Key Concept: Properties of Squares and Rectangles

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 its angles, which is true as per the properties of a square. The reason states that a square has all sides equal and diagonals perpendicular to each other, which is also true. However, the reason does not explain why the diagonals of a square bisect its angles. The correct explanation would involve the fact that the diagonals of a square are angle bisectors due to the congruence of triangles formed by the diagonals and sides of the square. Thus, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Properties of a Square

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the diagonals of a square are equal in length and bisect each other at 90°, which is true because it is one of the properties of a square. The reason states that a square is a quadrilateral with all sides equal and all angles equal to 90°, which is also true and explains why the diagonals have these properties. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Properties of a Rectangle

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that all angles in a rectangle are $90°$, which is a fundamental property of rectangles as per the syllabus. The reason provided is that a rectangle is a quadrilateral with opposite sides equal and parallel. While this is also a true statement, it does not explain why the angles are $90°$. Therefore, both statements are true, but the reason does not correctly explain the assertion.

Key Concept: Rectangle properties, Diagonal congruence

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that $\triangle AOB \cong \triangle COD$ by SAS congruence in rectangle $ABCD$.
Step 1: Properties of rectangle diagonals - Diagonals are equal in length ($AC = BD$) and bisect each other at point $O$. So, $AO = OC$ and $BO = OD$.
Step 2: Since opposite sides of rectangle are equal, $AB = CD$.
Step 3: Included angle between sides $AO$ and $BO$ is $\angle AOB$, which is equal to $\angle COD$ (vertically opposite angles).
Thus, by SAS criteria ($AO = OC$, $\angle AOB = \angle COD$, $BO = OD$), $\triangle AOB \cong \triangle COD$.
The reason correctly states that diagonals are equal and bisect each other, but this property alone does not directly explain why SAS congruence holds because it doesn't mention the vertically opposite angles or the equality of opposite sides. Hence, both assertion and reason are true, but reason is not the correct explanation.

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: Properties of a Square

c) 32 cm
[Solution Description]
Given the diagonal $d = 8\sqrt{2}$ cm. For a square, the relationship between the side $s$ and the diagonal $d$ is given by $d = s\sqrt{2}$. Solving for $s$, we get $s = \frac{d}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}} = 8$ cm. The perimeter of the square is $4s = 4 \times 8 = 32$ 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 the diagonals of rectangle $ABCD$, $AC$ and $BD$, are equal in length, which is a known property of rectangles. Therefore, the assertion is true.
The reason states that the diagonals of a rectangle bisect each other, which is also a known property of rectangles. However, this property does not explain why the diagonals are equal in length—it is an independent property. Thus, the reason is true, but it is not the correct explanation for the assertion.

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: 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: 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: Diagonals of Rectangle

c) Are equal in length and bisect each other
[Solution Description]
The diagonals of a rectangle are equal in length and bisect each other at their midpoints. They do not necessarily intersect at right angles unless the rectangle is a square.

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: Angles between diagonals of a rectangle

a) $50^\circ, 50^\circ, 130^\circ$
[Solution Description]
In a rectangle, the diagonals bisect each other and form two pairs of equal angles. Given one angle as $x = 50^\circ$, the other angles will be $50^\circ$, $180^\circ - 50^\circ = 130^\circ$, and $130^\circ$ as they must add up to $360^\circ$ in total.

Key Concept: Quadrilaterals, Diagonals of a Rectangle and Square

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that if diagonals of a quadrilateral bisect each other at $90^\circ$, the quadrilateral must be a rhombus or square. This is true because a rhombus and square have diagonals intersecting at right angles, while rectangles without equal adjacent sides (non-squares) do not necessarily meet this condition. However, the reason states that a quadrilateral with diagonals bisecting each other at $90^\circ$ has all sides equal, which is only true for rhombuses and squares but not for general quadrilaterals. Therefore, the assertion is true, but the reason is false because it incorrectly generalizes the property to all quadrilaterals instead of specific ones like rhombuses and squares.

