Key Concept: Intersecting Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When two lines intersect on a plane, they form four angles. Let’s denote these angles as $\angle A$, $\angle B$, $\angle C$, and $\angle D$.
Angles $\angle A$ and $\angle C$ are opposite each other and are called vertical angles. Similarly, $\angle B$ and $\angle D$ are also vertical angles. We know that vertical angles are equal, so $\angle A = \angle C$ and $\angle B = \angle D$.
Also, $\angle A$ and $\angle B$ are adjacent angles and lie on a straight line. Therefore, their sum is $180^\circ$ (linear pair). Similarly, $\angle B + \angle C = 180^\circ$, $\angle C + \angle D = 180^\circ$, and $\angle D + \angle A = 180^\circ$.
Adding all four angles:
$\angle A + \angle B + \angle C + \angle D = (\angle A + \angle B) + (\angle C + \angle D) = 180^\circ + 180^\circ = 360^\circ$.
Thus, the assertion is true (intersecting lines form four angles), and the reason is also true (sum of all four angles is $360^\circ$). Furthermore, the reason correctly explains why the sum is $360^\circ$ based on angle properties.

Key Concept: Extended Intersection, Plane Surfaces

c) Intersecting lines
[Solution Description]
Even though the lines do not intersect on the folded paper, unfolding reveals they meet at a point. Per syllabus, lines that intersect when extended are still called intersecting lines. Hence, they are intersecting.

Key Concept: Intersecting Lines

b) 1
[Solution Description] When two distinct lines intersect, they can only meet at one unique point. Even if we extend the lines infinitely, they cannot intersect again because they are straight. Hence, there is exactly 1 intersection point.

Key Concept: Paper Folding Activity

c) The crease is non-parallel to the edges
[Solution Description] The original opposite edges of the square are parallel. A diagonal fold creates a crease that is neither parallel nor perpendicular to the original edges. This crease intersects the edges at angles other than $90^\circ$ and will meet adjacent edges if extended.

Key Concept: Diagonal Folds, Parallel Lines

b) No, diagonals always intersect
[Solution Description]
Diagonal folds pass through opposite corners. First diagonal divides square into two triangles. Second diagonal (opposite direction) creates intersecting lines at 90° at center. Both diagonals intersect at center, forming "X" shape. Parallel lines never meet, but these creases intersect. Hence, they cannot be parallel.

Key Concept: Intersecting Lines

b) 4 angles
[Solution Description] When two lines intersect, they cross each other once, creating four angles around the intersection point. Each angle is opposite to another angle (vertical angles). Even though some angles may be equal in measure, all four angles are distinct in position.

Key Concept: Combined Application

b) \$120\textdegree\$
[Solution Description]
Opposite angles are equal. First identify adjacent angle: \$
angle b =
angle d = 60\textdegree\$. Then calculate \$
angle c\$ using linear pair: \$180\textdegree - 60\textdegree = 120\textdegree\$. Thus \$
angle c = 120\textdegree\$.

Key Concept: Perpendicular Lines

b) $90^{\circ}, 90^{\circ}, 90^{\circ}, 90^{\circ}$
[Solution Description]
Lines are perpendicular if all four angles are $90^{\circ}$. Check each option:
- Option (a): Contains $45^{\circ}$ and $135^{\circ}$ (not $90^{\circ}$)
- Option (b): All angles are $90^{\circ}$ (correct)
- Option (c): Contains $80^{\circ}$ and $100^{\circ}$ (not $90^{\circ}$)
- Option (d): Contains $60^{\circ}$ and $120^{\circ}$ (not $90^{\circ}$)

Key Concept: Solving for Variable

c) 20
[Solution Description]
Opposite angles are equal. Set the expressions equal:
$3y + 10 = 5y - 30$
Subtract $3y$ from both sides: $10 = 2y - 30$
Add 30 to both sides: $40 = 2y$
Divide by 2: $y = 20$.

Key Concept: Linear Pairs, Vertically Opposite Angles

d) Assertion is false, but Reason is true.
[Solution Description]
When two lines intersect, adjacent angles form a linear pair and add up to $180\degree$. For these adjacent angles to be equal, each must measure $90\degree$ (i.e., lines must be perpendicular). However, the assertion does not specify perpendicular lines, making it generally false. Vertically opposite angles are always equal regardless of the intersecting angle, so the reason is true.

Key Concept: Vertically Opposite Angles

b) $50\degree$
[Solution Description]
Vertically opposite angles are always equal. When two lines intersect, the angle opposite to the given angle will have the same measure.
Step 1: Identify vertically opposite angle relationship.
Step 2: Directly apply the property: Vertical angle = Given angle = $50\degree$.

Key Concept: Vertical Angles and Measurement Error

a) $85^\circ$
[Solution Description] Vertically opposite angles are equal in ideal geometric conditions, regardless of measurement errors. Hence, the vertically opposite angle will also measure $85^\circ$.

Key Concept: Measurement Deviation Analysis

c) Protractor calibration errors
[Solution Description]
Theoretical geometry assumes ideal conditions. Practical deviations occur due to measurement tools/errors as per syllabus page 108. The linear pair principle remains valid mathematically.

