Key Concept: Parallel Lines, Perpendicular Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if two lines are folded or constructed to be perpendicular to a common transversal, they must be parallel. This is true because in geometry, if two lines are both perpendicular to a third line, they cannot intersect each other (as intersecting would require at least one of them not to remain perpendicular to the transversal). Thus, they must be parallel.
The reason states that when two lines are perpendicular to a third line, the corresponding angles between them and the transversal are equal. This is also correct because perpendicular lines form $90^\circ$ angles with the transversal, making all corresponding angles equal ($90^\circ$).
Moreover, the reason correctly explains the assertion since the equality of corresponding angles is a fundamental property that ensures the two lines are parallel. Therefore, both the assertion and reason are true, and the reason is the correct explanation for the assertion.

Key Concept: Intersecting Lines and Angles

c) 4
[Solution Description]
When two lines intersect, they cross each other at a single point. This crossing divides the space around the point into four distinct regions. Each region corresponds to an angle. Observing Figure 5.2 in the syllabus, we see two intersecting lines $l$ and $m$ forming four angles.

Key Concept: Intersecting Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: When two lines intersect, they cross each other at exactly one point.
Step 2: At the point of intersection, adjacent angles are formed around the point where the lines meet.
Step 3: Since two lines intersect, they divide the plane into four distinct angles.
Step 4: Hence, the assertion about four angles being formed is true.
Step 5: The reason correctly states that intersecting lines meet at one point, and this meeting creates the angles, explaining why four angles form.
Therefore, both the assertion and reason are true, and the reason properly explains the assertion.

Key Concept: Paper Folding Patterns, Right Angles

c) 64
[Solution Description]
Horizontal folds create $2^3 = 8$ parallel lines. Vertical fold creates $2^1 = 2$ lines perpendicular to horizontals. Each horizontal line intersects 2 vertical lines. Total intersections: $8 \times 2 = 16$. Each intersection forms 4 right angles. Total right angles: $16 \times 4 = 64$.

Key Concept: Perpendicular Lines via Folding

c) $90^\circ$
[Solution Description]
Perpendicular lines form $90^\circ$ angles. Syllabus states adjacent edges become perpendicular after folding. Vertical/horizontal folds create right angles explicitly.

Key Concept: Plane Surfaces, Parallelism

b) No, they occupy different planes
[Solution Description]
Parallel lines must lie on the same plane. Table and wall are different planes. Though lines don’t meet, they belong to different surfaces. Per syllabus definition: lines on different planes cannot be parallel regardless of non-intersection.

Key Concept: Linear Pairs, Vertical Angles, Algebraic Equations

b) 90\textdegree
[Solution Description]

Key Concept: Intersecting lines, Linear pairs

c) Assertion is true, but Reason is false.
[Solution Description]
When two lines intersect, they form four angles. These angles can be grouped into two pairs of vertically opposite angles and two pairs of adjacent angles (linear pairs). The vertically opposite angles are equal because they are non-adjacent angles formed by the intersection of two lines, and their equality is derived from the fact that each pair shares a common vertex but no common arm. In contrast, adjacent angles (linear pairs) are supplementary (add up to $180^\circ$), not complementary (which would mean adding up to $90^\circ$). Therefore, the Assertion is true as vertically opposite angles are indeed equal, but the Reason is false because vertically opposite angles are not complementary; they are equal in measure.

Key Concept: Linear Pair Angles

c) \$135\textdegree\$
[Solution Description]
Linear pairs sum to \$180\textdegree\$. Subtract known angle: \$
angle y = 180\textdegree - 45\textdegree = 135\textdegree\$.

Key Concept: Perpendicular Lines

c) $90\degree$
[Solution Description]
Perpendicular lines form four right angles. By definition, each angle equals $90\degree$.
Step 1: Recall property of perpendicular lines: All angles = $90\degree$.
Step 2: Confirm all four angles are equal in this case.

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: Perpendicular Lines

a) $90^\circ$
[Solution Description] Perpendicular lines intersect at $90^\circ$, creating four right angles. All angles formed will therefore measure $90^\circ$.