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 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: Properties of square diagonals

c) $90^\circ$
[Solution Description]
In a square, the diagonals bisect each other at $90^\circ$. Therefore, the angle between the diagonals is $90^\circ$.

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: Construction of Square

c) $3\sqrt{2}$ cm
[Solution Description]
For a square, the relationship between the diagonal ($d$) and the side length ($s$) is given by $d = s\sqrt{2}$. Rearranging this, $s = \frac{d}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2}$ cm.

Key Concept: Construction and reasoning about squares

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion claims that a quadrilateral with equal diagonals bisecting each other at 90° must be a square. The reason states that such conditions ensure all sides are equal and all angles are 90°.
First, consider the properties of a square:
- Diagonals are equal in length.
- Diagonals bisect each other at 90°.
- All sides are equal.
- All angles are 90°.
However, a rhombus also has diagonals that bisect each other at 90° and all sides equal, but its angles are not necessarily 90°. Hence, the assertion is partially correct but not entirely, as the given conditions could also describe a rhombus unless the angles are specified as 90°. The reason correctly explains the properties but fails to distinguish between a square and a rhombus uniquely. Thus, the assertion is false because the conditions alone do not guarantee a square, while the reason is true.

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: 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: 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: 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: 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: Rectangle Properties

a) Diagonals bisect each other at $90^\circ$
[Solution Description]
A rectangle has the following properties: all angles are $90^\circ$, opposite sides are equal and parallel, and diagonals are equal and bisect each other. The option that does not match any of these properties is the correct answer.

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: 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: Properties of Rectangles and Squares

b) 10 cm
[Solution Description]
To find the diagonal of a rectangle, we can use the Pythagorean theorem. The diagonal divides the rectangle into two right-angled triangles.
Step 1: Let the sides of the rectangle be $a = 6$ cm and $b = 8$ cm.
Step 2: The diagonal $d$ can be calculated using the formula:
$d = \sqrt{a^2 + b^2}$
Step 3: Substituting the values:
$d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}$
Thus, the diagonal is $10$ cm.

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: Special Case: Square

d) Square
[Solution Description]
A quadrilateral with all sides equal and all angles equal to $90^\circ$ satisfies the definition of a square. Hence, it is a square.

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: 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: Properties of a rectangle

c) Rectangle
[Solution Description]
A quadrilateral with all four angles equal must have each angle equal to $90^\circ$ because the total sum of angles is $360^\circ$. Therefore, such a quadrilateral is a rectangle.

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: Sum of angles in a quadrilateral, diagonals creating triangles

b) 40°
[Solution Description]
Step 1: In triangle EFG, angles are 50°, 60°, and 70° (sum = 180°).
Step 2: Diagonal EG divides the quadrilateral into two triangles, EFG and EHG.
Step 3: Given one angle of triangle EHG is 40°. Let the other two angles be x and y ⇒ $40° + x + y = 180°$ ⇒ $x + y = 140°$.
Step 4: The quadrilateral's angles are the combination of both triangles' angles: 50°, 60°, 70°, 40°, x, y.
But since the quadrilateral has only four angles, they must be 50°, 60°, 70°, and 40° (the unique angles from both triangles). However, this leads to a sum of $50° + 60° + 70° + 40° = 220° ≠ 360°$, which is incorrect.
Upon reevaluating, the quadrilateral's angles are formed by combining adjacent angles from the two triangles. For example, $∠FEG + ∠HEG$ forms angle E, and similarly for others. Therefore, the quadrilateral's angles are:
$∠E = ∠FEG + ∠HEG$, $∠F = ∠EFG$, $∠G = ∠EGF + ∠EGH$, $∠H = ∠EHG$.
Given $∠F = 60°$ (from triangle EFG), $∠H = 40°$ (given in EHG), $∠FEG = 50°$, $∠EGF = 70°$, and let $∠HEG = z$, $∠EGH = w$.
Since $∠FEG + ∠HEG = ∠E ⇒ 50° + z = ∠E$.
And $∠EGF + ∠EGH = ∠G ⇒ 70° + w = ∠G$.
Also, in triangle EHG: $40° + z + w = 180° ⇒ z + w = 140°$.
Now, sum of quadrilateral angles: $∠E + ∠F + ∠G + ∠H = 360° ⇒ (50° + z) + 60° + (70° + w) + 40° = 360° ⇒ 220° + z + w = 360° ⇒ z + w = 140°$.
This confirms consistency but doesn't provide new information. To find the smallest angle, note that the quadrilateral's angles include $∠F = 60°$, $∠H = 40°$, and $∠E$ and $∠G$ are combinations of angles greater than or equal to their components. Thus, the smallest angle is $∠H = 40°$.