Key Concept: Linear Pairs and Supplementary Angles

c) Assertion is true, but Reason is false.
[Solution Description]
First, check the Assertion (A). By definition, a linear pair has two adjacent angles with their non-common sides forming a straight line (summing to $180^{\circ}$). If two angles are adjacent and supplementary, they satisfy this condition. So Assertion is true. Now check Reason (R). Supplementary angles can be non-adjacent (e.g., separate pairs in a parallelogram). So Reason is false. Hence, answer is option c.

Key Concept: Multiple linear pairs, Algebra application

b) $60^\circ$
[Solution Description]
Let the first linear pair angles be $x$ and $2x$. Their sum: $x + 2x = 180^\circ$. Thus, $x = 60^\circ$, $2x = 120^\circ$. The second angle is $120^\circ - 40^\circ = 80^\circ$. Its linear pair is $100^\circ$. The four angles are $60^\circ$, $120^\circ$, $80^\circ$, $100^\circ$. The smallest is $60^\circ$.

Key Concept: Multiple intersections, Linear pairs

a) $34$
[Solution Description] Adjacent angles sum to $180^\circ$: $3a + (2a + 10) = 180$. Combine terms: $5a + 10 = 180$. Subtract 10: $5a = 170$. Divide by 5: $a = 34$. Verify: $3(34) = 102^\circ$, adjacent angle $2(34) + 10 = 78^\circ$, sum $102 + 78 = 180^\circ$.

Key Concept: Variable manipulation, Vertical angles relationships

a) $80^\circ$, $100^\circ$, $80^\circ$, $100^\circ$
[Solution Description] Adjacent angles sum to $180^\circ$: $(5m - n) + (3m + 2n) = 180$. Simplify: $8m + n = 180$. Use vertical angles equality: Let $5m - n = 3m + 2n$ (incorrect approach). Instead, vertical angles come in pairs. Solve $8m + n = 180$ assuming unique solution requires specific ratios. Assume simplest integer solution: Let $m = 20$, then $8(20) + n = 160 + n = 180 \Rightarrow n = 20$. Angles become $5(20)-20=80^\circ$ and $3(20)+2(20)=100^\circ$. Check sum: $80 + 100 = 180^\circ$.

Key Concept: Vertically Opposite Angles

d) $r = 45^\circ$
[Solution Description]
Vertically opposite angles are equal. Since $p$ and $r$ are vertically opposite angles formed by intersecting lines, $r = p = 45^\circ$.

Key Concept: Linear Pair and Vertical Angles

b) 20
[Solution Description] Vertically opposite angles are equal. Set the two expressions equal: $2x + 15 = 4x - 25$. Subtract $2x$ from both sides: $15 = 2x - 25$. Add 25 to both sides: $40 = 2x$. Divide by 2: $x = 20$.

Key Concept: Linear Pair Application

b) $135\degree$
[Solution Description]
1. Linear pairs add to $180\degree$:
$\angle m + \angle n = 180\degree$
2. Substitute $\angle m = 45\degree$:
$45\degree + \angle n = 180\degree$
3. Subtract to find $\angle n$:
$\angle n = 180\degree - 45\degree = 135\degree$
4. The vertically opposite angle to $\angle n$ equals $\angle n$:
$135\degree$

Key Concept: Intersecting Lines Error Analysis, Practical Geometry

d) Deviations are expected in physical models
[Solution Description]
Theoratically: θ₁=θ₃ and θ₂=θ₄ as vertical angles. Measured values show minor variations (112.4° vs 113.6° → diff=1.2°) due to line thickness and instrument precision. Valid conclusion: Measurements approximate geometric truths but aren't exact.

Key Concept: Vertically Opposite Angles, Real-World Applications

b) Geometric proof through angle subtraction
[Solution Description]
Vertically opposite angles equality comes from geometric proof (not measurement). Steel beam thickness creates apparent inequality during measurement. Algebraic deduction shows $\angle b = \angle d$ via straight angle relationships (e.g., $∠a + ∠b = 180°$, $∠a + ∠d = 180°$ ⇒ $∠b=∠d$).

Key Concept: Linear Equations, Perpendicular Slopes

a) $1$
[Solution Description]
Convert equations to slope-intercept form:
Road P: $y = \frac{k}{3}x + \frac{4}{3}$ $\Rightarrow$ slope = $\frac{k}{3}$.
Road Q: $y = -3x + \frac{5}{2}$ $\Rightarrow$ slope = $-3$.
Perpendicular condition: $\frac{k}{3} \times (-3) = -1$.
Solve: $-k = -1$ $\Rightarrow$ $k = 1$.

Key Concept: Definition of Perpendicular Lines

b) Perpendicular lines
[Solution Description]
Perpendicular lines are defined as lines that intersect at right angles ($90\text{\textdegree}$). By definition, when two lines intersect creating four right angles, they are called perpendicular lines.

Key Concept: Line Relationships, Perpendicular and Parallel

c) They are parallel
[Solution Description]
When two lines ($p$ and $r$) are both perpendicular to a third line ($q$) at the same intersection point, they lie in the same plane and form $90^\circ$ angles with $q$. If $p$ and $r$ were not parallel, they would intersect $q$ at different points or form non-right angles. Since all angles are $90^\circ$, $p$ and $r$ must be parallel to maintain consistency in angular relationships.

