Key Concept: Properties of Rectangles

d) Diagonals intersect at $90^\circ$.
[Solution Description]
A rectangle has all angles equal to $90^\circ$, opposite sides equal and parallel, and diagonals that are equal and bisect each other. However, the diagonals do not necessarily bisect each other at $90^\circ$ in a rectangle (this is a property of a square).

Key Concept: Square Area, Rectangle Perimeter

b) 100 cm$^2$
[Solution Description]
1. First, find the perimeter of the rectangle: $P_{rect} = 2(l + w) = 2(12 + 8) = 40$ cm.
2. Since the perimeter of the square is equal to that of the rectangle, $P_{square} = 40$ cm.
3. For a square, the perimeter formula is $P = 4s$, where $s$ is the side length. Solving for $s$ gives $s = \frac{40}{4} = 10$ cm.
4. The area of the square is then $A = s^2 = 10^2 = 100$ cm$^2$.

Key Concept: Square properties

c) Square
[Solution Description]
A quadrilateral with all sides equal and one angle as 90° must have all angles as 90° (since opposite angles are equal and adjacent angles are supplementary). Hence, it is a square.

Key Concept: Rectangles, Squares, Properties

c) Both I and II
[Solution Description]
Statement I: A quadrilateral with all angles $90^\circ$ must have opposite sides equal and parallel, making it a rectangle. This statement is correct because these are the defining properties of a rectangle.
Statement II: A quadrilateral with diagonals that are equal and bisect each other at $90^\circ$ could be either a square or a rhombus (if sides are not equal). However, since the diagonals are equal, it must be a square (as rhombuses do not necessarily have equal diagonals unless they are squares). Therefore, this statement is also correct.
Hence, both statements I and II are correct.

Key Concept: Diagonals of a Square

c) The diagonals are not perpendicular to each other.
[Solution Description]
The properties of the diagonals of a square are: (1) They are equal in length, (2) They bisect each other at $90^{\circ}$, and (3) They bisect the angles of the square. The statement “The diagonals are not perpendicular to each other” is false because, in a square, the diagonals are always perpendicular.

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: Square properties

c) All its sides are equal.
[Solution Description]
A square has all the properties of a rectangle (right angles, equal opposite sides, and equal diagonals that bisect each other), but additionally, all its sides are equal, and its diagonals are perpendicular and bisect the angles.

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: 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: Diagonal Properties of Square and Rhombus, Conceptual Complexity

c) 32
[Solution Description]
Step 1: In a square, diagonals are equal and bisect each other at right angles. So, $PR = QS$ and $PT = TR = QT = TS = 4\sqrt{2}$. Thus, full diagonal $PR = 8\sqrt{2}$.
Step 2: Let side of square be s. By Pythagoras’ theorem in triangle PQR: $PR^2 = PQ^2 + QR^2$. But since all sides are equal, $PQ = QR = s$. So, $(8\sqrt{2})^2 = s^2 + s^2 = 2s^2$.
Step 3: Simplify: $128 = 2s^2$ → $s^2 = 64$ → $s = 8$.
Step 4: Perimeter = $4 \times 8 = 32$.

Key Concept: Diagonals of a Rhombus

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

Key Concept: Diagonals of 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: Assertion-Reason

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

Key Concept: 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: 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: 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 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 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: 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: Quadrilateral Properties, Congruence

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

Key Concept: Properties of Rectangle Diagonals

b) At the midpoint of both strips
[Solution Description]
The diagonals of a rectangle bisect each other at their midpoints. Therefore, the carpenter should join the strips at their midpoints.

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: Rectangle and Square Properties, Diagonal Congruence

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

Key Concept: Properties of Rectangle

c) Opposite sides must be equal
[Solution Description]
In a quadrilateral with all angles as 90\degree, drawing a diagonal divides it into two congruent right-angled triangles. Using properties of congruent triangles, we deduce that opposite sides must be equal in length. Thus, such a quadrilateral must have opposite sides equal.