Key Concept: Sum of angles in a quadrilateral, relationship between angles

d) 90°
[Solution Description]
Step 1: Given that angles P and R are supplementary, so $P + R = 180°$.
Step 2: Given $P = 110°$, so $R = 180° - 110° = 70°$.
Step 3: Angles Q and S are supplementary, so $Q + S = 180°$.
Step 4: Sum of all angles in quadrilateral PQRS is 360°.
Step 5: So, $P + Q + R + S = 360°$ ⇒ $110° + Q + 70° + S = 360°$.
Step 6: From step 3, substitute Q as $(180° - S)$ ⇒ $110° + (180° - S) + 70° + S = 360°$.
Step 7: Solve the equation: $110° + 180° - S + 70° + S = 360°$ ⇒ $360° = 360°$.
This shows that angle S can be any value as long as $Q + S = 180°$. However, since the options provided are specific, we need to consider possible constraints. Typically, for a valid quadrilateral, $S$ must be such that none of the angles exceed practical limits.

Key Concept: Sum of angles in a quadrilateral

b) $360^\circ$
[Solution Description]
The sum of the interior angles of any quadrilateral is always $360^\circ$. This is because splitting the quadrilateral into two triangles (by drawing a diagonal) gives two triangles, each with angles summing to $180^\circ$.

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]
To verify the assertion, let's calculate the fourth angle:
Sum of given angles = $80° + 100° + 90° = 270°$.
Since the total sum of angles in a quadrilateral is $360°$, the fourth angle must be $360° - 270° = 90°$. This matches the assertion.
The reason states that the sum of interior angles in any quadrilateral is $360°$, which is true and correctly explains why the fourth angle must be $90°$ in this case.
Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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: Quadrilateral construction, angle sum

c) $100^\circ$
[Solution Description]
Using the angle sum property of quadrilaterals:
$\angle A + \angle B + \angle C + \angle D = 360^\circ$
Substituting known values:
$80^\circ + 90^\circ + 90^\circ + \angle D = 360^\circ$
$260^\circ + \angle D = 360^\circ$
Therefore, $\angle D = 100^\circ$
The side length AB = 5 cm is given but doesn't affect the angle calculation as angle sum is independent of side lengths in this case.

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: Properties of quadrilaterals

c) $\angle S$ is a right angle
[Solution Description]
Since the sum of angles in a quadrilateral is $360^\circ$, we can calculate $\angle S$ as follows:
$\angle P + \angle Q + \angle R + \angle S = 360^\circ$
Substituting the given angles:
$90^\circ + 90^\circ + 90^\circ + \angle S = 360^\circ$
$270^\circ + \angle S = 360^\circ$
$\angle S = 90^\circ$
Thus, $\angle S$ must also be $90^\circ$.

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: Properties of a Parallelogram

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In a parallelogram, by definition, both pairs of opposite sides are parallel. The assertion states that the opposite sides are equal and parallel, which is true as per the properties of a parallelogram. The reason correctly defines a parallelogram as a quadrilateral with both pairs of opposite sides parallel. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Special Cases: Rhombus, Rectangle, Square

c) All sides are equal
[Solution Description] A rhombus has all sides equal, but its angles are not necessarily $90^\circ$. A rectangle has all angles equal to $90^\circ$, but its sides are not necessarily equal. Therefore, the property that applies to all rhombuses but not necessarily to all rectangles is "All sides are equal".