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: Linear pairs, Vertical angles

a) $x = 20$, $y = 20$
[Solution Description]
First, solve for $x$ using the linear pair relationship:
$3x + 40 + 5x - 20 = 180^\circ$
Combine like terms: $8x + 20 = 180$. Subtract 20: $8x = 160$. Divide by 8: $x = 20$. The vertical angle to $3x + 40^\circ$ is equal to it. Substitute $x = 20$ into $3(20) + 40 = 100^\circ$. Set $4y + 20 = 100$. Subtract 20: $4y = 80$. Divide by 4: $y = 20$.

Key Concept: Supplementary angles

b) 56°
[Solution Description]
Linear pairs sum to $180°$. Subtract given angle from 180°:
$180° - 124° = 56°$

Key Concept: Measurement Errors in Geometry

b) Instrument precision \& line thickness
[Solution Description]
As per syllabus, deviations occur due to protractor measurement errors or physical line thickness. Geometrically they remain equal under ideal conditions.

Key Concept: Vertical Angles and Linear Pairs

c) Assertion is true, but Reason is false.
[Solution Description]
- Vertically opposite angles ($\angle a$ and $\angle c$) formed by intersecting lines are always equal.
- Adjacent angles in a linear pair (like $\angle a$ and $\angle b$) add up to $180^{\circ}$, not necessarily equal unless each is $90^{\circ}$.
- Assertion correctly states vertical angle equality (True).
- Reason incorrectly claims linear pair angles are equal (False).
$\therefore$ Assertion is true, Reason is false.

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: Algebraic Angle Relationships

a) $-15^\circ$
[Solution Description]
Vertical angles are equal. Set $∠x = ∠z$:
$3y + 10 = 2y - 5$
Subtract $2y$ from both sides:
$y + 10 = -5$
Subtract 10 from both sides:
$y = -15^\circ$
Check validity: Negative angles indicate possible error in problem setup, but mathematically consistent.

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: 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: Real-world measurements vs ideal geometric properties

c) Instrument errors and line thickness
[Solution Description]
Vertically opposite angles are geometrically proven to be equal through reasoning. However, real-world measurements may differ due to two practical limitations: measurement errors from improper protractor use, and variation in drawn line thickness. These factors prevent perfect accuracy in physical measurements.

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: Angle Measurement

b) They are perpendicular
[Solution Description]
Perpendicular lines intersect to form four right angles ($90\degree$ each). If all four angles are equal to $90\degree$, the lines must be perpendicular by definition.

Key Concept: Supplementary and Equal Angles

c) $90^\circ$
[Solution Description]
Supplementary angles add to $180^\circ$. If adjacent angles are equal, each is $\frac{180^\circ}{2} = 90^\circ$. Vertically opposite angles are equal, making all four angles $90^\circ$.

Key Concept: Properties of Intersecting Lines

b) $90^\circ$, $90^\circ$, $90^\circ$
[Solution Description]
Vertically opposite angles are equal. If one angle is $90^\circ$, its vertically opposite angle is also $90^\circ$. Adjacent angles form a linear pair (sum to $180^\circ$). Remaining angles: $180^\circ - 90^\circ = 90^\circ$. All four angles are $90^\circ$.

Key Concept: Real-life Examples

b) Perpendicular intersection
[Solution Description]
The carpenter is verifying if the two structural elements form a right angle (90\textdegree). Perpendicular lines intersect at exactly 90\textdegree, which is essential for stability in constructions like bookshelves.

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: Intersection Points and Angles

c) Intersect at midpoint M at $85.4^{\circ}$
[Solution Description]
AB and CD intersect at midpoint M. The angle at the intersection is given as $85.4^{\circ}$. Since intersection occurs at M (midpoint of AB), the correct statement must include both intersection at midpoint and the specified angle. Other options either refer to endpoints or incorrect angles.

