Class 6 Mathematics Chapter 2 Lines and Angles

Key Concept: Point, Infinite Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that through a single point, infinitely many lines can be drawn. This is true because a line can pass through a point at any angle, leading to infinite possibilities. The reason explains that a point has no dimensions (length, breadth, or height) and only marks a location. This property of a point allows for the infinite number of lines passing through it. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Definition of a Point

b) A precise location with no size
[Solution Description]
A point is a fundamental object in geometry that represents a precise location but has no size, such as length, breadth, or height. It is often denoted by a single capital letter and visualized as an invisibly thin dot.

Key Concept: Angle Naming Convention

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, when naming an angle, the vertex must always be written as the middle letter. For example, for an angle formed by points $A$, $B$, and $C$, the angle is named $∠ABC$ where $B$ is the vertex. Thus, the assertion is correct, and the reason correctly explains why the assertion is true.

Key Concept: Real-life examples: Tip of a compass, pencil, needle

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the tip of a compass is an example of a point. This is true as per the syllabus which mentions that models for a point include the tip of a compass, pencil, and needle. The reason states that a point determines a precise location but has no dimensions. This is also true as described in the syllabus. Moreover, the reason correctly explains why the tip of a compass is considered a point, as it precisely marks a location without any dimensions.

Key Concept: Line Segment Properties

c) Line segment
[Solution Description]
The shortest path between two points is known as a line segment. This property defines what a line segment is in geometry.

Key Concept: Lines and Angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the line segment $AB$ is the shortest path between two points $A$ and $B$. This is true according to the syllabus, which defines a line segment as the shortest path between two points, including the endpoints. The reason states that a line segment has two distinct endpoints and includes all points between them. This is also true as per the syllabus definition of a line segment. Moreover, the reason correctly explains why the line segment $AB$ is the shortest path between two points—because it includes all intermediate points without any deviation. Therefore, both are true, and the reason explains the assertion.

Key Concept: Line Segment Definition

c) Line segment
[Solution Description]
The shortest path between two points A and B is called a line segment, as it directly connects the two points without any curves or bends.

Key Concept: Order of Points

a) Yes, they represent the same line segment
[Solution Description]
In geometry, $PQ$ and $QP$ both refer to the same line segment because the order of points does not change the physical line segment. However, in other contexts like vectors, order may matter.

Key Concept: Line Segment and Ray

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that a line segment $AB$ can be extended beyond point $B$ to form a ray $AB$. This is true because extending a line segment from one of its endpoints creates a ray with the starting point at $A$ and passing through $B$.
The Reason states that a ray has one endpoint, while a line segment has two endpoints. This is also true as per the definitions: a line segment is bounded by two endpoints ($A$ and $B$), whereas a ray starts at one endpoint and extends infinitely in one direction.
However, the Reason does not explain why extending a line segment beyond one of its endpoints forms a ray. It merely states the difference between a ray and a line segment. Therefore, both statements are true, but the Reason is not the correct explanation of the Assertion.

Key Concept: Line Segment and Endpoints

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that a line segment has exactly two endpoints, which is true as per the syllabus definition of a line segment. The reason states that a line segment is the shortest distance between two points, which is also true according to the syllabus. However, the reason does not explain why a line segment has two endpoints; it describes a different property of a line segment. Therefore, both statements are true, but the reason is not the correct explanation of the assertion.

Key Concept: Lines through non-collinear points

d) Six
[Solution Description]
The number of lines is equal to the number of ways you can select 2 distinct points out of 4. This is calculated using combination formula $C(4,2) = \frac{4!}{2!(4-2)!} = 6$.

Key Concept: Lines through a single point

c) Infinite
[Solution Description]
A line is determined by two points. However, if there is only one point, infinitely many lines can pass through that point in different directions.

Key Concept: Infinite Extension

c) Because it extends infinitely in both directions
[Solution Description]
As mentioned in the syllabus, a line extends infinitely in both directions without any end. Since our drawing space and tools are limited, it is impossible to represent the infinite extension fully.

Key Concept: Infinite extension of a line, Line properties

a) Zero
[Solution Description]
A line extending infinitely in both directions implies it covers all possible directions from any point not on it. Since any line passing through $P$ must either intersect $AB$ or be parallel to it, but for lines in a plane, there’s only one line through $P$ that does not intersect $AB$ (parallel line). However, the syllabus does not cover planes, and in 1D, no such line exists. Given the context, the answer is zero as per basic line properties.

Key Concept: Basic Definition

c) Because it has no endpoints and extends forever
[Solution Description]
A line extends infinitely in both directions, so it cannot be fully drawn on paper or any finite surface. Therefore, the correct answer is that a line has no endpoints and extends forever.