Key Concept: Diagonals of Rhombus

c) $10$ cm
[Solution Description]
The diagonals of a rhombus bisect each other at $90^\circ$. So, half-diagonals are $6$ cm and $8$ cm. Using the Pythagorean theorem:
Side length = $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm.

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: Opposite sides and angles of a parallelogram

b) $110^\circ$
[Solution Description]
In a parallelogram, opposite angles are equal. Here, $\angle P$ and $\angle R$ are opposite angles, while $\angle Q$ and $\angle S$ are opposite angles. Therefore, $\angle S = \angle Q$. However, adjacent angles in a parallelogram add up to $180^\circ$, so $\angle Q = 180^\circ - \angle P = 180^\circ - 70^\circ = 110^\circ$. Thus, $\angle S = 110^\circ$.

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: 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: Diagonal Properties

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

Key Concept: Classification of Quadrilaterals

c) Every rectangle is a parallelogram, but not vice versa.
[Solution Description]
Every rectangle is a parallelogram since it has opposite sides parallel and equal. However, not every parallelogram is a rectangle because parallelograms do not necessarily have all angles equal to $90^{\circ}$.

Key Concept: Rectangle as a Parallelogram

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
A rectangle satisfies the definition of a parallelogram because it has two pairs of opposite sides that are parallel and equal in length (Property 3 of Rectangle). Additionally, the diagonals of a rectangle bisect each other (Property 4 of Rectangle), which is also true for any parallelogram. However, the Reason does not explain why a rectangle is a "special" type of parallelogram (i.e., due to its all angles being $90^\circ$). Thus, while both Assertion and Reason are true, the Reason does not fully explain the Assertion.

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: 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: 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: Venn Diagram: Relationship between Rectangles, Rhombuses, Squares, and Parallelograms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that all squares are rectangles, which is true because a square is a special type of rectangle where all sides are equal and all angles are $90^\circ$. The reason given is that a square has all angles equal to $90^\circ$, which is also true and correctly explains why all squares are rectangles, as this property is essential for any quadrilateral to be classified as a rectangle.

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: Comparison with Square

c) A square has all angles equal to $90^\circ$
[Solution Description]
A square is a special case of a rhombus where all angles are $90^\circ$. Therefore, the key difference lies in the angle measures.

Key Concept: Properties of Rhombus, Diagonal properties

b) 20 cm
[Solution Description]
A rhombus has diagonals that bisect each other at $90^\circ$. Given $AC = 8$ cm, so $AO = CO = 4$ cm. Given $BD = 6$ cm, so $BO = DO = 3$ cm. Using the Pythagorean theorem in $\triangle AOB$:
$AB = \sqrt{AO^2 + BO^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ cm.
Since all sides of a rhombus are equal, perimeter $= 4 \times AB = 4 \times 5 = 20$ cm.

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, 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: Relationship with Other Quadrilaterals

d) A quadrilateral with all sides equal
[Solution Description]
We need to identify the quadrilateral that is always a rhombus but not necessarily a rectangle. Let's evaluate the options:
1) "Square" is incorrect because a square is both a rhombus and a rectangle.
2) "Parallelogram" is incorrect because not all parallelograms have all sides equal (only rhombuses do).
3) "Rectangle" is incorrect because a rectangle does not necessarily have all sides equal unless it is a square.
4) "A quadrilateral with all sides equal" is correct because it defines a rhombus, which may or may not be a rectangle (unless it is a square).
The correct answer is option d).