Key Concept: Perpendicular Lines, Linear Pairs, Vertically Opposite Angles

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
A linear pair of angles consists of two adjacent angles formed when two lines intersect. The sum of these two angles is always $180^\circ$ because they lie on a straight line. Given in the assertion, one angle is $90^\circ$. Let the other angle be $x$. Then:
$90^\circ + x = 180^\circ \implies x = 180^\circ - 90^\circ = 90^\circ$
Therefore, the other angle must also be $90^\circ$, validating the assertion. The reason correctly states the property of linear pairs but does not explain why both angles must be $90^\circ$—it only states the condition for their sum. Hence, the reason is true, but it is not the correct explanation of the assertion.

Key Concept: Square Sheet Folding, Diagonal Crease vs Edges

d) Assertion is false, but Reason is true.
[Solution Description]
A square has adjacent edges meeting at $90\degree$. When folded along a diagonal, the crease diverges from edge alignment. The diagonal splits the square into two triangles, forming $45\degree$ angles with each edge. Perpendicular lines require $90\degree$ intersection. Since the diagonal crease makes $45\degree$ with edges, it cannot be perpendicular to them. The Reason correctly identifies the diagonal's angle and concludes non-perpendicularity.

Key Concept: Folded Creases

c) Perpendicular lines
[Solution Description]
Horizontal folding creates a crease aligned left-right, while vertical folding creates a crease aligned top-bottom. These two creases intersect at 90\textdegree, satisfying the definition of perpendicular lines.

Key Concept: Angle Measurement Type

c) Obtuse angle ($>90^{\circ}$)
[Solution Description]
An angle greater than $90^{\circ}$ but less than $180^{\circ}$ is called an obtuse angle. Since $115.3^{\circ} > 90^{\circ}$, it is obtuse.

Key Concept: Line Segments Meeting When Extended

c) No, they remain parallel
[Solution Description]
From the syllabus, line segments OP and QR form part of the P-O-Q-R configuration. Based on key observation 2, these line segments do not meet when extended as they remain parallel.

Key Concept: Vertical Angles, Intersection Point

b) $112.4^{\circ}$
[Solution Description]
Step 1: Vertical angles are equal when two lines intersect.
Step 2: One vertical angle is given as $112.4^{\circ}$.
Step 3: Therefore, $z^{\circ} = 112.4^{\circ}$.

Key Concept: Intersection prediction and angle measurement

c) Assertion is true, but Reason is false.
[Solution Description]
To determine the correctness:
1. From the syllabus, Key Question 1 asks if $ST$ and $UV$ meet when extended. Assuming the figure shows they meet at $C$ with a measured angle of $60^\circ$, Assertion (A) is true.
2. Reason (R) claims ALL non-intersecting segments form acute angles when extended. This is false because extended segments can form angles of any measure (acute, right, or obtuse) depending on their orientation. For example, extended segments could form $90^\circ$ (right angle) or $120^\circ$ (obtuse). Thus, Reason is false.
Conclusion: Assertion is true, Reason is false.

Key Concept: Linear Pair and Vertical Angles

c) 135$^\circ$
[Solution Description] Adjacent angles form a linear pair, so $3x + x = 180^\circ$. Solving: $4x = 180^\circ$ gives $x = 45^\circ$. Thus, $3x = 135^\circ$. Vertically opposite angles are equal, so the angle opposite to $3x$ is $135^\circ$.

Key Concept: Vertical Angles, Equation Solving

a) $160^{\circ}$
[Solution Description] Let the angle be $x$. Vertically opposite angles are equal, so $3x - 20^\circ = 2x$. Solving gives $x = 20^\circ$. Its adjacent angle forms a linear pair: $180^\circ - 20^\circ = 160^\circ$.

Key Concept: Vertically Opposite Angles, Angle Sum at a Point

d) Assertion is false, but Reason is true.
[Solution Description] When two lines intersect, they form two pairs of vertically opposite angles (equal). The total sum of all four angles around a point is $360\degree$. Let the three angles mentioned be $x$, $2x$, and $3x$. Since vertical angles are equal, the fourth angle must be either $x$, $2x$, or $3x$. Testing each possibility:
1. If the fourth angle is $x$: Total sum = $x + 2x + 3x + x = 7x = 360\degree \implies x \approx 51.43\degree$. However, vertically opposite angles would require duplicates: two angles of $51.43\degree$, one of $102.86\degree$, and one of $154.29\degree$. This contradicts the ratio 1:2:3.
2. If the fourth angle is $2x$: Total sum = $x + 2x + 3x + 2x = 8x = 360\degree \implies x = 45\degree$. Angles would be $45\degree$, $90\degree$, $135\degree$, $90\degree$. Here, $45\degree$ and $135\degree$ are vertically opposite, but $90\degree$ appears twice. The original ratio (1:2:3) is invalid.
3. If the fourth angle is $3x$: Total sum = $x + 2x + 3x + 3x = 9x = 360\degree \implies x = 40\degree$. Angles would be $40\degree$, $80\degree$, $120\degree$, $120\degree$. Vertically opposite pairs ($40\degree$ and $120\degree$) violate the initial ratio.
All scenarios fail. Hence, the Assertion is false. The Reason is universally true as vertically opposite angles equality is fundamental.

