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: Venn Diagram Representation

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

Key Concept: Properties of Rectangle

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that in a rectangle, the diagonals are equal and bisect each other. This is true because it is one of the key properties of a rectangle as per the syllabus.
The reason claims that if the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral must be a rectangle. While this is true for parallelograms (since rectangles are a type of parallelogram), it is not necessarily true for all quadrilaterals. For example, an isosceles trapezoid has equal diagonals but is not a rectangle. Hence, the reason is false.
Therefore, Assertion is true, but Reason is false.

Key Concept: Properties of a Rectangle

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

Key Concept: Square vs Rectangle

a) A square has all sides equal, while a rectangle does not.
[Solution Description]
A square is a special type of rectangle where all sides are equal. While all squares are rectangles, not all rectangles are squares. Rectangles only require opposite sides to be equal and all angles to be $90^\circ$.

Key Concept: Properties of Squares

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the diagonals of a square bisect each other at $90^\circ$, which is true because the diagonals of a square are perpendicular bisectors. The reason states that the diagonals are equal in length and bisect the angles, which is also true as it's a property of squares. However, the reason does not correctly explain why the diagonals bisect each other at $90^\circ$ (it only mentions their equality and angle bisection). Thus, both statements are true, but the reason is not the correct explanation for the assertion.

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 of a Rectangle

b) 5 cm
[Solution Description]
The diagonals of a rectangle are equal in length and bisect each other. Given $PR = 10$ cm, which means each half-diagonal $PO$ and $OR$ will be half of $PR$.
$PO = \frac{PR}{2} = \frac{10}{2} = 5 \text{ cm}$
Therefore, the correct answer is 5 cm.

Key Concept: Diagonals of Rectangle and 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 rectangle always intersect at their midpoints. This is true because, in any rectangle, the diagonals are equal in length and they bisect each other at their midpoints. The reason states that a rectangle is a type of parallelogram where the diagonals bisect each other. This is also true as all rectangles are parallelograms with congruent diagonals that bisect each other. Moreover, the reason correctly explains why the assertion is true since it provides the underlying property (parallelogram diagonals bisecting each other) that leads to the assertion being valid.
Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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 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: Diagonals of a Rectangle

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

Key Concept: 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: Angles between diagonals of a rectangle

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

Key Concept: 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: Properties of diagonals in a square

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

Key Concept: Properties of Rectangles

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

Key Concept: Square Construction, Diagonals

c) $90°$
[Solution Description]
In a square, the diagonals are equal in length, bisect each other, and intersect at right angles (90°). Thus, to construct a square with a diagonal of 6 cm, the angle between the diagonals must be 90°.

Key Concept: Properties of Rectangle

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

Key Concept: Properties of Rectangles, 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: 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: Properties of Rectangles

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
From the syllabus, we know that if the diagonals of a quadrilateral are equal and bisect each other, then it must be a rectangle. So, the assertion (A) is true. However, the reason (R) states that the diagonals bisect at $90^\circ$, which is only true for squares, not all rectangles. Therefore, while both statements are individually correct, the reason does not correctly explain the 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: 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: Rectangle Construction

d) Connect sticks at midpoints and join ends with thread
[Solution Description]
To construct a right angle using two equal-length sticks and thread:
1) Join the sticks at their midpoints (5 cm from either end) forming an X-shape.
2) Use the thread to connect all four ends of the sticks sequentially - this forms a quadrilateral.
3) Since diagonals are equal and bisect each other, the quadrilateral is a rectangle (Property 4).
4) All angles in a rectangle are $90°$, so the angle formed by adjacent sides is a right angle.
Key points:
- Equal-length sticks act as diagonals.
- Bisecting them ensures rectangle formation.
- Rectangle guarantees right angles at vertices.

Key Concept: Quadrilateral Construction

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

Key Concept: Angle Sum Property of Quadrilateral

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

Key Concept: Square Properties

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

Key Concept: 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: Venn Diagram Representation

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

Key Concept: Square and Rectangle Properties

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

Key Concept: 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: Parallelogram angles

d) $120^\circ$
[Solution Description]
In a parallelogram, adjacent angles are supplementary, meaning they add up to $180^\circ$.
Given $\angle P = 60^\circ$, then $\angle Q = 180^\circ - 60^\circ = 120^\circ$.

