sides). This correctly explains why the assertion is true. Hence, both statements are true, and the reason correctly explains 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, Variable Substitution

b) $50^\circ$
[Solution Description]
The sum of all angles in a quadrilateral is $360^\circ$. Let the fourth angle be $x$.
We can write the equation: $70^\circ + 110^\circ + 130^\circ + x = 360^\circ$.
Adding the known angles: $310^\circ + x = 360^\circ$.
Subtract $310^\circ$ from both sides: $x = 50^\circ$.
Therefore, the fourth angle is $50^\circ$.

Key Concept: Sum of angles in a quadrilateral

b) $110^\circ$
[Solution Description]
The sum of all angles in a quadrilateral is $360^\circ$. Given three angles: $85^\circ$, $95^\circ$, and $70^\circ$.
Let the fourth angle be $x$.
Therefore, $85^\circ + 95^\circ + 70^\circ + x = 360^\circ$.
Simplify: $250^\circ + x = 360^\circ$.
Subtract $250^\circ$ from both sides to find $x = 110^\circ$.

Key Concept: Angles in a Quadrilateral

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The sum of the given angles is calculated as follows:
$70^\circ + 80^\circ + 100^\circ + 110^\circ = 360^\circ$.
Since the total equals $360^\circ$, the quadrilateral is valid, confirming the Assertion (A).
The Reason (R) states the fundamental property that the sum of interior angles in any quadrilateral is always $360^\circ$, which justifies why the given quadrilateral is valid. Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

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: Rectangle properties

d) Rectangle
[Solution Description]
A rectangle is defined as a quadrilateral with all four angles equal to $90^\circ$, opposite sides equal, and opposite sides parallel.

Key Concept: Sum of angles in a quadrilateral, properties of diagonals

c) 110°
[Solution Description]
Step 1: The sum of all angles in a quadrilateral is 360°.
Step 2: Given angles A, B, and C are 80°, 100°, and 70° respectively.
Step 3: Calculate the sum of known angles: $80° + 100° + 70° = 250°$.
Step 4: Subtract the sum from 360° to find angle D: $360° - 250° = 110°$.

Key Concept: Impossible quadrilateral configuration

c) 120°, 110°, 100°, 20°
[Solution Description]
For a valid quadrilateral, the sum of all four angles must be $360°$.
Checking each option:
Option a) $90° + 90° + 90° + 90° = 360°$ (valid).
Option b) $100° + 80° + 70° + 110° = 360°$ (valid).
Option c) $120° + 110° + 100° + 20° = 350° \neq 360°$ (invalid).
Option d) $95° + 85° + 90° + 90° = 360°$ (valid).
Thus, option c does not satisfy the sum condition.

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: 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 it is impossible to have a quadrilateral with three right angles and one non-right angle. This is true because, as per the Reason, the sum of all angles in a quadrilateral must be $360^\circ$. If three angles are each $90^\circ$, their total would be $270^\circ$, leaving the fourth angle as $90^\circ$ ($360^\circ - 270^\circ = 90^\circ$). Therefore, the fourth angle cannot be anything other than $90^\circ$. Here, the Reason correctly explains why the Assertion is true.

Key Concept: Angles in a Quadrilateral, Sum of Angles

d) Assertion is false, but Reason is true.
[Solution Description]
To verify, assume a quadrilateral has three right angles ($90^{\circ}$ each). Let the fourth angle be $x$. According to the property of quadrilaterals, the sum of all four angles must be $360^{\circ}$. Thus:
$90^{\circ} + 90^{\circ} + 90^{\circ} + x = 360^{\circ}.$
Simplifying:
$270^{\circ} + x = 360^{\circ},$
which gives:
$x = 90^{\circ}.$
Therefore, if three angles are $90^{\circ}$, the fourth must also be $90^{\circ}$. Hence, it is impossible for a quadrilateral to have exactly three right angles and one non-right angle. The Assertion is false, but the Reason is true.

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: Quadrilateral Angle Sum Property

d) Assertion is false, but Reason is true.
[Solution Description]
We know from the properties of quadrilaterals that the sum of all four interior angles is always $360^\circ$.
If three angles are each $90^\circ$, their sum is $3 \times 90^\circ = 270^\circ$. If the fourth angle were not $90^\circ$, say $x$, then the total sum would be $270^\circ + x$. For this to equal $360^\circ$, $x$ must satisfy:
$270^\circ + x = 360^\circ$
$x = 360^\circ - 270^\circ$
$x = 90^\circ$
This means the fourth angle must also be $90^\circ$ to satisfy the angle sum property. Hence, it is impossible for a quadrilateral to have three right angles and a non-right fourth angle.
The assertion claims it is possible, which is false, but the reason correctly states the angle sum property of quadrilaterals.