Key Concept: Line, Uniqueness of a line through two points

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a line can be uniquely determined by any two points lying on it. This is true because, according to the syllabus, any two points determine a unique line that passes through both of them. The reason also correctly states the same concept, confirming that any two distinct points determine a unique line. Therefore, the reason is the correct explanation of the assertion.

Key Concept: Uniqueness of a Line

b) 1
[Solution Description]
Any two distinct points determine a unique line that passes through both of them. Hence, only one unique line can pass through $P$ and $Q$.

Key Concept: Naming Lines and Rays

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a line denoted by $AB$ is uniquely determined when points $A$ and $B$ are given. This aligns with the geometric principle mentioned in the syllabus: any two distinct points determine a unique line passing through them. Therefore, the reason provided correctly explains why the assertion holds true.

Key Concept: Unique line through two points

b) 1
[Solution Description]
The syllabus states that a unique straight line passes through any two distinct points. Hence, Sheetal can draw only one straight line through both P and Q.

Key Concept: Point-Line Relationship

c) Three
[Solution Description]
The question specifies three non-colinear points. From each pair of points, we can draw one unique line:
1. Line through $X$ and $Y$,
2. Line through $Y$ and $Z$,
3. Line through $X$ and $Z$.
Thus, there are exactly three distinct lines possible with three non-colinear points.

Key Concept: Identifying Rays

c) A straight line segment connecting two cities on a map
[Solution Description]
A ray starts at a point and extends infinitely in one direction. A line segment has two endpoints and does not extend infinitely in either direction, so it does not represent a ray.

Key Concept: Ray properties, counting

c) 5
[Solution Description]
For $n$ rays emanating from a single point with no three concurrent, the maximum number of regions created is given by $R(n) = n^2 – n + 2$. We set up the equation $n^2 – n + 2 = 22$. Solving: $n^2 – n – 20 = 0$. Using the quadratic formula: $n = [1 \pm \sqrt{1 + 80}]/2 = [1 \pm 9]/2$. Taking the positive root: $n = (1 + 9)/2 = 5$.

Key Concept: Properties of Rays, Multiple Points

c) The ray can be named as $XZ$
[Solution Description]
The ray starts at $X$ and passes through $Y$ and $Z$. According to the definition, the vertex (starting point) must always be written first when naming a ray. Therefore, the ray can be named as $XY$ or $XZ$, but not as $YX$ or $ZX$ since those would imply the ray starts at $Y$ or $Z$, respectively.
Among the given options, the correct naming convention is followed only in $XZ$.

Key Concept: Ray Definition

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that a ray $AB$ and a ray $BA$ are the same, which is false. A ray has a starting point and extends infinitely in one direction only. For example, ray $AB$ starts at point $A$ and goes through point $B$ infinitely beyond $B$, whereas ray $BA$ starts at point $B$ and goes through point $A$ infinitely beyond $A$. These two rays are not the same. However, the reason given is true as it correctly defines what a ray is.

Key Concept: Angles

c) An angle that measures exactly $180^\circ$
[Solution Description]
A straight angle is formed when the arms of the angle lie in a straight line, making an angle of $180^\circ$. For example, opening a book cover fully on a table forms a straight angle.

Key Concept: Practical Angle Application, Rotation

b) $22.5^\circ/\text{hour}$
[Solution Description]
The shadow moves from north ($0^\circ$) to northeast ($45^\circ$), covering $45^\circ$ in 2 hours. The angular displacement per hour is calculated as $\frac{45^\circ}{2} = 22.5^\circ/\text{hour}$.

Key Concept: Ray vs Line, Multiple Concepts

c) There is exactly one line containing both points
[Solution Description]
Between two points $P$ and $Q$: (1) There is exactly one line passing through both points, (2) There are two distinct rays (one starting at $P$ going through $Q$ and one starting at $Q$ going through $P$), and (3) These rays are different because they have different starting points. Therefore, only the statement about there being exactly one line is universally true.

Key Concept: Ray Properties

d) A ray has a starting point and extends infinitely in one direction.
[Solution Description]
A ray has a starting point and extends infinitely in one direction. Hence, a ray can pass through multiple points but its name is determined by its starting point and any other point on its path.

Key Concept: Real-Life Example

b) 3 o’clock
[Solution Description]
The angle between the hour and minute hands of a clock changes as time passes. A $90^\circ$ angle occurs when the hands are perpendicular to each other, such as at 3 o’clock or 9 o’clock.

Key Concept: Special Types of Angles

d) 6:00
[Solution Description]
A straight angle measures 180° and occurs when the two hands of the clock are in opposite directions. On a clock, this happens every 30 minutes past the hour (e.g., at 6:00, 12:30, etc.). For example, at 6:00, the hour hand is on 6 and the minute hand is on 12, forming a straight angle.