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: 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: 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: 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: 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: Special Rectangle and Square Properties

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

Key Concept: 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: 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: 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: 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 Angle Sum, Algebraic Equation

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

Key Concept: Sum of angles in a quadrilateral

c) 360$^\circ$
[Solution Description]
The sum of all interior angles of any quadrilateral is always $360^\circ$, as per the angle sum property of quadrilaterals.

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 quadrilateral, properties of triangles

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

Key Concept: Sum of angles in a quadrilateral

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

Key Concept: Sum of angles in 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: 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: Sum of angles in a quadrilateral

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

Key Concept: 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: 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: Properties of Parallelograms

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

Key Concept: Adjacent Angles in Parallelogram

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

Key Concept: Diagonals of 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: 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: 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: Adjacent Angles in Parallelogram

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

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

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

Key Concept: Rectangles as a type of parallelogram

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

Key Concept: Diagonal Properties

c) Diagonals are equal and bisect each other.
[Solution Description]
The diagonals of a rectangle are equal in length and bisect each other. This is a key property that distinguishes rectangles from general parallelograms.

Key Concept: Diagonals, Angles

b) 35°
[Solution Description]
Step 1: Draw rectangle $PQRS$ with diagonals $PR$ and $QS$ intersecting at $O$.
Step 2: In rectangles, diagonals are equal and bisect each other. Thus, $OP = OR = OQ = OS$.
Step 3: Triangle $POS$ has sides $OP = OS$, making it an isosceles triangle.
Step 4: Given $\angle POS = 70°$, the remaining angles in $\triangle POS$ must sum to $110°$. Since $OP = OS$, $\angle PSO = \angle OPS = 55°$ each.
Step 5: Now, $\angle OSR = 90° – \angle PSO = 35°$ because $PSR$ is a right angle ($90°$) in the rectangle.

Key Concept: Venn Diagram: Relationship between squares, rectangles, and rhombuses

d) Nested circles: Square inside both Rectangle and Rhombus, which are inside Parallelogram.
[Solution Description]
A square is both a rectangle (all angles are $90^\circ$) and a rhombus (all sides equal). A rectangle has opposite sides equal and all angles $90^\circ$, while a rhombus has all sides equal but angles not necessarily $90^\circ$. Thus, the set of squares lies within both the sets of rectangles and rhombuses, which are subsets of parallelograms.

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: Special Cases of Parallelograms

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

Key Concept: Properties of Quadrilaterals, Venn Diagram

d) Square
[Solution Description]
A quadrilateral with diagonals that are equal in length is either a rectangle or a square. Additionally, if the diagonals bisect each other at $90^\circ$, it must be a square because rectangles have equal diagonals but they do not necessarily bisect each other at $90^\circ$. Only squares satisfy both conditions: equal diagonals and perpendicular bisectors.

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

a) 50°, 50°, 130°, 130°
[Solution Description]
In a rhombus, opposite angles are equal, and adjacent angles add up to 180°. Given one angle is 50°, its opposite angle is also 50°. The adjacent angle will be $180° – 50° = 130°$. Therefore, the angles are 50°, 130°, 50°, and 130°.

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: Angle Properties of Rhombus

c) $120^\circ$
[Solution Description]
Adjacent angles in a rhombus sum to $180^\circ$ (Property 3). Given $\angle A = 60^\circ$, then $\angle B = 180^\circ – 60^\circ = 120^\circ$.

Key Concept: Properties of Rhombus, Angle Relationships

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

Key Concept: Properties of 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: Rhombus Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that all sides of a rhombus are equal, which is true because one of the defining properties of a rhombus is that all its sides have the same length. The reason correctly explains this property by stating that a quadrilateral with all sides equal is defined as a rhombus. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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: Diagonal Properties

c) They intersect at $90^\circ$.
[Solution Description]
The diagonals of a rhombus bisect each other at right angles ($90^\circ$) and also bisect the angles of the rhombus. Thus, the correct statement is that they intersect at $90^\circ$.