Key Concept: Parallel Lines Behavior

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Parallel lines are defined as lines that do not intersect even when extended infinitely. The assertion correctly states that $OP$ and $QR$ (if parallel) will not meet. The reason describes a property of parallel lines (constant separation distance), which is true. However, the non-intersecting nature is the defining characteristic of parallel lines, while equidistance is a derived property. Hence, the reason does not directly explain why they do not meet.

Key Concept: Vertically Opposite Angles

b) $72.8\degree$
[Solution Description]
Vertically opposite angles are equal when two lines intersect. Since angle $a$ is given as $72.8\degree$, its vertically opposite angle will have the same measure. No calculation needed beyond applying this property.

Key Concept: Line segment extension possibilities

b) They will never meet
[Solution Description] Line segments $OP$ and $QR$ are shown in Figure 5.5 as non-intersecting and parallel. When two parallel line segments are extended infinitely in both directions, they remain equidistant and never meet. This makes them parallel lines by definition.

Key Concept: Vertically Opposite Angles, Measurement Errors

d) Assertion is false, but Reason is true.
[Solution Description]
Step 1: According to the geometric principle, vertically opposite angles (e.g., \$∠b\$ and \$∠d\$) are always equal regardless of the lines' orientation.
Step 2: The student’s measured values (121° and 119°) differ slightly. Such discrepancies typically arise from measurement errors (e.g., protractor precision or line thickness), which are acknowledged in the syllabus.
Step 3: The assertion falsely interprets measurement errors as proof against the geometric principle.
Step 4: The reason correctly states the theoretical equality of vertical angles and identifies error sources.
Conclusion: Assertion is false (theoretical equality holds), Reason is true.

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: Corresponding Angles, Vertical Angles, Supplementary Angles

c) 25°
[Solution Description]
Corresponding angles between parallels: $∠EFG = ∠FHD = 3y$. Line GH creates vertical angle $∠IGH = y+50°$ which equals vertical opposite $∠FHD$. Hence $y+50° = 3y$. Solve: $50° = 2y → y = 25°$.

Key Concept: Vertical Angles, Linear Pairs, Parallel Lines

b) 70°
[Solution Description]
First find co-interior angle: Since lines are parallel, consecutive interior angles sum to 180°. Original $∠1=70°$, its co-interior angle ($∠6$) would be $180° - 70° = 110°$. Vertically opposite angle to $∠6$ would also be 110°, making $∠x=110°$. Adjacent angle (linear pair) would be $180° - 110° = 70°$.

Key Concept: Parallel lines definition, Plane dependency

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion claims non-intersecting lines are always parallel. According to the syllabus, parallel lines lie on the same plane and never meet. However, two lines may not intersect if they are on different planes (e.g., a line on a table and a line on a wall), but these are not parallel. Hence, the Assertion is false. The Reason correctly states that parallelism requires the lines to be on the same plane. Thus, the Reason is true.

Key Concept: Transversal Angles

b) No, since corresponding angles should be equal
[Solution Description]
Corresponding angles must be equal for parallel lines. Here $65^\circ$ and $115^\circ$ are supplementary (65 + 115 = 180). This indicates consecutive interior angles - meaning lines are NOT parallel as corresponding angles aren't equal.

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: Definition Recognition

c) Lines running in same direction without meeting
[Solution Description]
Parallel lines are defined as two lines on the same plane that never meet no matter how far they are extended. They maintain constant distance between them and are marked with matching arrow symbols. Lines not on same plane can't be parallel even if they don't intersect.

Key Concept: Parallel Lines via Perpendicular Folds, Corresponding Angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: When we fold a perpendicular line $t$ to line $l$ through point $A$, $t \perp l$ by definition. Step 2: Folding line $m$ perpendicular to $t$ through $A$ gives $m \perp t$. Since both $l$ and $m$ are perpendicular to the same transversal $t$, they form 90\textdegree{} corresponding angles with $t$. Step 3: According to the syllabus, equality of corresponding angles guarantees parallelism. Here $\angle$ between $t$-$l$ and $t$-$m$ are equal (both 90°), hence $l \parallel m$. Both Assertion and Reason are true, and Reason correctly explains why folding creates parallel lines.