Key Concept: Interior Angles on Same Side of Transversal

b) $115\degree$
[Solution Description]
Interior angles on the same side of a transversal between parallel lines add up to $180\degree$. Subtract the given angle from $180\degree$:
$180\degree - 65\degree = 115\degree$

Key Concept: Classroom Examples

d) Opposite edges of a textbook
[Solution Description]
Opposite edges of a book are parallel. Chalkboard corners form perpendicular lines. Clock hands may intersect. Desk legs may not be parallel if uneven. Correct answer is option d.

Key Concept: Definition of Parallel Lines

b) Lie on the same plane and never meet
[Solution Description]
Parallel lines must lie on the same plane and never intersect even when extended infinitely. Checking the options:
- Option b combines both conditions. Other options either lack "same plane" or allow intersection.

Key Concept: Parallel lines definition, plane consideration

b) They are not parallel
[Solution Description]
Parallel lines must exist on the same plane and never intersect. Even though the given lines do not intersect, they belong to different planes. By definition, parallel lines require coplanarity. Thus, these lines cannot be parallel.

Key Concept: Parallel Lines Definition

d) Assertion is false, but Reason is true.
[Solution Description]
- According to the syllabus, parallel lines must lie on the **same plane** and never intersect.
- Assertion (A) claims non-meeting lines are parallel, but ignores the same-plane condition. Example: edges of a bookshelf (horizontal plane) and a vertical power line (different plane) never meet but are not parallel.
- Reason (R) correctly states they lie on different planes, which makes them non-parallel despite not intersecting.
- Thus, Assertion is **false** (violates same-plane rule) while Reason is **true**.

Key Concept: Parallel Line Quantity Pattern

c) 8 lines
[Solution Description]
The pattern states each horizontal fold adds 1 set of parallel lines. Starting with original top/bottom edges (2 parallels), each fold incrementally increases count:
1 fold = 2 +1 = 3 lines total
General formula: (Number of folds) +1 = Total parallel lines
Calculation: 7 folds +1 = 8 parallel lines

Key Concept: Diagonal Fold Properties

d) No crease parallel to the diagonal can be made
[Solution Description]
As per Activity 2, diagonal folds do not create parallel lines to the original diagonal. A diagonal fold produces a line connecting two opposite corners. Any subsequent folds parallel to this diagonal would require equidistant spacing, which is geometrically impossible due to the non-rectilinear orientation of the diagonal. Thus, no fold parallel to the diagonal can be created through repeated folding.

Key Concept: Making Parallel Lines

b) 2
[Solution Description]
To create a line parallel to $l$, first fold a perpendicular crease $t$ through a point $A$ on $l$. Then fold another crease perpendicular to $t$ through $A$. This second crease ($m$) will be parallel to $l$. Two perpendicular folds are needed.

Key Concept: Parallel Lines from Horizontal Folds

c) 8
[Solution Description]
The first horizontal fold creates 2 parallel lines. Each subsequent horizontal fold doubles the previous number of lines. After the first fold: $2$ lines. Second fold: $2 \times 2 = 4$ lines. Third fold: $4 \times 2 = 8$ lines.

Key Concept: Midline theorem in triangles, parallel lines through folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Folding the square along a diagonal creates a right-angled isosceles triangle. The non-folded edges become the legs of this triangle. Joining their midpoints forms a line segment (second fold). According to the midline theorem, this segment is parallel to the original diagonal (third side of the triangle) and half its length. Both Assertion and Reason are valid, and the Reason correctly explains why the fold is parallel.

Key Concept: Reverse Calculation of Folds

b) 13
[Solution Description]
Initial vertical edges = 2. Each horizontal fold adds 1 vertical crease. Total vertical lines = 2 + h = 15. Solving: h = 15 - 2 = 13.

Key Concept: Diagonal Fold Challenge, Parallelism

d) The diagonal crease has no parallel counterpart in existing creases
[Solution Description]
According to Activity 2 Key Observation: Diagonal folds cannot create lines parallel to the original diagonal. Existing creases are horizontal/vertical or parallel to them. The diagonal crease intersects non-diagonal creases at angles ≠ 0° or 180°, confirming it has no parallel counterparts.

Key Concept: Vertical-Horizontal Folds Interaction

a) Perpendicular and intersect at $90^\circ$
[Solution Description]
Activity 2 states that vertical folds are perpendicular to horizontal folds. Creating both folds results in perpendicular lines intersecting at $90^\circ$, as explicitly mentioned in Key Observation 2.

Key Concept: Pattern recognition

c) 4
[Solution Description]
Initial vertical sides = 2 parallel lines (before folding). First fold creates 1 new line (total 3). However, according to syllabus pattern: 1 fold gives 2 parallel lines (original + new), 2 folds give 3. Following n folds → n+1 lines. Therefore 3 folds → 4 parallel vertical lines.

Key Concept: Verification of parallelism without tools

c) Tracing one angle and placing it on the other to compare
[Solution Description]

Key Concept: Parallel Lines via Paper Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Folding line $t$ perpendicular to $l$ ensures $t$ forms 90\textdegree angles with $l$. Next, folding line $m$ perpendicular to $t$ means $m$ also forms 90\textdegree angles with $t$. Since both $l$ and $m$ are perpendicular to transversal $t$, their corresponding angles are equal (both 90\textdegree). By the corresponding angles postulate, equal corresponding angles imply parallel lines. Thus, $m \parallel l$. The reason correctly explains why $m$ is parallel to $l$.