Key Concept: Angle Naming

c) $\angle QPR$
[Solution Description]
The angle is named using the common point (vertex) and one point from each ray. Here, the vertex is $P$ and the rays are $\overrightarrow{PQ}$ and $\overrightarrow{PR}$. According to the naming convention, the vertex must be the middle letter. Therefore, the angle can be named as $\angle QPR$ or $\angle RPQ$.

Key Concept: Arms of Angle

c) $\overrightarrow{YX}$ and $\overrightarrow{YZ}$
[Solution Description]
The arms of an angle are the two rays that form the angle, originating from the vertex. For $\angle XYZ$, the vertex is $Y$. Therefore, the two arms are the rays $\overrightarrow{YX}$ and $\overrightarrow{YZ}$.

Key Concept: Parts of an angle: Vertex and arms, Naming angles

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
When naming an angle formed by two rays meeting at a common vertex, it is conventional to represent the vertex as the middle letter in the angle’s name (e.g., $\angle ABC$ where $B$ is the vertex). However, the assertion states that the angle is “always” named with the vertex as the middle letter, which is not strictly necessary—alternative notations might use just the vertex (e.g., $\angle B$) if clarity is maintained. The reason correctly explains the standard practice but does not justify the absolute necessity claimed in the assertion.

Key Concept: Identifying Arms of an Angle

c) $PQ$ and $PR$
[Solution Description]
In a pair of scissors, the two blades act as the arms of the angle, and the joining point acts as the vertex. Given that the vertex is $P$, the arms are the lines from $P$ to $Q$ and from $P$ to $R$. Therefore, the arms are $PQ$ and $PR$.

Key Concept: Angle Classification

d) Reflex angle
[Solution Description]
Angles can be classified as follows:
– Acute: less than $90^\circ$
– Right: exactly $90^\circ$
– Obtuse: between $90^\circ$ and $180^\circ$
– Reflex: greater than $180^\circ$
Since $195^\circ$ is greater than $180^\circ$, it is a reflex angle.

Key Concept: Angle Measurement

c) $60^\circ$
[Solution Description]
An angle bisector divides the angle into two equal parts. Given $\angle ABC = 120^\circ$, the bisector $BD$ will split it into two angles of equal measure:
$\angle ABD = \frac{120^\circ}{2} = 60^\circ$
So, $\angle ABD = 60^\circ$.

Key Concept: Acute Angle Puzzle

b) $35^\circ$
[Solution Description]
Let the angle be $x^\circ$. Since doubling it must still be acute, $2x < 90^\circ$, so $x 90^\circ$, meaning $x > 30^\circ$. Thus, $x$ must lie between $30^\circ$ and $45^\circ$.

Key Concept: Common Angle Measures

c) 90°
[Solution Description]
The syllabus states that a right angle is half of a straight angle ($180°$). So, we calculate it as $180° / 2 = 90°$. Hence, a right angle measures $90°$.

Key Concept: Real-life applications of angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that opening scissors increases the angle between their blades, which is true because rotating one blade away from the other increases the angular separation. The reason explains that angle size depends on rotational measure between arms around a vertex – this correctly explains why opening scissors increases the angle. Both statements are factually correct and the reason properly justifies the assertion.

Key Concept: Angle identification in real-life objects

b) The point where the two blades are joined
[Solution Description]
The vertex of an angle is the point where two arms meet. In a pair of scissors, the vertex of the angle formed when opening or closing the scissors is at the point where the two blades are joined together.

Key Concept: Comparing angles without superimposition

a) The angle with arc AB is larger
[Solution Description]
When using a transparent circle to compare angles, the angle with a longer arc between the marked points is larger. Since arc AB is longer than arc CD, the angle corresponding to arc AB is larger.

Key Concept: Comparing angles without superimposition

c) $∠LMN$ is larger than $∠XYZ$
[Solution Description]
The arc length intercepted by an angle on a circle is directly proportional to the size of the angle. Since the arc length for $∠LMN$ (from $L’$ to $N’$) is longer than that for $∠XYZ$ (from $X’$ to $Z’$), $∠LMN$ is larger than $∠XYZ$.

Key Concept: Comparing Angles by Superimposition

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when two angles are superimposed with coinciding vertices, the angle with arms lying outside the other is larger. This is true because superimposition directly compares the extent of rotation between the arms. The reason explains that angle size depends on rotation, not arm length, which is also true and correctly explains why the assertion holds.
Both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Superimposition, Circular Comparison

b) $∠PQR > ∠XYZ$
[Solution Description]
First, understand that the size of an angle is proportional to the arc length it subtends on a circle centered at its vertex. Here, since arc AB is longer than arc CD for the same radius (same circle), the angle corresponding to arc AB ($∠PQR$) must be larger than the angle corresponding to arc CD ($∠XYZ$).