Key Concept: Horizontal Fold Parallelism

c) The crease is parallel to the vertical edges
[Solution Description]
According to Activity 2, folding horizontally creates a line segment parallel to the vertical sides of the square. The original vertical sides remain unchanged. Since the fold divides the paper into two equal horizontal halves, the crease runs midway between the top and bottom edges. These edges are initially horizontal, so their perpendicular direction is vertical. Hence, the crease aligns with vertical sides.

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: Diagonal Fold Parallelism, Perpendicular Construction

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
- **Assertion Analysis**: Following the Parallel Line Construction Method (Step 1: Fold perpendicular to \$d\$ to get \$t\$. Step 2: Fold perpendicular to \$t\$ through \$A\$ to get \$m\$. Since both \$m\$ and \$d\$ are perpendicular to \$t\$, by geometry, \$m \parallel d\$. Hence, Assertion (A) is true.
- **Reason Analysis**: While equality of corresponding angles (verified via protractor or tracing) confirms parallelism, the direct cause of \$m \parallel d\$ is their mutual perpendicularity to \$t\$, not the angle equality itself. Thus, Reason (R) is true but does not correctly explain (A).

Key Concept: Repeated Horizontal Folds

c) 9
[Solution Description]
The original square has 2 parallel edges (top and bottom). Each horizontal fold adds $2^n - 1$ creases where $n$ is the number of folds. For 3 folds: Number of creases $= 2^3 - 1 = 7$. Total parallel lines $= 2 \,(\text{original}) + 7 \,(\text{creases}) = 9$.

Key Concept: Parallel Lines via Perpendicular Folds

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- Original edge $l$ (horizontal) has crease $t$ folded vertically (perpendicular to $l$).
- Crease $m$ is then folded perpendicular to $t$ (horizontal again).
- Mathematical rule states: If two lines ($l$ and $m$) are both $\perp$ to a third line ($t$), they are parallel.
- Assertion describes exact method from syllabus (Fig. 5.24).
- Reason provides the geometric principle behind this observation.

Key Concept: Adjacent edges perpendicular, horizontal fold parallel lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- A square sheet’s original top and bottom edges are parallel horizontal lines.
- The first horizontal fold splits the sheet into two equal parts, creating a new line parallel to the top/bottom edges (Key Observation 3).
- The second horizontal fold creates another line parallel to the original edges. Since all three lines (original top, fold 1, fold 2) are parallel to the bottom edge, they remain parallel to each other. The Reason correctly explains why the folds are parallel.
- Assertion is true (three parallel lines). Reason is true and directly explains Assertion.

Key Concept: Perpendicular folds verification

b) Perpendicular to horizontal edges
[Solution Description]
Vertical folding creates a line parallel to vertical edges. Horizontal edges are the top and bottom sides of the square. Perpendicular lines intersect at $90\degree$. Since vertical fold runs from top to bottom, it intersects horizontal edges (top/bottom) at right angles. Hence, the fold is perpendicular to horizontal edges.

Key Concept: Parallel Line Construction Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: When we fold perpendicular line $t$ through point $A$ on original line $l$, line $t$ forms a 90° angle with $l$.
Step 2: Folding another perpendicular line $m$ to $t$ through $A$ ensures $m$ is at 90° to $t$.
Step 3: Since both $l$ and $m$ are perpendicular to transversal $t$, by geometric rules, they remain equidistant and never intersect, making them parallel ($l \parallel m$).
Conclusion: Assertion is true. The reason correctly states the geometric principle explaining parallelism via shared perpendicular transversal.

Key Concept: Creating Parallel Lines via Perpendicular Folds

c) They are parallel
[Solution Description]
Both $l$ and $m$ are perpendicular to line $t$. Two distinct lines perpendicular to the same line are parallel. Hence $l \parallel m$.

Key Concept: Corresponding Angles

c) Their corresponding angles are equal.
[Solution Description]
As per the syllabus note, equality of corresponding angles is necessary and sufficient for two lines to be parallel. Perpendicularity to a third line is a construction method but not the defining condition mentioned here.