Key Concept: Definition of Parallelogram

c) Both pairs of opposite sides are parallel.
[Solution Description]
A parallelogram is a quadrilateral with both pairs of opposite sides parallel and equal, opposite angles equal, adjacent angles supplementary, and diagonals bisecting each other.

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: Opposite Angles in Parallelogram

a) $70^\circ$
[Solution Description]
In a parallelogram, opposite angles are equal. So, if $\angle A = 70^\circ$, then $\angle C$ will also be $70^\circ$ because they are opposite angles.

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: Definition of Parallelogram

c) A quadrilateral in which opposite sides are parallel
[Solution Description]
A parallelogram is defined as a quadrilateral in which opposite sides are parallel.

Key Concept: Opposite sides and diagonals of parallelogram

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion using the properties of parallelograms. Consider a parallelogram with adjacent sides $a$ and $b$. Let $d_1$ and $d_2$ be its diagonals. By the property of parallelograms, the diagonals bisect each other. Using the law of cosines in one of the triangles formed by the diagonals, we get:
For triangle formed by half-diagonals and side $a$:
$\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 - 2 \cdot \frac{d_1}{2} \cdot \frac{d_2}{2} \cos(\theta) = a^2$
Similarly for side $b$:
$\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 - 2 \cdot \frac{d_1}{2} \cdot \frac{d_2}{2} \cos(180^\circ - \theta) = b^2$
Adding these two equations and simplifying:
$2\left(\frac{d_1^2}{4} + \frac{d_2^2}{4}\right) + \frac{d_1 d_2 \cos(\theta)}{2} - \frac{d_1 d_2 \cos(\theta)}{2} = a^2 + b^2$
Which simplifies to:
$\frac{d_1^2 + d_2^2}{2} = a^2 + b^2 \implies d_1^2 + d_2^2 = 2(a^2 + b^2)$
Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Assertion-Reason Question

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the opposite sides of a parallelogram are equal, which is one of the fundamental properties of a parallelogram as given in the syllabus. The Reason states that a parallelogram has its opposite sides parallel, which is the definition of a parallelogram. While both statements are true, the Reason does not explain why the opposite sides are equal; it merely defines what a parallelogram is. Therefore, the correct option is b).

Key Concept: Congruent triangles in a parallelogram

c) $\Delta ABD \cong \Delta BDC$
[Solution Description]
In parallelogram ABCD, opposite sides are equal ($AB = CD$ and $AD = CB$). By the SSS (Side-Side-Side) congruence condition, $\Delta ABD \cong \Delta CDB$ because $AB = CD$, $AD = CB$, and the common side BD is equal to itself. Hence, these two triangles are congruent.

Key Concept: Properties, Angles

b) Rhombus
[Solution Description]
Step 1: Quadrilateral with diagonals bisecting each other is a parallelogram. So, $ABCD$ is initially a parallelogram.
Step 2: Since $\angle AOB = 90°$, the diagonals intersect at right angles. A parallelogram with perpendicular diagonals is either a rhombus or a square.
Step 3: However, without knowing whether all sides are equal or angles are $90°$, we cannot confirm if it is a square. But the given condition guarantees at least rhombus properties.
Step 4: Therefore, $ABCD$ must be a rhombus among the options provided.

Key Concept: Properties of Rectangles

c) All angles are equal to $90^{\circ}$ and diagonals are equal.
[Solution Description]
A rectangle is a parallelogram with all angles equal to $90^{\circ}$. It satisfies all properties of a parallelogram, and additionally, its diagonals are equal in length.

Key Concept: Properties of Rectangles

d) The diagonals bisect each other at $90^\circ$
[Solution Description]
A rectangle has all angles equal to $90^\circ$, opposite sides equal and parallel, and diagonals that bisect each other. However, its diagonals do not necessarily bisect at $90^\circ$ unless it is a square. Thus, the incorrect property among the options is that the diagonals bisect at $90^\circ$.

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: Venn Diagram Relationships

d) Square
[Solution Description]
A square has all the properties of both a rectangle and a rhombus. It has all angles equal to $90^\circ$ (like a rectangle) and all sides equal (like a rhombus). Therefore, the correct answer is Square.