Key Concept: Superimposing angles

b) $\angle ABC$ is larger than $\angle PQR$
[Solution Description]
Superimposing angles involves overlapping their vertices and arms. If one angle ($\angle ABC$) completely covers another angle ($\angle PQR$) but extends further, it means $\angle ABC$ has a larger measure than $\angle PQR$.

Key Concept: Comparing angles using a transparent circle

b) $\angle A$ is larger than $\angle B$
[Solution Description]
The transparent circle method measures the angle by marking points where the arms pass through the circle. The distance between these points in degrees directly represents the measure of the angle. So, $\angle A = 120^\circ$ and $\angle B = 90^\circ$. Therefore, $\angle A$ is larger than $\angle B$.

Key Concept: Comparing Angles by Superimposition

c) The angles are equal in size
[Solution Description]
When two angles are superimposed and their vertices and arms overlap exactly, the sizes of the angles are equal. This means they have the same amount of rotation.

Key Concept: Equal angles through superimposition

d) $∠AOB$ is equal to $∠XOY$
[Solution Description]
When two angles are superimposed and both their vertices and arms coincide exactly, the angles are equal in size. Here, since $OA$ overlaps with $OX$ and $OB$ overlaps with $OY$, the amount of rotation from $OA$ to $OB$ is the same as from $OX$ to $OY$. Therefore, the angles must be equal.

Key Concept: Transparent Circle Angle Comparison

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Using a transparent circle to measure angles involves placing the center at the vertex and marking points where the arms intersect the circumference. The assertion is true because this method allows visual comparison of angles. The reason is also true as larger angles subtend longer arcs; however, it does not directly explain why the method works, since we are comparing angles visually rather than measuring arc lengths. Thus, the reason is not the correct explanation for the assertion.

Key Concept: Transparent Circle Method

b) To compare two angles without superimposition
[Solution Description]
The transparent circle helps in comparing two angles without superimposing them physically. By placing the center of the circle at the vertex of the angle, the points where the arms intersect the circle can be marked. These points help in determining which angle is larger by comparing their positions on the circle.

Key Concept: Angle Size and Arm Length

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion (A) claims that increasing the length of the arms of an angle increases the angle’s size, which is false. The size of an angle depends on the rotation between its arms, not their lengths. The Reason (R) correctly states that angle size is determined by the rotation between the arms, making it true. Therefore, Assertion is false, but Reason is true.

Key Concept: Superimposition of angles

c) The vertices and both arms must overlap exactly.
[Solution Description]
For two angles to be equal in size after superimposition, their vertices and both arms must overlap exactly. This means there is no gap between the arms when placed one over the other.

Key Concept: Making Rotating Arms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, a rotating arm can pass through a slit only when the angle between its arms is exactly equal to the angle of the slit. If the angles are different (greater or smaller), the arm will not pass through. Additionally, the length of the arms does not affect whether they can pass through the slit, as long as the arms are shorter than the length of the slit. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Making Rotating Arms, Angle Comparison

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, for a rotating arm to pass through a slit, the angle between the arms of the rotating arm must be equal to the angle of the slit. If the rotating arm’s angle ($45^\circ$) is greater than the slit’s angle ($30^\circ$), it will not pass through the slit. The reason correctly states that the passage depends only on the angle and not on the length of the arms (as long as they are shorter than the slit’s length). Therefore, both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.

Key Concept: Components of an Angle

b) Two straws and one clip
[Solution Description]
An angle is formed by two arms (straws) joined at a common point (vertex). In this activity, the paper clip acts as the vertex where the two straws (arms) are inserted to create the angle.

Key Concept: Rotating Arms and Slit Angle

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The activity explains that only a rotating arm with an angle equal to the slit’s angle will pass through it, regardless of the lengths of the rotating arms (as long as they are shorter than the length of the slit). Thus, the assertion is true. The reason given is also true because the activity explicitly states that the passage through the slit depends only on the angle and not on the lengths. Furthermore, the reason correctly explains why the rotating arm with the same angle as the slit can pass through it.

Key Concept: Using slits in cardboard to compare angles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, only the pair of rotating arms where the angle is equal to that of the slit passes through the slit. The length of the arms does not matter as long as they are shorter than the length of the slit. Hence, the Assertion is true. The Reason is also true and correctly explains why the Assertion is true, because it highlights that the passing condition depends only on the angle and not the length.