Key Concept: Double perpendicular folding, parallelism reasoning

c) Both are perpendicular to line $t$
[Solution Description]
Perpendicularity ensures $l \perp t$ and $m \perp t$. Two lines perpendicular to the same line ($t$) are parallel. This follows from the equality of corresponding angles (both $90^\circ$), satisfying the parallel condition.

Key Concept: Diagonal Fold Perpendicularity

c) Perpendicular to the first fold at midpoint
[Solution Description]
A diagonal fold divides the square into two triangles. To create a perpendicular intersection, the second fold must cross the diagonal at 90°. This requires folding along a line perpendicular to the diagonal (not the other diagonal). Such a fold intersects the original diagonal at one point, forming a right angle marked with a square symbol.

Key Concept: Horizontal-Vertical Folds Interaction

d) Assertion is false, but Reason is true.
[Solution Description]
Two horizontal folds create parallel lines. A vertical fold intersects each horizontal fold at one point. Each intersection forms a right angle (90°), marked with a small square. There are only two intersections between the vertical and horizontal folds, resulting in two small squares. The original sheet's corners retain their small squares, but the assertion specifically refers to fold intersections. Thus, the Assertion (claiming three) is false. The Reason is true because horizontal and vertical folds are perpendicular.

Key Concept: Corresponding Angles

b) $110^\circ$
[Solution Description]
For parallel lines, corresponding angles are equal. $\angle 2$ and $\angle 6$ are corresponding angles. Subtract from $180^\circ$ for adjacent angles:
Since $\angle 2 = 110^\circ$, $\angle 6$ must also be $110^\circ$ due to the corresponding angles property.

Key Concept: Parallel Line Test

b) No, because they are not equal
[Solution Description]
Lines are parallel only if corresponding angles are equal. Compare $\angle b$ and $\angle f$:
$75^\circ \neq 70^\circ$. Therefore, the lines are not parallel. No calculations beyond comparison.

Key Concept: Vertically Opposite Angles

c) $\angle 4 = 60^\circ$
[Solution Description]
Vertically opposite angles are equal. From the figure, $\angle 2$ and $\angle 4$ are vertically opposite angles. Therefore, $\angle 2 = \angle 4$. Given $\angle 4 = 60^\circ$, $\angle 2 = 60^\circ$. This corresponds to option c).

Key Concept: Vertical Angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Vertical angles are pairs of opposite angles formed by two intersecting lines. By definition, vertical angles are equal. When transversal $t$ cuts lines $l$ and $m$, each intersection point creates two pairs of vertical angles. For example, at line $l$, $\angle1$ = $\angle3$ and $\angle2$ = $\angle4$. Similarly, at line $m$, another set of vertical angles exists. This equality forces at least four angles to repeat their measures. Hence, the maximum distinct angles possible are four. Both Assertion and Reason are true; the Reason directly explains why the Assertion holds.

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, Corresponding Angles

d) Assertion is false, but Reason is true.
[Solution Description]
The vertical angles formed by intersecting lines are always equal (here, $45^\circ$). However, corresponding angles created by a transversal depend on whether the intersected lines are parallel. Since the given lines intersect (forming $45^\circ$ vertical angles), they are not parallel. The equality of corresponding angles ($45^\circ$) would ONLY occur if the lines were parallel, which they are not. The Assertion assumes correspondence without parallelism, making it false. The Reason correctly states the fundamental condition for equal corresponding angles (parallelism), making it true.

Key Concept: Parallel lines with transversal

a) 2
[Solution Description]
Since \$l \parallel m\$, corresponding angles are equal. \$∠2 = 50^\circ\$ implies \$∠4 = 50^\circ\$ (vertically opposite). Adjacent angles: \$∠1 = 130^\circ, ∠3 = 130^\circ\$. Corresponding angles on \$m\$: \$∠6 = 50^\circ, ∠8 = 130^\circ\$, and their vertical opposites. This results in only 2 distinct measures: \$50^\circ\$ and \$130^\circ\$.