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 Rhombus, Diagonals

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, analyze the Assertion: A rhombus has diagonals that are perpendicular bisectors and divide the rhombus into four congruent right-angled triangles. However, unless it is a square, the diagonals are unequal. So, the Assertion is true.
Next, examine the Reason: It states that the diagonals bisect each other at right angles, which is a fundamental property of a rhombus. Hence, the Reason is also true.
Now, check if the Reason explains the Assertion: The fact that diagonals bisect at right angles does not imply they must be unequal (e.g., in a square, they are equal). Therefore, although both statements are true, the Reason does not explain why the diagonals are unequal.
Thus, the correct option is: Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Adjacent Angles of Rhombus

b) $110^\circ$
[Solution Description]
Property 3 mentions that adjacent angles in a rhombus add up to $180^\circ$. If one angle is $70^\circ$, then its adjacent angle will be $180^\circ - 70^\circ = 110^\circ$.

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: Angle properties, External points

c) 60$^\circ$
[Solution Description]
Since $XWZB$ is a straight line, $\angle BXW = 180^\circ$. Triangle $AXB$ is equilateral (since $XA = XB$ and $\angle AXB = 60^\circ$). In rhombus $XYZW$, adjacent angles add up to $180^\circ$ and opposite angles are equal. Let $\angle YXW = y$. Then $\angle WXZ = 180^\circ - y$. Also, $\angle AXW = 60^\circ + y$ (exterior angle). But $\angle AXW$ must equal $\angle BXW - \angle BXA = 180^\circ - 60^\circ = 120^\circ$. Thus $60^\circ + y = 120^\circ$ → $y = 60^\circ$.

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: Folding Paper to Form Quadrilaterals

d) Kite
[Solution Description]
Folding the sheet into half creates a straight line (crease). Folding it again into a quarter creates another perpendicular straight line. Making a triangular crease at the middle corner introduces diagonal folds. When these creases are opened, they form intersecting lines within the paper. Specifically:
- Two main creases from folding into quarters (vertical and horizontal).
- Diagonal creases from the triangular fold.
The resulting pattern forms a kite shape due to the intersection of the diagonal folds with the main creases, creating two pairs of adjacent sides of unequal length meeting at the central point.

Key Concept: Diagonals of Quadrilaterals, Properties of Squares and Rhombuses

c) Rhombus
[Solution Description]
The properties described here are:
- Diagonals are perpendicular to each other.
- Diagonals bisect each other.
- Diagonals are not equal in length.
From the given conditions, the quadrilateral cannot be a square because squares have equal-length diagonals. It also cannot be a rectangle since rectangles do not necessarily have perpendicular diagonals. However, a rhombus satisfies all the given conditions: its diagonals are perpendicular bisectors but are not equal in length. Therefore, the quadrilateral must be a rhombus.

Key Concept: Quadrilateral Transformation

a) Kite

[Solution Description]

Extending one diagonal changes the property of equal diagonals. Now, the figure has diagonals that are perpendicular but not equal, which defines a kite.

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: Joining Triangles to Form Quadrilaterals

c) Rhombus
[Solution Description]
When two equilateral triangles are joined along one side, the resulting figure will have four sides. Each angle of an equilateral triangle is $60^\circ$. When two such triangles are joined, the adjacent angles add up to $120^\circ$. The opposite angles will also be $120^\circ$ each, making the quadrilateral a rhombus but not a square, as the angles are not $90^\circ$.

Key Concept: Properties of Quadrilaterals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a quadrilateral formed by two perpendicular diagonals of equal length is a rectangle. This is true because when diagonals are equal and intersect at $90^\circ$, they form a rectangle as all angles become right angles and opposite sides become equal. The reason states that diagonals of a rectangle are equal in length and bisect each other at $90^\circ$. This is also true as it is a property of rectangles. Further, the reason correctly explains the assertion because the properties mentioned in the reason lead to the conclusion in the assertion.