Key Concept: Angle Properties in Rhombus

d) $120^\circ$
[Solution Description]
Adjacent angles in a rhombus are supplementary (add up to $180^\circ$). Given $\angle A = 60^\circ$, we can find $\angle B$ using the property:
$\angle A + \angle B = 180^\circ$
$60^\circ + \angle B = 180^\circ$
$\angle B = 180^\circ - 60^\circ = 120^\circ$.
The opposite angles in a rhombus are equal, so $\angle D = \angle B = 120^\circ$.

Key Concept: Venn Diagram, Properties of Quadrilaterals

d) Parallelogram
[Solution Description]
The quadrilateral has both pairs of opposite sides equal and parallel, which means it is a parallelogram. Since the diagonals are not perpendicular and do not bisect the angles, it cannot be a rhombus or square. It also does not have all angles equal to $90^\circ$, so it is not a rectangle. Therefore, it is a general parallelogram.

Key Concept: Venn Diagram of Quadrilaterals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that a square is both a rhombus and a rectangle because it meets the criteria of all three types of quadrilaterals. This is true since a square has all sides equal (rhombus property), all angles equal to $90^\circ$ (rectangle property), and opposite sides parallel (parallelogram property). The Reason explains that in a Venn diagram, squares lie in the intersection of rhombuses and rectangles, which correctly represents their hierarchical relationship. Since a square inherits properties from both rhombuses and rectangles, the Venn diagram must show this overlap, making the Reason the correct explanation of the Assertion.

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: Properties of Rhombus and Square

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that every square is a rhombus, which is true because a square has all sides equal, satisfying the definition of a rhombus. The Reason states that a rhombus with all angles equal to $90^\circ$ is a square, which is also true as this defines a square. Moreover, the Reason correctly explains why the Assertion is true, since it describes the condition under which a rhombus becomes a square.

Key Concept: Types of Quadrilaterals

c) Rhombus
[Solution Description]
A rhombus has all sides equal, but its angles are not necessarily equal (unless it is a square). Therefore, a rhombus fits this description.

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: Basic Properties of a Rhombus

c) Opposite sides are parallel
[Solution Description]
A rhombus has all sides equal, opposite sides parallel, adjacent angles supplementary, and diagonals bisecting each other at right angles. Among the given options, only one correctly states a property that applies to all rhombuses.
Checking each option:
1) "All angles are 90 degrees" is incorrect because a rhombus does not necessarily have all angles as 90 degrees (that would make it a square).
2) "Diagonals are equal in length" is incorrect because diagonals of a rhombus are generally not equal unless it is a square.
3) "Opposite sides are parallel" is correct as this is a fundamental property of all rhombuses.
4) "Only two sides are equal" is incorrect because all four sides must be equal in a rhombus.
Thus, the correct answer is option c).

Key Concept: Diagonals of a Rhombus

c) $90^\circ$
[Solution Description]
The diagonals of a rhombus bisect each other at right angles ($90^\circ$).

Key Concept: Properties of a Rhombus

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, let's examine Assertion (A): "The diagonals of a rhombus bisect each other at right angles." This is true because one of the properties of a rhombus states that its diagonals bisect each other at $90^\circ$.
Next, let's check Reason (R): "In a rhombus, the adjacent angles add up to $180^\circ$ and the opposite angles are equal." This is also true as it is another property of a rhombus mentioned in the syllabus.
However, while both Assertion and Reason are true, they describe different properties of a rhombus. The fact that adjacent angles sum to $180^\circ$ and opposite angles are equal does not directly explain why the diagonals bisect at right angles. These are separate properties.
Therefore, both statements are correct, but the reason does not explain the assertion.

Key Concept: Adjacent Angles in a Rhombus

d) $110^\circ$
[Solution Description]
In a rhombus, adjacent angles are supplementary, meaning they add up to $180^\circ$. Given $\angle A = 70^\circ$, we can find $\angle B$ using the property of adjacent angles:
$\angle A + \angle B = 180^\circ$
$70^\circ + \angle B = 180^\circ$
$\angle B = 180^\circ - 70^\circ = 110^\circ$
Therefore, the correct answer is option d).