Key Concept: Using slits in cardboard to compare angles

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that a rotating arm with angle $45^\circ$ will pass through a slit of angle $60^\circ$ if its length is shorter than the slit’s length. However, as per the syllabus, the arm will only pass through the slit if its angle is equal to the slit’s angle. Since $45^\circ \neq 60^\circ$, the assertion is false.
The reason correctly states that the possibility of passing through the slit depends only on the angle between the rotating arms and not on their lengths. This is directly supported by the syllabus statement: “Note that the possibility of passing through the slit depends only on the angle between the rotating arms and not on their lengths (as long as they are shorter than the length of the slit).” Therefore, the reason is true.
Hence, the correct answer is that the assertion is false, but the reason is true.

Key Concept: Real-life Example

b) No, the angle remains 60 degrees regardless of arm length
[Solution Description]
The angle between the arms of a compass is determined by how much they are rotated apart from each other, not by their lengths. Even if the lengths of the arms are changed, as long as the rotation remains 60 degrees, the angle stays unchanged.

Key Concept: Angle measurement without superimposition

b) Second crane’s angle is larger
[Solution Description]
The size of an angle is proportional to the arc length it intercepts on a circle centered at its vertex. Since the arc $AB$ is shorter than $CD$, the angle made by the first crane ($∠AOB$) must be smaller than the angle made by the second crane ($∠COD$).

Key Concept: Reflex Angle, Straight Angle

b) $60^\circ$
[Solution Description]
Let the acute angle be $\alpha^\circ$. The reflex angle is then $(360^\circ – \alpha^\circ)$. According to the problem:
$360^\circ – \alpha = 2\alpha \Rightarrow 360^\circ = 3\alpha \Rightarrow \alpha = 120^\circ$
However, $\alpha$ must be acute ($<90^\circ$). Thus, there is an inconsistency here since $120^\circ$ is not acute. Rechecking the problem interpretation: A reflex angle is greater than $180^\circ$ but less than $360^\circ$. The "corresponding" acute angle is likely the angle formed by the same arms but measured on the other side of the vertex. So, the correct equation should be:
$\text{Reflex angle} = 360^\circ – \text{acute angle}$
Given that the reflex angle is twice the acute angle:
$360^\circ – \alpha = 2\alpha \Rightarrow 360^\circ = 3\alpha \Rightarrow \alpha = 120^\circ$
This leads to the same contradiction. Hence, the problem may require rephrasing or reevaluating the definition of "corresponding." Alternatively, if the reflex angle is considered as $(180^\circ + \beta)$, where $\beta$ is another angle, the solution changes.
Assuming a different interpretation: The reflex angle is $2\beta$, where $\beta$ is the acute angle, and they are supplementary:
$2\beta + \beta = 360^\circ \Rightarrow 3\beta = 360^\circ \Rightarrow \beta = 120^\circ$
Again, this contradicts the acute condition. Thus, no valid acute angle satisfies the given condition. The closest plausible option from the choices below is $60^\circ$ (though it doesn't solve the equation correctly).

Key Concept: Acute and Obtuse Angles

d) Between $35^\circ$ and $90^\circ$
[Solution Description]
An angle greater than $0^\circ$ but less than $90^\circ$ is an acute angle. Since the measure lies between $35^\circ$ and $90^\circ$, it is an acute angle.

Key Concept: Fraction of Full Turn

c) $\frac{1}{2}$
[Solution Description]
Given:
Step 1: A full turn is $360^\circ$.
Step 2: A straight angle is $180^\circ$.
To find the fraction, divide the measure of the straight angle by the full turn:
$\frac{180^\circ}{360^\circ} = \frac{1}{2}$
Thus, a straight angle is $\frac{1}{2}$ of a full turn.

Key Concept: Basic Concept: Right Angles in a Straight Angle

b) 2
[Solution Description]
A right angle is $90^\circ$ and a straight angle is $180^\circ$. So, the number of right angles in a straight angle is calculated as $\frac{180^\circ}{90^\circ} = 2$.

Key Concept: Straight Angle Composition

b) 2
[Solution Description]
A straight angle measures $180^\circ$ and each right angle measures $90^\circ$. To find how many right angles make a straight angle, we divide $180^\circ$ by $90^\circ$: $180^\circ ÷ 90^\circ = 2$. So two right angles make one straight angle.

Key Concept: Right Angle and Straight Angle

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

Two lines intersecting at $90°$ are indeed perpendicular, which matches the definition in the syllabus. Additionally, a right angle ($90°$) is half of a straight angle ($180°$), as stated in the syllabus. The Reason correctly explains why perpendicular lines form a right angle because it defines the measure of a right angle relative to a straight angle.

Key Concept: Perpendicular lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that two lines meeting at $90^\circ$ are perpendicular, which is true based on the definition of perpendicular lines in the syllabus. The reason explains why a right angle is formed by dividing a straight angle ($180^\circ$) into two equal parts ($90^\circ$ each). This directly supports the assertion since perpendicular lines form right angles by this exact division. Hence, both the assertion and reason are correct, and the reason correctly explains the assertion.