Key Concept: Maximum Distinct Angles

b) 4
[Solution Description]
Vertical angles are equal ($\angle 1 = \angle 3$, $\angle 2 = \angle 4$, $\angle 5 = \angle 7$, $\angle 6 = \angle 8$). Adjacent angles form linear pairs summing to $180^\circ$. Even if lines are not parallel, this creates exactly 4 distinct measures.

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: Algebraic Application

b) $20$
[Solution Description]
Corresponding angles are equal: $2x + 15 = 3x - 5$. Subtract $2x$ from both sides: $15 = x - 5$. Add 5: $x = 20$.

Key Concept: Identifying Parallel Lines using Corresponding Angles

c) The lines are parallel
[Solution Description] If the corresponding angles formed by a transversal cutting two lines are equal, the lines must be parallel according to the syllabus. Both corresponding angles being $120^\circ$ satisfy this condition.

Key Concept: Corresponding angles, Supplementary angles

b) $m \parallel n$
[Solution Description] Line $l$ and $m$ form angles with transversal $t$. Since $∠1 + ∠2 = 112^\circ + 68^\circ = 180^\circ$, they are supplementary, meaning $t$ cuts through $l$ and $m$ non-parallelly. However, transversal $s$ creates $∠3 = 68^\circ$ with $m$. If $l \parallel n$, their corresponding angles via $s$ must match. Assume $∠4$ is the corresponding angle to $∠3$ on line $n$. If $∠4 = 68^\circ$, then $m \parallel n$. Checking transversal $t$: If $m \parallel n$, their corresponding angles via $t$ (like $∠2$ and $∠5$) must equal. Since $∠2 = 68^\circ$, $∠5$ should also be $68^\circ$, making $m \parallel n$.

Key Concept: Paper Folding Verification, Corresponding Angle Creation

c) Angles on same side of transversal intersecting both creases
[Solution Description] Perpendicular folds create right angles. Drawing any transversal will form corresponding angles. Measure any pair (e.g., top-left angles on same side of transversal). If equal to $90^{\circ}$ (within protractor error), creases are parallel.

Key Concept: Verifying Corresponding Angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Measuring both corresponding angles with a protractor gives equal measurements ($60^\circ$ each). According to the syllabus key observation, equal corresponding angles prove lines are parallel. The Reason states the general rule that validates the Assertion's specific case. Hence both are true and Reason correctly explains Assertion.

Key Concept: Corresponding Angle Equality Test

d) If all pairs of corresponding angles are equal
[Solution Description]
According to the syllabus: If corresponding angles formed by a transversal are equal ($\angle 1 = \angle 2$, $\angle 3 = \angle 4$), the lines are parallel. This condition is both necessary and sufficient.

Key Concept: Parallel Lines Criteria, Paper Folding Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The paper folding method creates line $m$ perpendicular to transversal $t$ through point $A$, and line $l$ already perpendicular to $t$. Since both lines form $90^{\circ}$ angles with $t$, their corresponding angles are equal ($90^{\circ} = 90^{\circ}$). By the theorem, equal corresponding angles $\Leftrightarrow$ parallel lines. Thus both statements are true AND the reason correctly explains the assertion.

Key Concept: Parallel lines using set square

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- When we draw two lines perpendicular to line $l$ using a set square, they both form $90^\circ$ angles with $l$.
- Line $l$ acts as a transversal cutting these two lines.
- Since the corresponding angles (both $90^\circ$) formed by the transversal with the two lines are equal, the lines are parallel.
- Therefore, both Assertion (A) and Reason (R) are true, and R correctly explains A.