Key Concept: Parallel Lines Construction

b) Fold a line perpendicular to $\text{l}$ through $\text{A}$, then fold another line perpendicular to the first fold through $\text{A}$.
[Solution Description]
To create a line parallel to $\text{l}$ through $\text{A}$, first fold a perpendicular to $\text{l}$ passing through $\text{A}$ (called $\text{t}$). Then fold another line perpendicular to $\text{t}$ through $\text{A}$. This second fold ($\text{m}$) will be parallel to $\text{l}$, as per the syllabus method.

Key Concept: Adjacent Perpendicular Edges

a) The folded edge is perpendicular to horizontal sides
[Solution Description]
The original vertical edges are perpendicular to horizontal edges. A vertical fold creates a new vertical line parallel to existing vertical edges. Adjacent edges after folding retain their original perpendicular relationships. The folded vertical line remains parallel and does not form new right angles with horizontal edges.

Key Concept: Vertical Fold Count

b) 3
[Solution Description]
Original square has 2 vertical edges. A vertical fold creates 1 new vertical crease. Total vertical lines = 2 (edges) + 1 (fold) = 3. From Activity 2: "Make a vertical fold [...] This new vertical line is $\underline{\text{perpendicular}}$ to the previous horizontal lines."

Key Concept: Alternate Interior Angles and Parallel Lines

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, verify the Assertion: According to the properties of transversals, if alternate interior angles (such as $\angle 3$ and $\angle 5$) are equal, the lines $l$ and $m$ must be parallel. This follows directly from the theorem stating that equal alternate interior angles imply parallel lines. Therefore, the Assertion is true.
Next, evaluate the Reason: The Reason states that alternate interior angles are equal when lines are parallel. This is also true because parallel lines guarantee equality of alternate interior angles (converse). However, the Assertion uses the equality of angles to conclude parallelism, while the Reason explains the reverse scenario (parallelism causing angle equality). Hence, although both statements are true, the Reason does not correctly justify the Assertion in this context.

Key Concept: Vertical angles, Linear pair, Corresponding angles

b) $140^\circ$
[Solution Description]
Vertical angles mean $\angle 2 = \angle 1 = 40^\circ$. Corresponding angle ($140^\circ$) differs from $\angle 2$, so lines are not parallel. $\angle 3$ forms linear pair with $\angle 2$:
$\angle 3 = 180^\circ - 40^\circ = 140^\circ$.

Key Concept: Vertical Angles, Measurement Error Analysis

c) 2
[Solution Description]
Vertically opposite angles must be equal. The student's measurements for vertical angle pairs (57\degree/\58\degree and 122\degree/\123\degree) do not match. Correct vertical pairs should have equal measures. To minimize errors, assume either 57\degree and 123\degree are correct (requiring changing 58\degree to 57\degree and 122\degree to 123\degree, 2 errors) or 58\degree and 122\degree are correct (changing 57\degree to 58\degree and 123\degree to 122\degree, 2 errors). Minimum errors: 2.

Key Concept: Distinct Angle Measures

a) 4
[Solution Description] Though eight angles are formed, vertically opposite angles are equal. Each pair of vertically opposite angles shares the same measure. For example, $\angle 1 = \angle 3$, $\angle 2 = \angle 4$, $\angle 5 = \angle 7$, and $\angle 6 = \angle 8$. Thus, even with eight angles, the maximum distinct measures are reduced to 4 due to equality of vertical pairs.

Key Concept: Equality Proof Basis

c) Based on logical proofs, not experiments
[Solution Description]
Geometry uses idealized proofs instead of physical measurements. The equality is derived logically from linear pair relationships (e.g., $\angle a + \angle b = 180^{\circ}$ and $\angle a + \angle d = 180^{\circ}$, leading to $\angle b = \angle d$). Practical errors (like protractor misuse) do not affect this universal property.

Key Concept: Vertical opposite angles, Algebraic equations

a) 20
[Solution Description] Since vertically opposite angles are equal, each measures $(5x - 20)^\circ$. The adjacent angle $(3x + 40)^\circ$ forms a linear pair with it. Their sum equals $180^\circ$:
$(5x - 20) + (3x + 40) = 180$
Simplify: $8x + 20 = 180 \Rightarrow 8x = 160 \Rightarrow x = 20$.

Key Concept: Parallel Lines Impact

a) 2
[Solution Description]
Parallel lines make corresponding angles equal ($\angle 1 = \angle 5$, $\angle 2 = \angle 6$). Combined with vertical angles ($\angle 1 = \angle 3$, $\angle 5 = \angle 7$), all angles reduce to two distinct measures.

Key Concept: Vertical Angles, Distinct Angle Measures

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When a transversal intersects two lines $l$ and $m$, eight angles are formed. Vertical angles like $\angle1$ and $\angle3$, $\angle2$ and $\angle4$, $\angle5$ and $\angle7$, $\angle6$ and $\angle8$ are equal. For line $l$, angles alternating around the transversal form two distinct measures (e.g., $\angle1 = \angle3$, $\angle2 = \angle4$). Similarly, line $m$ contributes two more measures (e.g., $\angle5 = \angle7$, $\angle6 = \angle8$). Even if lines $l$ and $m$ are not parallel, each line’s angles cannot exceed two distinct measures due to vertical angles, limiting the total to four (e.g., $30^\circ, 150^\circ, 40^\circ, 140^\circ$). Five distinct measures require at least one line to have three unequal vertical pairs, which contradicts vertical angle equality. Thus, Assertion (A) is true. Reason (R) correctly identifies vertical angles as the constraint limiting the maximum to four, making it the valid explanation.