Key Concept: Perpendicular lines, Coordinates

b) 6
[Solution Description]
First, rewrite both equations in slope-intercept form ($y = mx + b$) to identify their slopes.
For first equation: $3x – 4y + 5 = 0$ → $-4y = -3x – 5$ → $y = \frac{3}{4}x + \frac{5}{4}$. Thus, slope ($m_1$) = $\frac{3}{4}$.
For second equation: $8x + ky – 7 = 0$ → $ky = -8x + 7$ → $y = -\frac{8}{k}x + \frac{7}{k}$. Thus, slope ($m_2$) = $-\frac{8}{k}$.
For two lines to be perpendicular, the product of their slopes must be $-1$: $\frac{3}{4} \times -\frac{8}{k} = -1$.
Solve for $k$: $-\frac{24}{4k} = -1$ → $-\frac{6}{k} = -1$ → $\frac{6}{k} = 1$ → $k = 6$.

Key Concept: Folding sequences, complementary angles

c) $67.5^{\circ}$
[Solution Description]
Folding an angle of $135^{\circ}$ in half implies dividing it into two equal parts.
$\frac{135^{\circ}}{2} = 67.5^{\circ}$
$135 \div 2 = 67.5$

Key Concept: Supplementary angles in paper folding

c) $135^\circ$
[Solution Description]
Supplementary angles add up to $180^\circ$. Given one angle is $45^\circ$, the other angle is:
$180^\circ – 45^\circ = 135^\circ$
So, the supplementary angle is $135^\circ$.

Key Concept: Acute, Obtuse, and Reflex Angles

c) $340^\circ$ (reflex)
[Solution Description]
At 3:20 PM, the hour hand moves a third of the way from 3 to 4 because 20 minutes is $\frac{1}{3}$ of an hour. The angle between each number on the clock is $30^\circ$ since a full rotation is $360^\circ$ divided by 12 numbers. So, the hour hand is at $90^\circ + \left(\frac{1}{3} \times 30^\circ\right) = 100^\circ$. The minute hand is at $120^\circ$ because it points to the 4. The difference between them is $120^\circ – 100^\circ = 20^\circ$. However, the smaller angle is $20^\circ$, but we must consider the reflex angle as well. The reflex angle is $360^\circ – 20^\circ = 340^\circ$.

Key Concept: Multiple Angle Types

b) $30^\circ$, $90^\circ$, $150^\circ$
[Solution Description]
The three angles must sum to $270^\circ$ and must all be different types. Possible combinations where each angle is a multiple of $15^\circ$:
1. Acute ($30^\circ$), Right ($90^\circ$), Obtuse ($150^\circ$): $30 + 90 + 150 = 270^\circ$.
2. Acute ($45^\circ$), Right ($90^\circ$), Reflex ($135^\circ$): But $135^\circ$ is not reflex.
3. Acute ($60^\circ$), Obtuse ($120^\circ$), Right ($90^\circ$): $60 + 120 + 90 = 270^\circ$.
Only the first combination satisfies the condition.

Key Concept: Acute Angle Classification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a $30^\circ$ angle is an acute angle. According to the syllabus, an acute angle is defined as any angle greater than $0^\circ$ and less than $90^\circ$. Since $30^\circ$ falls within this range, the assertion is true. The reason correctly defines an acute angle as per the syllabus, and it also explains why the assertion is true. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Acute Angle Multiplier Puzzle

b) $25°$
[Solution Description]
Let the angle be $x$. According to the question:
$0° < x < 90°$
Doubling it: $2x$ must be acute, so $2x < 90° \Rightarrow x < 45°$.
Tripling it: $3x$ must be acute, so $3x < 90° \Rightarrow x 90°$, so $x > 22.5°$.
Combining these inequalities:
$22.5° < x < 30°$
Thus, any angle between $22.5°$ and $30°$ satisfies the condition.

Key Concept: Right Angle and Straight Angle

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion (A) states that a right angle measures exactly $90^{\circ}$, which is correct as per the syllabus. The Reason (R) provides an explanation for this by stating that two right angles form a straight angle ($180^{\circ}$). This means each right angle must be half of $180^{\circ}$, i.e., $90^{\circ}$. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Right Angle in Full Turn

c) 1/4
[Solution Description]
A full turn is $360°$, and a right angle is $90°$. To find the fraction, divide the right angle by the full turn: $90° / 360° = 1/4$.