Key Concept: Parallel lines and corresponding angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that two lines drawn perpendicular to line $l$ using a set square are parallel. Using a set square ensures each constructed line forms a $90^\circ$ angle with $l$. When $l$ acts as a transversal cutting these two lines, the corresponding angles at both intersections are $90^\circ$. Since corresponding angles are equal, the two lines must be parallel. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Parallel Line Properties

d) They are parallel to each other
[Solution Description]
By syllabus Key Observation 1: Lines drawn perpendicular to same baseline (here $MN$) using set square are parallel. All other options contradict basic properties.
Step 1: Perpendicular lines create $90^\circ$ angles with $MN$. Step 2: Baseline $MN$ acts as transversal. Step 3: Corresponding angles (both $90^\circ$) are equal. Step 4: Therefore, lines are parallel. Incorrect options mention intersection or non-90° angles.

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: 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: Parallel Line Verification

d) Comparing corresponding angles made by transversal using protractor
[Solution Description]
According to syllabus, parallelism can be verified by checking corresponding angles using protractor or tracing methods. Equal angles formed by transversal prove parallelism more directly than distance measurements which require multiple checks.

Key Concept: Method Verification

b) Reena’s method is correct since corresponding angles are equal.
[Solution Description]
Equal corresponding angles (both $55^\circ$) formed by transversal $t$ with lines $k$ and $n$ guarantee $k \parallel n$. Steps follow the syllabus procedure.

Key Concept: Set Square Alignment, Corresponding Angles

b) Rotation of set square during sliding
[Solution Description]
To maintain parallelism during set square sliding, the angle between ruler and set square must remain constant (90$^\circ$). If the set square rotates slightly while sliding, the resulting lines will form unequal corresponding angles with the transversal (ruler), breaking parallelism.

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: Paper Folding Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: To create line $m$ parallel to $l$, first fold a perpendicular crease $t$ to line $l$ passing through $A$. This makes $t$ a common transversal.
Step 2: Then fold another line $m$ perpendicular to $t$ through $A$. Since both $l$ and $m$ are perpendicular to transversal $t$, their corresponding angles are equal (90$\degree$).
Step 3: By the key observation in the syllabus, equality of corresponding angles guarantees parallelism. Hence, $m \parallel l$. Both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Essential Tool

c) Square paper folding
[Solution Description]
Syllabus mentions "square paper folding" as key tool. Compass/ruler are not involved in paper-folding method described.

Key Concept: Parallel lines, Perpendicular folds

b) Parallel to $t$
[Solution Description]
Since $m \parallel l$ and both are perpendicular to crease $t$, folding perpendicular to $m$ through $A$ creates a new line. Any line perpendicular to $m$ at $A$ must be parallel to $t$ because $t$ is also perpendicular to $m$. This follows from the property that two lines perpendicular to the same line are parallel.

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 Angles

c) $110°$
[Solution Description]
The question requires understanding of alternate angles and vertically opposite angles.
Step 1: Given that an alternate angle is $110°$. Since alternate angles are equal in parallel lines, the other alternate angle is also $110°$.
Step 2: The alternate angle's vertically opposite angle is equal to it because vertically opposite angles are always equal.
Step 3: Therefore, the vertically opposite angle is also $110°$.
The measure is $110°$.

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 Exterior Angles, Triangle Angle Sum

b) 55 degrees
[Solution Description]

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 Equality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When a transversal intersects two parallel lines, corresponding angles (e.g., $\angle f$ and $\angle b$) are equal. Vertically opposite angles (e.g., $\angle b$ and $\angle d$) are also equal. Therefore, alternate angles ($\angle f$ and $\angle d$) must be equal since $\angle f = \angle b$ (corresponding) and $\angle b = \angle d$ (vertically opposite). Both Assertion and Reason are true, and the Reason correctly explains why the Assertion holds.

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: Application of Alternate Angles

c) $110^{\circ}$
[Solution Description]
Alternate angles lie on opposite sides of the transversal and between the parallel lines. The alternate angle of $\angle 6$ will be the angle positioned diagonally opposite across the transversal. Using the alternate angles property, this angle has the same measure: $110^{\circ}$.