Key Concept: Angle Calculation in Rhombus

d) $120^\circ$
[Solution Description]
In a rhombus, adjacent angles add up to $180^\circ$ and opposite angles are equal. Given $\angle A = 60^\circ$, its adjacent angle $\angle B$ can be found as:
$\angle B = 180^\circ - \angle A = 180^\circ - 60^\circ = 120^\circ$

Key Concept: Construction of Rhombus

c) All sides must be equal.
[Solution Description]
A rhombus has all four sides of equal length. Therefore, for construction, we must ensure that all sides are equal.

Key Concept: Opposite angles, Diagonal properties

b) 60
[Solution Description]
In a rhombus, diagonals bisect each other at $90^\circ$ and also bisect the angles. So $\angle QPR = 2 \times \angle QPO = 60^\circ$. Since all sides are equal, triangle $PQR$ is isosceles with $PQ = QR$. Therefore, $\angle PQR = 180^\circ - 2(30^\circ) = 120^\circ$. The diagonals intersect at right angles, so in triangle $ROQ$, $\angle ROQ = 180^\circ - 90^\circ - 30^\circ = 60^\circ$.

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: Parallelograms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states the definition of a parallelogram, which is correct. The reason states that rectangles and squares are special cases of parallelograms, which is also true. Furthermore, the reason correctly explains why the assertion holds because it provides additional classification information about parallelograms. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Quadrilateral Identification

d) Square
[Solution Description]
When two diagonals are perpendicular and of equal length, the quadrilateral formed is a square because all sides and angles are equal.

Key Concept: Folding Paper Activity

c) Kite

[Solution Description]

Folding the paper into quarters and making a triangular crease results in a kite shape due to the symmetry of the folds and the crease.

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: Properties of Diagonals in Quadrilaterals

c) Square
[Solution Description]
If the diagonals of a quadrilateral are equal in length, bisect each other, and intersect at $90^\circ$, the quadrilateral must be a square. This is because these conditions ensure all sides are equal and all angles are $90^\circ$.

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: 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: Geoboard Activity, Quadrilateral Properties

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that such a quadrilateral must be a square, but the reason only guarantees it is a rectangle (as per syllabus observations). For it to be a square, additional conditions (all sides equal) are needed. Thus, the assertion is false, but the reason is true as it correctly describes properties leading to a rectangle.

Key Concept: Constructing quadrilaterals using scalene triangles

d) Parallelogram
[Solution Description]
When two identical scalene triangles (with sides 6 cm, 9 cm, and 12 cm) are joined along their 6 cm sides, the resulting quadrilateral will have sides measuring 9 cm, 12 cm, 12 cm, and 9 cm. Since both pairs of opposite sides are equal, it forms a parallelogram.

Key Concept: Joining Triangles, Kite Properties

d) Assertion is false, but Reason is true.
[Solution Description]
Consider two scalene triangles ABC and CDA with sides 6 cm, 9 cm, and 12 cm. When joined along the 9 cm side (AC), the quadrilateral ABCD will have AB = CD = 6 cm and BC = DA = 12 cm. This satisfies the definition of a kite ($AB = BC$ is false here; correct condition for kite is $AB = AD$ and $CB = CD$). The assertion incorrectly states the condition for a kite, while the reason provides the correct property. Thus, Assertion is false but Reason is true.

Key Concept: Joining Triangles, Trapezium Properties

c) Isosceles trapezium
[Solution Description]
When two isosceles triangles with sides 8 cm, 8 cm, and 6 cm are joined along their unequal sides (6 cm), the resulting quadrilateral is an isosceles trapezium. In an isosceles trapezium, the non-parallel sides are equal, and the base angles are equal. The diagonals are also equal in length.

Key Concept: Kite, Diagonals

b) They bisect each other perpendicularly
[Solution Description]
In a kite ABCD with AB = BC and CD = DA, diagonal BD bisects the other diagonal AC perpendicularly. This means BD divides AC into two equal parts (AO = OC) and forms right angles (90 degrees) with AC. Additionally, BD also bisects the angles ABC and ADC.

Key Concept: Kite and Trapezium combined, Angle calculation

c) 120$^\circ$
[Solution Description]
The quadrilateral described is a combination of kite and trapezium properties. As it has two pairs of adjacent sides equal, it must be a kite. But it also has one pair of parallel sides, making it a trapezium. Hence, it is an isosceles trapezium where non-parallel sides are equal. Given one angle as $120^\circ$, the adjacent angle on the same side of the transversal will be $60^\circ$ (since they add up to $180^\circ$). The opposite angles in an isosceles trapezium are supplementary, so the largest angle will be $120^\circ$.