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: 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: 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: 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: Construction of Parallelograms

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

Key Concept: 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: Properties of Quadrilaterals

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that a quadrilateral with diagonals intersecting at right angles and equal in length is a square. This is true because such properties are unique to squares among quadrilaterals. However, the reason claims that a quadrilateral with all sides equal and diagonals equal in length must be a rectangle. This is false because rhombuses also have all sides equal, but their diagonals are not necessarily equal unless they are squares.
Therefore, the assertion is true, but the reason is false.

Key Concept: Quadrilateral Transformation

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

Key Concept: Quadrilateral Properties, Angle Sum Property

c) $95^\circ$
[Solution Description]
The sum of the interior angles of a quadrilateral is always $360^\circ$. Given three angles as $90^\circ$, $90^\circ$, and $85^\circ$, we can find the fourth angle by subtracting the sum of these angles from $360^\circ$.
Calculation: $360^\circ – (90^\circ + 90^\circ + 85^\circ) = 95^\circ$.

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: 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: Construction of quadrilaterals using triangles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When two equilateral triangles are joined along a common side, the resulting quadrilateral will have all four sides equal because each side of the quadrilateral is either a side of the original equilateral triangle or the newly formed side from joining. Since an equilateral triangle has all three sides equal, this property ensures that all sides of the quadrilateral are equal. Moreover, the opposite angles of the quadrilateral will be equal due to the symmetry of the equilateral triangles. Therefore, the quadrilateral formed is a rhombus as it satisfies the conditions mentioned in the Reason. Thus, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Constructing quadrilaterals using isosceles triangles

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

Key Concept: Kite Properties

b) Kite
[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 has adjacent sides equal (8 cm each). This satisfies the definition of a kite, where two pairs of adjacent sides are equal. Thus, the quadrilateral formed is a kite.

Key Concept: Trapezium Angles

d) $110^\circ$
[Solution Description]
In an isosceles trapezium, the base angles on the same side are supplementary ($\angle U + \angle W = 180^\circ$). Given $\angle U = 70^\circ$, we can find $\angle W$ as follows:
$\angle W = 180^\circ – \angle U = 180^\circ – 70^\circ = 110^\circ$

Key Concept: Isosceles Trapezium Properties

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

Key Concept: Isosceles Trapezium

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

Key Concept: Kite Properties

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: Kite Properties

d) All of the above.
[Solution Description]
According to the properties of a kite, diagonal $BD$ bisects angles $\angle ABC$ and $\angle ADC$, bisects diagonal $AC$ at right angles, and makes $AO = OC$. Thus, the correct statement must reflect these properties.

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: Classification of Quadrilaterals

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

Key Concept: 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: Kite Properties

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

Key Concept: Trapezium, Isosceles Trapezium, Angle Properties

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Given an isosceles trapezium $UVWX$ with $UV \parallel XW$ and $UX = VW$.
Step 1: According to the property of a trapezium, since $UV \parallel XW$, the consecutive angles on the same side of the legs are supplementary. Therefore, $\angle U + \angle W = 180^\circ$ and $\angle V + \angle X = 180^\circ$. This makes the Assertion (A) true.
Step 2: For an isosceles trapezium, the non-parallel sides are equal by definition, i.e., $UX = VW$. This makes the Reason (R) true.
Step 3: However, the Reason (R) states a defining property of an isosceles trapezium but does not explain why $\angle U$ and $\angle W$ are supplementary. The supplementary nature of these angles arises from the parallel sides and the properties of a trapezium, not directly from the equality of the non-parallel sides.
Conclusion: Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation for Assertion (A).

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: Sum of angles in a quadrilateral

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

Key Concept: 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: 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 $OC = 4\,cm$.

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: Diagonals of an Isosceles Trapezium

d) Diagonals are equal in length
[Solution Description]
One of the key properties of an isosceles trapezium is that its diagonals are equal in length. Therefore, $UW = VX$.

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).