Key Concept: Obtuse angle identification, angle sum property

b) $70^\circ$
[Solution Description]
The sum of all interior angles in a quadrilateral is $360^\circ$. We know three angles: A = $95^\circ$, B = $85^\circ$, C = $110^\circ$. To find angle D, subtract the sum of known angles from $360^\circ$.
Calculation: $D = 360^\circ – (95^\circ + 85^\circ + 110^\circ) = 360^\circ – 290^\circ = 70^\circ$.
Since $70^\circ < 90^\circ$, it's not obtuse. Therefore, quadrilateral ABCD has exactly one obtuse angle ($110^\circ$).

Key Concept: Angle Classification

c) 120°
[Solution Description]
An obtuse angle is between 90° and 180°. Let’s check each option:
– 45° is acute (<90°)
– 90° is right angle
– 120° is between 90°-180° so it's obtuse
– 180° is straight angle (not obtuse)
The correct answer is 120°.

Key Concept: Using medium and large marks on a protractor

c) 35$^\circ$
[Solution Description]
To measure the angle, subtract the smaller degree mark from the larger one. Here, the arms pass through $10^\circ$ and $45^\circ$. The calculation is:
$45^\circ – 10^\circ = 35^\circ$
Thus, the angle measures $35^\circ$.

Key Concept: Measuring Angles

c) $35^\circ$
[Solution Description]
To find the measure of $\angle TOS$, subtract the smaller angle from the larger angle: $55^\circ – 20^\circ = 35^\circ$. Therefore, the measure of $\angle TOS$ is $35^\circ$.

Key Concept: Historical Context

b) Babylonia
[Solution Description]
The Babylonians are known for their sexagesimal (base-60) numbering system, which led them to frequently use divisions of 60 and 360. This system influenced modern time and angle measurements.
Step-by-step reasoning:
1. The Babylonian civilization used base-60 for calculations.
2. Their mathematical texts show consistent use of 60 and its multiples, including 360.
3. This historical usage is documented in surviving clay tablets and astronomical records.

Key Concept: Divisibility of 360

c) It can be evenly divided by all numbers up to 10, except 7.
[Solution Description]
The number 360 is historically significant because it can be evenly divided by all numbers from 1 to 10, except 7. It is also divisible by 12 and 24, making it useful for dividing circles and time measurement.
Step-by-step explanation:
1. Check divisibility of 360 by numbers 1 to 10: $360 \div 1 = 360$, $360 \div 2 = 180$, $360 \div 3 = 120$, $360 \div 4 = 90$, $360 \div 5 = 72$, $360 \div 6 = 60$, $360 \div 8 = 45$, $360 \div 9 = 40$, $360 \div 10 = 36$.
2. Observing that all divisions result in whole numbers except when divided by 7 ($360 \div 7 \approx 51.428$).
3. Additionally, 360 is divisible by 12 ($360 \div 12 = 30$) and 24 ($360 \div 24 = 15$), which are important for calendars and timekeeping.

Key Concept: Scale Reading

b) $60^\circ$
[Solution Description]
Protractors have two scales: inner and outer. If the inner scale shows $120^\circ$, the outer scale will show $180^\circ – 120^\circ = 60^\circ$. Therefore, misreading the outer scale would give an incorrect measurement of $60^\circ$.

Key Concept: Unlabelled Protractor, Common Mistakes

c) 51.3$^\circ$
[Solution Description]
Each small unit on a standard unlabelled protractor represents $1^\circ$.
The measured angle is $45$ units, so it should be $45^\circ$.
But every $5^\circ$ mark is shifted by $0.7^\circ$.
The number of $5^\circ$ marks in $45^\circ$ is $9$, so total shift is $9 \times 0.7^\circ = 6.3^\circ$.
The actual angle is $45^\circ + 6.3^\circ = 51.3^\circ$ (assuming the shift adds to the angle).

Key Concept: Labelled Protractor

b) 120 degrees
[Solution Description]
To find the measure of the angle, subtract the smaller degree measurement from the larger one: 150 – 30 = 120 degrees.

Key Concept: Combined Unlabelled and Labelled, Non-Straightforward Path

b) 125°
[Solution Description]
First, determine the angle using the unlabelled protractor: Each long mark is 10°, so the third long mark is 30° and the fourth is 40°. A medium-sized mark between them represents 35° (halfway between 30° and 40°). When measuring this on the labelled protractor, aligning one arm with 20° on the outer scale means the other arm will align with $20° + 35° = 55°$ on the outer scale or $180° – 55° = 125°$ on the inner scale.

Key Concept: Measuring angles using a protractor

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A protractor has two scales: an inner scale and an outer scale. This is designed so that when one arm of the angle aligns with the $0^\circ$ mark (either left or right), the other arm can be measured directly using the corresponding scale without needing to flip the protractor. Hence, both Assertion (A) and Reason (R) are true, and Reason correctly explains why the protractor has two scales.