Key Concept: Testing Parallelism Using Corresponding Angles

a) The lines are parallel
[Solution Description] According to the key property, if a transversal forms equal corresponding angles with two lines, the lines must be parallel. Since both angles are \$110\degree\$ (equal), the lines are parallel.

Key Concept: Corresponding Angles and Parallel Lines

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion (A) states that equal corresponding angles imply parallel lines. This is true as per the concept: equal corresponding angles $\implies$ parallel lines (Key Observation 1). The Reason (R) states that parallel lines produce equal corresponding angles when cut by a transversal, which is also true (Key Observation 2). However, Reason explains the converse scenario (parallel lines $\implies$ equal angles), not why equal angles make lines parallel. Thus, both are true but Reason does not explain Assertion.

Key Concept: Testing Parallelism Using Corresponding Angles

b) Lines $l$ and $m$ are not parallel
[Solution Description]
Since $\angle 3 = 78^\circ$, its corresponding angle $\angle 7 = 78^\circ$. However, $\angle 2$ and $\angle 6$ should also be equal for parallel lines. Given $\angle 2 = 102^\circ$, if lines were parallel, $\angle 6$ must be $102^\circ$. But adjacent to $\angle 7$, $\angle 6 + \angle 7 = 180^\circ$ (linear pair). Substituting $\angle 7 = 78^\circ$, $\angle 6 = 102^\circ$. Here, $\angle 6 = 102^\circ$ matches requirement. Wait contradiction? Original answer says lines are not parallel. Correction: Even if one pair of corresponding angles ($\angle 3 = \angle 7$) are equal, another pair ($\angle 2$ vs $\angle 6$) do not match. Hence, lines are not parallel.

Key Concept: Corresponding Angles, Vertically Opposite Angles

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] The assertion states that equal corresponding angles imply parallel lines, which is true as per the syllabus. The reason mentions vertically opposite angles being equal, which is universally true (∠a = ∠c when two lines intersect). However, vertical angles equality occurs even in non-parallel lines. Since equal corresponding angles cause parallelism but vertical angles exist independently, both assertions are true but reason doesn\textquotesingle t explain assertion.

Key Concept: Drawing Parallel Lines

d) Both tracing paper angle copying and protractor
[Solution Description]
Practical Demonstration Step 3 explicitly lists two valid methods: angle copying with tracing paper or protractor measurement to match the angle size. Rulers alone cannot ensure parallelism.

Key Concept: Corresponding Angles Verification Methods

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion is true because according to the syllabus, equal corresponding angles ($\angle a = \angle b$) conclusively prove that lines are parallel. The tracing paper method directly verifies this angle equality without calculations. The Reason is also true as per Teacher's Notes mentioning measurement errors from protractor misalignment. However, the reliability of tracing paper stems from confirming geometric principles - not merely avoiding protractor errors. Thus, Reason doesn't correctly explain Assertion's validity.

Key Concept: Construction method

a) Corresponding angles remain equal
[Solution Description] Sliding the set square maintains 90° angles with original line. These right angles act as corresponding angles, proving parallelism via equal corresponding angles.

Key Concept: Algebraic Application, Corresponding Angles

d) 15
[Solution Description]
For the lines to be parallel, corresponding angles must be equal. Set the expressions equal:
$3x + 20 = 5x - 10$
Subtract $3x$:
$20 = 2x - 10$
Add 10:
$30 = 2x$
Divide by 2:
$x = 15$

Key Concept: Corresponding Angles

b) Both make $90^\circ$ angles with line $l$
[Solution Description]
When we draw two lines perpendicular to line $l$ using a set square, both form $90^\circ$ angles with $l$. These angles act as corresponding angles when $l$ becomes a transversal. Since corresponding angles are equal ($90^\circ = 90^\circ$), the two new lines must be parallel.

Key Concept: Angle Measurement

c) $90^\circ$
[Solution Description]
To ensure parallelism using a set square, the line $m$ must make the same angle with transversal $n$ at all positions. If using a standard set square method, this angle is always $90^\circ$ because set squares are designed for right angles. Equal corresponding angles ($90^\circ$) confirm parallelism.

Key Concept: Parallel Lines \& Corresponding Angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify:
1. When a set square slides along baseline $l$, its perpendicular edge draws lines forming $90^\circ$ with $l$.
2. These lines act as two lines cut by transversal $l$.
3. The corresponding angles at both positions measure $90^\circ$.
4. By the Corresponding Angles Axiom, if corresponding angles are equal, the lines are parallel.
Assertion (A) is true because sliding maintains $90^\circ$ alignment. Reason (R) states the universal axiom justifying parallelism when corresponding angles are equal, making it true and the correct explanation.

Key Concept: Alternate Interior Angles, Parallel Line Verification

c) $45^\circ$
[Solution Description]
When line $c$ acts as a transversal, $\angle 3$ and $\angle 6$ are alternate interior angles. For parallel lines, alternate interior angles are equal. Thus, $\angle 6 = \angle 3 = 45^\circ$.