Key Concept: Interior angles, Angle Bisector

a) $75^\circ$
[Solution Description]
Let the smaller interior angle be $x$. The larger angle is $5x$. Using the interior angles property:
$x + 5x = 180^\circ \Rightarrow 6x = 180^\circ \Rightarrow x = 30^\circ$
The angles are $30^\circ$ and $150^\circ$. Bisecting the $150^\circ$ angle gives two $75^\circ$ angles. The acute angle between the bisector and the transversal is $75^\circ$.

Key Concept: Interior angles, Triangle Angle Sum

a) $90^\circ$
[Solution Description]
The other interior angle is $180^\circ - 115^\circ = 65^\circ$. In the triangle, two angles are $65^\circ$ (from transversal) and $25^\circ$. Using triangle angle sum:
$65^\circ + 25^\circ + x = 180^\circ \Rightarrow x = 90^\circ$
The angle opposing the transversal is $90^\circ$.

Key Concept: Parallel Lines \& Transversal

c)
[Solution Description]
Assertion (A) refers to the key property of parallel lines: interior angles on the same side of the transversal add up to $180^{\circ}$ (true). Reason (R) claims the sum is always $180^{\circ}$ even without parallel lines. This is false because the sum only equals $180^{\circ}$ when lines are parallel. For non-parallel lines, the sum varies. Hence, Assertion is true, but Reason is false.

Key Concept: Alternate Interior Angles with Non-Parallel Lines

c) 100$\degree$
[Solution Description] When two lines are not parallel, alternate interior angles formed by a transversal are supplementary (sum to $180^\circ$). Subtract the given angle from $180^\circ$: $180^\circ - 80^\circ = 100^\circ$.

Key Concept: Art application, Perceptual distortion

b) Diagonal marks create virtual intersection points
[Solution Description]
Contextual elements like intersecting hatch marks create false depth cues. Adjacent angles/patterns distort spatial perception, making parallel lines appear skewed. True parallelism is independent of surrounding visual clutter but requires verification through measurement or infinite extension.

Key Concept: Same-plane condition, Infinite extension test

a) $l$ and $m$ since both are vertical
[Solution Description]
Parallel lines must lie on the same plane AND never meet when extended infinitely. While $l$ and $m$ are both vertical (same direction), their infinite extension would maintain equal spacing. Line $n$ curves away, violating linearity. Only co-planar straight lines can be parallel.

Key Concept: Optical Illusion Patterns

b) Radiating lines trick depth perception
[Solution Description]
The radiating diagonal lines create a false perspective effect. The brain interprets these diagonals as depth cues, making the horizontal lines seem distorted. This is a classic example of the Hering illusion where parallel lines appear curved due to intersecting background patterns.

Key Concept: Checkerboard contrast, Oblique lines grid

b) Cumulative effect of square edges disrupting continuity perception
[Solution Description]
High-contrast square edges create an "edge displacement" effect. Where a vertical line crosses from a black to white square, the sharp luminance contrast exaggerates small alignment differences between segments. The brain interprets these exaggerated offsets as bends, overriding the actual straightness of the continuous vertical lines.

Key Concept: Vertical Angle Measurement Error

c) Errors inherent in practical measurement techniques
[Solution Description]
Vertically opposite angles should theoretically be equal. The discrepancy arises from measurement limitations: (1) Protractor placement errors (2) Difficulty aligning thick lines' edges precisely at the vertex.
Step 1: Calculate difference: $61^\circ - 58^\circ = 3^\circ$.
Step 2: Even 1° protractor division misalignment compounded during two angle measurements can create this gap.
Step 3: Thick lines make identifying true vertex positions harder, amplifying measurement errors.

Key Concept: Geometry Application Justification

c) Theoretical results closely match real observations
[Solution Description]
Real-world measurements closely approximate geometric predictions despite imperfections. Small errors caused by instrument limitations or line thickness do not negate mathematical relationships derived through logical reasoning.
Step 1: Theoretical vertically opposite angles are always equal.
Step 2: Practical measurements show near-equality (e.g., $89^\circ$ vs $91^\circ$) due to measurable but small errors.
Step 3: Consistency in proximity between theory and practice makes geometry usable across fields.