Key Concept: Inner and Outer Scales

b) To measure clockwise and counterclockwise angles
[Solution Description]
A standard protractor has two sets of numbers—inner and outer scales—to allow measurement of angles in both clockwise and counterclockwise directions. This flexibility ensures accuracy regardless of which direction the angle opens.

Key Concept: Angle Types

c) Obtuse angle
[Solution Description]
According to the syllabus, an obtuse angle is defined as an angle greater than $90°$ but less than $180°$.

Key Concept: Basic Angle Measures

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a right angle measures $90^\circ$, which is correct as per the syllabus. The reason states that two right angles form a straight angle of $180^\circ$, which is also correct. Additionally, the reason correctly explains why a right angle is $90^\circ$ because two such angles make a straight angle ($90^\circ + 90^\circ = 180^\circ$).

Key Concept: Angle Bisector, Protractor Construction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, a circle is folded to get a semicircle representing $180^\circ$. The first fold divides this semicircle into two equal angles of $90^\circ$ each. The second fold further divides these $90^\circ$ angles into two equal parts, resulting in four angles of $45^\circ$ each. The assertion correctly states that folding a semicircle twice marks four equally spaced angles of $45^\circ$. The reason explains the process using the angle bisector concept, which is the correct explanation for the assertion. Thus, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Angle Bisector, Protractor Division

d) $22.5^\circ$
[Solution Description]
A full circle is $360^\circ$. When folded into a semicircle, it becomes $180^\circ$. Dividing this semicircle into 8 equal angles means each angle will be $180^\circ / 8 = 22.5^\circ$.

Key Concept: Angle Bisector Concept

c) 90 degrees
[Solution Description]
A straight angle measures $180^\circ$. An angle bisector divides it into two equal angles. To find the measure of each angle, divide $180^\circ$ by 2:
$180^\circ / 2 = 90^\circ$
Each angle measures $90^\circ$.

Key Concept: Angle Estimation

b) 30°
[Solution Description]
At 1 o’clock, the hour hand is at 1 and the minute hand is at 12. Since each hour represents 30° (360° ÷ 12 hours), the angle between them is 30°.

Key Concept: Folding Paper to Understand Line Segments, Creating Angles Using Grid Points

d) $90°$
[Solution Description]
First, find the length of each side of triangle $ABC$. Use the distance formula:
– $AC = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$
– $BC = \sqrt{(2-4)^2 + (2-0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$
– $AB = 4$.
Now, observe that triangle $ABC$ is isosceles with $AC = BC$. Using the cosine rule for angle $C$:
$AB^2 = AC^2 + BC^2 – 2 \cdot AC \cdot BC \cdot \cos(∠ACB)$
Substituting known values:
$16 = 8 + 8 – 2 \cdot 8 \cdot \cos(∠ACB)$
Simplify:
$16 = 16 – 16\cos(∠ACB)$
This gives $\cos(∠ACB) = 0$, so $∠ACB = 90°$.

Key Concept: Angle Guessing Intuition

b) 92°
[Solution Description]
A right angle measures exactly 90°. Among the given options, 92° is the closest to 90°, with a difference of only $92° – 90° = 2°$.

Key Concept: Reflex Angle

c) 190°
[Solution Description]
A reflex angle is defined as an angle measuring greater than $180°$ and less than $360°$.
Step-by-Step Explanation:
1. Examine each option to determine if it fits within the range of $180°$ to $360°$.
2. Only one of the options satisfies this condition.

Key Concept: Line Segment, Point

b) 8.5 cm
[Solution Description]
The total length of the line segment $PR = PQ + QR = 5$ cm $+$ 7 cm $= 12$ cm. The midpoint of $QR$ divides it into two equal parts, each of length $3.5$ cm. Thus, the distance from $P$ to the midpoint of $QR$ equals $PQ$ plus half of $QR$, which is $5$ cm $+$ $3.5$ cm $= 8.5$ cm.

Key Concept: Angle: arms, vertex, and size

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that two rays sharing a common starting point form an angle, and the size of the angle is determined by the rotation required to align one ray with the other. This is true as per the syllabus, which defines an angle as formed by two rays with a common starting point (vertex) and describes the size of the angle as the amount of rotation needed to move one ray to the other. The reason correctly identifies the vertex as the common starting point of the two arms (rays), which is also true as per the syllabus. Furthermore, the reason provides the correct explanation for why the assertion is true, as the vertex and arms are fundamental to defining both the existence and measurement of the angle.

Key Concept: Acute Angle

a) 45°
[Solution Description]
An acute angle is any angle that measures between $0°$ and $90°$. Therefore, among the given options, only $45°$ falls within this range.