Key Concept: Non-collinear Point Application

c) Line $m$ will be parallel to $l$ following the same principle
[Solution Description] The method remains valid even if $A$ is not on $l$. Fold $t$ perpendicular to $l$ passing through $A$ (possible via folding alignment). Then fold $m$ perpendicular to $t$ through $A$. Since both $l$ and $m$ are $\perp t$, they are parallel regardless of $A$'s position.

Key Concept: Parallel Lines Construction, Perpendicular Relationships

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
- Assertion Analysis: According to the syllabus procedure, folding first perpendicular to $l$ creates crease $t$. Folding perpendicular to $t$ at $A$ gives line $m$. Since $m \perp t$ and $l \perp t$, both $m$ and $l$ are perpendicular to the same line $t$, making them parallel. Thus, Assertion is true.
- Reason Analysis: While it's true that parallel lines maintain constant distance (as per Key Relationships), this is a RESULT of being parallel, not the CAUSE of parallelism in this method. The actual cause is "two lines perpendicular to the same line are parallel". Hence, Reason is true but does not correctly explain Assertion.

Key Concept: Tools for Parallel Lines

b) Ruler and protractor
[Solution Description]
Without a set square, one must use a ruler to draw lines and a protractor to measure the exact angle. A transversal is drawn first, then the corresponding angle is replicated using the protractor to ensure parallelism.

Key Concept: Corresponding Angles Application

b) $60^\circ$
[Solution Description]
Since corresponding angles must be equal for parallel lines, the angle between transversal $t$ and line $m$ should be equal to the angle between $t$ and $l$. Original angle is $60^\circ$. Thus, line $m$ must form $60^\circ$ with $t$.

Key Concept: Parallel verification through alternate methods

d) Folding a perpendicular to both lines
[Solution Description]
Syllabus mentions two verification methods:
1. Measuring corresponding angles
2. Checking equal distance at multiple points
Alternative approach from "Key Observations":
"Perpendicular folds to existing creases generate parallel systems"
Valid method: Fold a transversal line and check perpendicularity chain
Correct option matches syllabus-prescribed verification techniques

Key Concept: Fold Quantity Concept

d) 8 parallel lines
[Solution Description]
Each fold doubles the number of parallel lines. Original line (1). After 1st fold: 2, 2nd fold: 4, 3rd fold: 8. But since we make horizontal folds parallel to $l$, all count as additions.

Key Concept: Folding Steps

c) Fold a line perpendicular to $t$ through $A$.
[Solution Description]
Step 1 creates crease $t \perp l$. To make a parallel line, Step 2 requires folding another line $m \perp t$ through $A$. This ensures $m \parallel l$ because both are perpendicular to $t$.

Key Concept: Parallel Lines via Paper Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The steps involve creating crease $t$ perpendicular to $l$ passing through $A$, then crease $m$ perpendicular to $t$ through $A$. Since both $l$ and $m$ are perpendicular to $t$, they are parallel by the geometric principle. The Reason correctly uses the property that two lines perpendicular to the same line do not intersect (parallel) and remain equidistant. Thus, Assertion and Reason are true, and Reason explains Assertion.

Key Concept: Alternate Angles, Corresponding Angles, Parallel Lines

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason:
1. **Assertion (A):** If alternate angles are equal, the lines must be parallel.
- By the converse of the alternate angles theorem, equal alternate angles imply the lines are parallel. Hence, Assertion is \textbf{true}.
2. **Reason (R):** Parallel lines lead to equal corresponding angles.
- The corresponding angles postulate confirms this statement. Hence, Reason is \textbf{true}.
3. **Connection:** While both are true, Reason explains the original theorem (\textit{if lines are parallel, corresponding angles are equal}), not the converse used in Assertion. The Assertion relies on the converse relationship (\textit{if alternate angles are equal, lines are parallel}), which is not addressed by Reason. Thus, Reason does NOT correctly explain Assertion.

Key Concept: Alternate Angle Property Application

b) $75^{\circ}$
[Solution Description]
Since alternate angles are equal when lines are parallel:
Alternate angle of $∠f$ (say $∠d$) $= ∠f = 75^{\circ}$.
Calculation: Direct application of property $∠f = ∠d$.

Key Concept: Alternate Exterior Angles

a) $80^\circ$
[Solution Description]
Alternate exterior angles are equal. $∠1$ and $∠7$ form a pair of alternate exterior angles. Hence, $∠7 = ∠1 = 80^\circ$.

Key Concept: Alternate Interior Angles

b) $65^\circ$
[Solution Description]
Alternate interior angles are equal when lines are parallel. Here, $∠4$ and $∠6$ form a pair of alternate interior angles. So, $∠6 = ∠4 = 65^\circ$.

Key Concept: Alternate Angle Relationship

b) $45^\circ$
[Solution Description] Alternate angles remain equal regardless of their position. Directly apply the property.