Key Concept: Sum of angles in a quadrilateral

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

Key Concept: Assertion-Reason

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Let's verify both statements:
1. Assertion: In a parallelogram, adjacent angles are supplementary (sum to $180^\circ$) because they are consecutive interior angles formed by a transversal intersecting parallel sides. This is true.
2. Reason: Opposite angles in a parallelogram are equal due to its properties. This is also true.
However, the equality of opposite angles does not explain why adjacent angles are supplementary. They are separate properties derived from different geometric principles.

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, 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: Impossible quadrilateral configuration

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

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

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

Key Concept: Rectangle properties

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

Key Concept: 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: Angles in a Quadrilateral, Sum of Angles

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

Key Concept: 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: Adjacent Angles in Parallelogram

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

Key Concept: Diagonals of Parallelogram, Bisection Property

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

Key Concept: 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: 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: Opposite sides and diagonals of parallelogram

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

Key Concept: Parallelogram Properties

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

Key Concept: 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: Properties of Parallelogram, Diagonal Bisection

a) $\frac{20}{3}$
[Solution Description]
The diagonals of a parallelogram bisect each other. Therefore, in parallelogram $PQRS$, $PO = OR$.
Given:
$PO = 4x - 5$
$OR = x + 15$
Since $PO = OR$, we set them equal:
$4x - 5 = x + 15$
Subtract $x$ from both sides:
$3x - 5 = 15$
Add $5$ to both sides:
$3x = 20$
Divide by $3$ on both sides:
$x = \frac{20}{3}$
The value of $x$ is $\frac{20}{3}$.

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: 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: 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: 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: 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 Relationships

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

Key Concept: 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: Venn Diagram: Relationship between quadrilaterals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that every square is a rhombus. This is true because a square has all sides equal, which is the defining property of a rhombus. The Reason states that a square has all sides equal and all angles equal to $90^\circ$, satisfying the definition of a rhombus. This is also correct because having all sides equal is sufficient to be a rhombus, regardless of angle measures. Additionally, the Reason correctly explains why the Assertion is true. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Properties of Rhombus

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

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

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

Key Concept: Adjacent angles, Opposite angles

c) $130^\circ$
[Solution Description]
In a rhombus, adjacent angles add up to $180^\circ$. If one angle is $50^\circ$, then the adjacent angle will be $180^\circ - 50^\circ = 130^\circ$.

Key Concept: Types of Quadrilaterals

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

Key Concept: 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: Adjacent Angles of a Rhombus

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

Key Concept: Properties of Rhombus, 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: 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

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

Key Concept: 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: Side length, Diagonal relationship

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

Key Concept: Properties of Rhombus

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the diagonals of a rhombus bisect each other at $90^\circ$, which is Property 4 from the syllabus, so it is true. The reason states that a rhombus is a quadrilateral with all sides equal in length, which is the definition of a rhombus (Property 1), so it is also true. However, the property of diagonals bisecting at $90^\circ$ does not depend on all sides being equal but is a fundamental geometric property, so the Reason does not explain the Assertion. Thus, both are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Sum of Angles in a Quadrilateral

b) $100^{\circ}$
[Solution Description]
The sum of all interior angles of a quadrilateral is $360^{\circ}$. Given three angles:
$70^{\circ} + 110^{\circ} + 80^{\circ} = 260^{\circ}$
The fourth angle is calculated as:
$360^{\circ} - 260^{\circ} = 100^{\circ}$

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: 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: Parallelograms

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

Key Concept: 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: Diagonals of Quadrilaterals, Rectangle Formation

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

Key Concept: 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: Constructing quadrilaterals using equilateral triangles

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

Key Concept: Constructing quadrilaterals by joining triangles of equal sides

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

Key Concept: Trapezium Angles

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: 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: 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: 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: 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

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: 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: Classification of Quadrilaterals

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

Key Concept: Kite 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: 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: 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: 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: 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: 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 Trapezium - Angles

c) $110^\circ$
[Solution Description]
In a trapezium with one pair of sides parallel (PQ$\parallel$SR), the sum of consecutive angles on the same side of the legs is $180^\circ$. Therefore:
$\angle P + \angle S = 180^\circ$
Substituting the given value:
$70^\circ + \angle S = 180^\circ \implies \angle S = 110^\circ$
Hence, the measure of $\angle S$ is $110^\circ$.

Key Concept: Kite properties, Angle relationships

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

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 $115^\circ$.