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

c) $120^\circ$
[Solution Description]
The properties of a kite state that the angles opposite to the equal sides are equal. Given $\angle ABC = 120^\circ$, its opposite angle $\angle ADC$ will also be $120^\circ$.

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: Definition of Kite

c) A quadrilateral with two pairs of adjacent sides equal
[Solution Description]
A kite is a quadrilateral with two distinct pairs of adjacent sides equal. By definition, if a quadrilateral ABCD has $AB = BC$ and $CD = DA$, then it is a kite.

Key Concept: Kite Properties, Congruent Triangles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given that $\Delta AOB \cong \Delta COB$ in the kite ABCD, this implies that the corresponding angles $\angle ABO = \angle CBO$ and $\angle ADO = \angle CDO$ due to congruency. Therefore, the diagonal BD indeed bisects $\angle ABC$ and $\angle ADC$. The Reason correctly explains why the Assertion is true because congruent triangles ensure equal corresponding angles, which leads to the bisection of the angles by the diagonal BD.

Key Concept: Kite Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In a kite ABCD, where AB = BC and CD = DA, the diagonal BD divides the kite into two congruent triangles $\Delta ABD$ and $\Delta CBD$ by SSS congruency (since AB = BC, AD = DC, and BD is common). This congruence implies that $\angle ABD = \angle CBD$ and $\angle ADB = \angle CDB$, meaning BD bisects both $\angle ABC$ and $\angle ADC$. Additionally, since $\Delta AOB$ and $\Delta COB$ are congruent (by SAS congruency as AB = CB, $\angle ABO = \angle CBO$, and BO is common), AO = OC, proving BD bisects AC. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Congruency in Kite

c) Both sides and angles are equal.
[Solution Description]
In kite ABCD, diagonal BD bisects $\angle ABC$ and $\angle ADC$. Hence, $\angle ABO = \angle CBO$. Also, $AB = BC$ by definition. Since BD is common to both $\Delta AOB$ and $\Delta COB$, they are congruent by SAS (Side-Angle-Side) rule. Therefore, corresponding parts like $AO = OC$ and $\angle AOB = \angle COB$ (both right angles).

Key Concept: Trapezium angles, Isosceles trapezium properties

b) $105^\circ$
[Solution Description]
Property of isosceles trapezium states that angles adjacent to each base are equal. Here, $\angle A = \angle B = 75^\circ$. Sum of angles on same side of leg in trapezium is $180^\circ$, so $\angle D = 180^\circ - 75^\circ = 105^\circ$. Since it's isosceles, $\angle C = \angle D = 105^\circ$.

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: Isosceles trapezium properties

b) $70^\circ$
[Solution Description]
In an isosceles trapezium, the angles opposite to the equal sides are equal. Therefore, $\angle U = \angle V$. Given that $\angle U = 70^\circ$, it follows that $\angle V$ is also $70^\circ$.

Key Concept: Trapezium Angle Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the property of trapezium, when one pair of opposite sides is parallel (here $AB \parallel CD$), the sum of the angles on the same side of the transversal is $180^\circ$. Therefore, $\angle A + \angle D = 180^\circ$ and $\angle B + \angle C = 180^\circ$. The reason given is also true as it states the general property of angles formed by a transversal intersecting two parallel lines. Further, the reason correctly explains why the assertion is true.

Key Concept: Properties of Isosceles Trapezium - Angles

b) $115^\circ$
[Solution Description]
In an isosceles trapezium, the angles opposite to the equal sides are equal. Let these angles be $65^\circ$ each. The sum of all angles in any quadrilateral is $360^\circ$. Let the remaining two angles be $x$ each since they must also be equal.
Calculation:
$65^\circ + 65^\circ + x + x = 360^\circ \implies 130^\circ + 2x = 360^\circ \implies 2x = 230^\circ \implies x = 115^\circ$
Thus, each of the other two angles measures $115^\circ$.

Key Concept: Angle Relationships in Trapezium

b) $110^\circ$
[Solution Description]
A property of a trapezium states that the sum of adjacent angles on the same side of the non-parallel sides is $180^\circ$. Thus, $\angle P + \angle S = 180^\circ$. Given $\angle P = 70^\circ$, we can calculate $\angle S$ as follows:
$\angle S = 180^\circ - 70^\circ = 110^\circ$

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 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.