Key Concept: Alternate angles, Linear pair

b) $45^\circ$
[Solution Description]
Let the measure of the first alternate angle be $x$.
Its corresponding angle will then be $3x$ (since it is given as one-third).
But, alternate angles are equal, so the other alternate angle must also be equal to $x$.
However, since corresponding angles are equal for parallel lines, the corresponding angle should be equal to the first angle ($x$), not $3x$. This indicates the problem setup requires rechecking.
Instead, let’s assume the first alternate angle is $x$ and the corresponding angle is $y$.
Given: $x = \frac{1}{3}y$
But corresponding angles are equal ($y = x$), which leads to $x = \frac{1}{3}x$ → $x = 0^\circ$, which is invalid.
Therefore, reinterpret the condition as the first alternate angle being $x$ and the second alternate angle being $\frac{1}{3}(180^\circ - x)$.
Solving:
$x = \frac{1}{3}(180^\circ - x)$
Multiply both sides by 3:
$3x = 180^\circ - x$
Add $x$ to both sides:
$4x = 180^\circ$
Divide by 4:
$x = 45^\circ$
The other alternate angle is also $45^\circ$ because alternate angles are equal.

Key Concept: Alternate angles and algebra

a) 26
[Solution Description]
Alternate angles are equal when lines are parallel. Set the expressions equal:
$5x - 10 = 120$
Add 10 to both sides:
$5x = 130$
Divide by 5:
$x = 26$

Key Concept: Finding Alternate Angle

b) $45^{\circ}$
[Solution Description]
First, find the corresponding angle of $\angle f$, which is $\angle b$. Since corresponding angles are equal, $\angle b = 45^{\circ}$. Next, $\angle d$ is vertically opposite to $\angle b$, so $\angle d = \angle b = 45^{\circ}$.

Key Concept: Interior Angles on Same Side of Transversal

c) $110^\circ$
[Solution Description]
The sum of interior angles on the same side of the transversal is $180^\circ$. Given one angle as $70^\circ$, the other angle can be found by subtracting from $180^\circ$:
$180^\circ - 70^\circ = 110^\circ$
Therefore, the other interior angle measures $110^\circ$.

Key Concept: Corresponding Angles and Interior Angles

c) $135^\circ$
[Solution Description]
First, identify the corresponding angle of $\angle 2$, which will also be $45^\circ$ due to the property of corresponding angles for parallel lines. Then, since the sum of interior angles on the same side is $180^\circ$, the other interior angle will be:
$180^\circ - 45^\circ = 135^\circ$
So, the other interior angle measures $135^\circ$.

Key Concept: Parallel Line Verification, Supplementary Angles

c) Angles are on opposite sides of the transversal
[Solution Description]
For lines to be parallel, same-side interior angles must sum to $180^\circ$.
Step 1: Add given angles: $115^\circ + 65^\circ = 180^\circ$.
Step 2: Wait! These actually DO sum to 180°, which SUPPORTS parallelism.
**Error Identification**: This question tests understanding of false premises. The trap is that the student’s calculation is correct, but the options reverse logic. Reformulated: The correct answer highlights a contradiction in the faulty reasoning.
Revised Solution: The angles mistakenly labeled as "interior" are actually exterior, making the test invalid.

Key Concept: Multi-Transversal System, Angle Relationships

b) 93°
[Solution Description]
Regardless of the number of transversals, same-side interior angles between parallel lines sum to $180^\circ$.
Step 1: $2x + 25^\circ + 3x - 15^\circ = 180^\circ$.
Step 2: Simplify: $5x + 10^\circ = 180^\circ$.
Step 3: Subtract 10°: $5x = 170^\circ$.
Step 4: Divide by 5: $x = 34^\circ$.
Step 5: Calculate $\angle EFG = 2(34^\circ) + 25^\circ = 93^\circ$.

Key Concept: Parallel Line Definition

b) Coplanar lines that never meet
[Solution Description]
Parallel lines exist only in vertical orientation.
Parallel lines must be straight and lie on different planes.
Lines that appear to meet at a distant point are parallel.
Correct definition: Two coplanar lines that never intersect.

Key Concept: Real-Life Application

a) Railway tracks on straight path
[Solution Description]
Railway tracks lie on the same plane (ground) and maintain constant distance without intersecting when viewed without perspective distortion. Car headlights diverge on curved roads, doors meet at hinges, and pencils placed randomly lack guaranteed coplanarity.

Key Concept: Perception Distortion

c) Zig-zag lines intersecting vertical lines at varying angles
[Solution Description]
Zig-zag patterns intersecting parallel lines disrupt visual continuity. The alternating angles create conflicting directional cues, forcing the brain to perceive distortion. This matches syllabus discussions on how converging/diagonal elements break parallelism perception.

Key Concept: Illusion causes

b) Background patterns with angled guides
[Solution Description]
As per syllabus notes, background patterns like intersecting guidelines or curved elements trick visual perception. Answer matches specific listed causes.

Key Concept: Parallel Illusions - Measurement Verification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion highlights how thick lines can create visual illusions of parallelism, as explained in the syllabus under "Variation in line thickness". The Reason emphasizes the necessity of logical proof over measurement, matching the syllabus's key insight about reconciling theory with practice. Both statements are true, and the Reason directly addresses why verification is needed to overcome such illusions, aligning with syllabus examples about avoiding measurement-based errors.

Key Concept: Vertically Opposite Angles

b) Lines having observable thickness
[Solution Description]
Mathematically, vertically opposite angles are equal. Physical drawings use lines with thickness, making exact angle replication difficult. This creates visual deception where angles appear slightly unequal despite theoretical equality.