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: 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: 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: 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: 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 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: 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: Properties of a Kite

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A kite is defined as a quadrilateral with two distinct pairs of adjacent sides equal. One of its properties is that the diagonals intersect at right angles (perpendicular). Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Trapezium angles, Isosceles trapezium properties

c) 80$^\circ$
[Solution Description]
In an isosceles trapezium, the angles adjacent to each base are equal. Thus, $\angle P = \angle Q = 100^\circ$. Since the sum of angles on the same side of a transversal in a trapezium is $180^\circ$, we have $\angle P + \angle S = 180^\circ$. Substituting $\angle P = 100^\circ$, we get $\angle S = 80^\circ$.

Key Concept: Kite properties, Diagonal bisection

b) 60 cm$^2$
[Solution Description]
The diagonal $BD$ bisects $AC$ perpendicularly, so $AO = OC = 6$ cm. Since $\Delta AOB \cong \Delta COB$, we can calculate the area of one triangle and multiply by 2.
Area of $\Delta AOB = \frac{1}{2} \times AO \times BO$. To find $BO$, use the property that $BD = 10$ cm and $BD$ is split into $BO$ and $OD$. However, without additional information, we recognize that the area of the kite is $\frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 10 \times 12 = 60$ cm$^2$.

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, angle bisector

b) $60^\circ$
[Solution Description]
First, since $BD$ bisects $\angle ABC$ and $\angle ABD = 30^\circ$, then $\angle ABC = 60^\circ$ because the angle bisector splits $\angle ABC$ into two equal angles.
Now, in a kite, one pair of opposite angles are equal. Therefore, $\angle ABC = \angle ADC = 60^\circ$.

Key Concept: Relationship Between Kite and Rhombus

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

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: 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: Isosceles Trapezium: Property Verification

c) The non-parallel sides are equal and the base angles are equal.
[Solution Description]
In an isosceles trapezium, the non-parallel sides are equal in length, and the angles adjacent to each base are equal. This means that if $UVWX$ is an isosceles trapezium with $UV \parallel XW$ and $UX = VW$, then $\angle U = \angle V$.

Key Concept: Isosceles Trapezium Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
An isosceles trapezium has two parallel sides (UV$\parallel$XW) and two non-parallel sides (UX = VW). According to Property 2 of the syllabus, in an isosceles trapezium, angles opposite to equal sides are equal ($\angle$U = $\angle$V). This equality arises because the trapezium can be divided into congruent triangles (e.g., $\triangle UXY \cong \triangle VWZ$), ensuring the base angles ($\angle$U and $\angle$V) are equal. The Reason correctly states the defining property of an isosceles trapezium (equal non-parallel sides), which directly leads to the Assertion.

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: Properties of Kite - Diagonal Bisection

b) $40^\circ$
[Solution Description]
Given that diagonal BD bisects $\angle ABC$, this means it splits $\angle ABC$ into two equal angles. Since $\angle ABC = 80^\circ$, $\angle ABD$ will be half of $\angle ABC$.
Calculation:
$\angle ABD = \frac{\angle ABC}{2} = \frac{80^\circ}{2} = 40^\circ$
Thus, the measure of $\angle ABD$ is $40^\circ$.

Key Concept: Properties of Kite

c) Assertion is true, but Reason is false.
[Solution Description]
**Assertion Analysis:**
According to the properties of a kite, one diagonal ($BD$) bisects the angles at its vertices. So, in kite $ABCD$, $BD$ bisects $\angle ABC$ and $\angle ADC$. Thus, Assertion (A) is true.
**Reason Analysis:**
The Reason states that the diagonals of a kite always bisect each other at right angles. However, this is not accurate. Only one diagonal ($BD$) bisects the other ($AC$) at right angles; the reverse is not necessarily true. Therefore, Reason (R) is false.
**Conclusion:**
Since Assertion (A) is true but Reason (R) is false, the correct answer corresponds to option c).

Key Concept: Diagonal Properties of a Kite

b) $4\,cm$
[Solution Description]
According to the properties of a kite, the diagonal $BD$ bisects the diagonal $AC$ at right angles. Hence, $AO = OC$. Given $AO = 4\,cm$, it follows that $OC = 4\,cm$.