Line: A straight path extending in both directions without end. Line Segment: A part of a line with two fixed endpoints. Ray: A segment of a line that begins at a specific point and stretches endlessly in one direction.
An angle is formed when two rays meet at a common endpoint, called the vertex.
The sum of angles on a straight line is always 180 degrees
A protractor is a tool used to measure and draw angles.
Angles on a straight line add up to 180 degrees. Angles around a point add up to 360 degrees. Vertically opposite angles are equal.
Acute Angle: Less than 90 degrees. Right Angle: Exactly 90 degrees. Obtuse Angle: Greater than 90 degrees but less than 180 degrees. Straight Angle: Exactly 180 degrees.
CBSE Notes Class 6 Maths Chapter 2 Lines and Angles
In Chapter 2 of CBSE Class 6 Maths, we explore the concepts of Points, Line Segments, and Rays. A Line Segment is a part of a line with two endpoints, while a Ray has one fixed endpoint and extends infinitely in one direction. Angles are formed when two rays meet at a point. We study Types of Angles such as acute, right, obtuse, and reflex. The Degree is used to measure angles, and a Protractor is a tool used to measure angles accurately.A PDF of the chapter is available below, making it easy to revise the key concepts and refer to them when needed.
Chapter 2 Lines and Angles CBSE Notes
Points
A point determines a precise location, but it has no height.
Ex: Point A, Point B, Point C, etc
Line Segments
It has two end points. It is denoted by either . The points A and B are called the end points of the line segment AB.
Line
It has no end points. It is denoted by AB.
Ray
A ray is a portion of a line that starts at one point (called the starting Point or initial point) and goes on endlessly in a direction.
It is denoted as AP.
Angle
An angle is formed when 2 rays meet at a common endpoint called the vertex. Angles are measured in degrees (∘).
Parts of an Angle:
Arms: The two rays that form the angle.
Vertex: The common endpoint of the 2 rays.
For example, if two rays OA and OB meet at point O, the angle is represented as ∠AOB.
Types of Angles
Angles are classified based on their measurement:
Acute Angle: An acute angle measures less than90∘. These angles are small and sharp. It is always smaller than a right angle.
Example: 45∘,60∘.
Real-life Example: The hands of a clock at 10:10.
Right Angle: A right angle is an angle that measures exactly 90∘. It forms a perfect "L" shape. It is the baseline for measuring other types of angles.
Example: The corner of a square or rectangle.
Real-life Example: The corner of a notebook, book, or a square, The edge of a table.
Obtuse Angle: An obtuse angle measures more than 90∘ but less than 180∘. These angles are broad and open. This is the angle formed when two opposite rays meet at a point.
Example: 120∘,150∘.
Real-life Example: The angle of an open door when it’s slightly ajar.
Straight Angle: A straight angle measures exactly 180∘. It looks like a straight line. It is equal to two right angles.
Example: A straight line.
Real-life Example: The angle formed by the hands of a clock at 6:00. A flat bridge or the horizon.
Reflex Angle: A reflex angle measures more than 180∘ but less than 360∘. These angles are very large and wrap around a significant part of a circle.
Example: 270∘.
Real-life Example: The hands of a clock at 10:10 (taking the outer part of the angle).
Complete Angle: A complete angle measures exactly 360∘. It represents one full rotation or revolution. It forms a perfect circle.
Example: A full circle.
Real-life Example: The wheels of a car rotating one full turn, A merry-go-round making one complete spin.
Degree and Protractor
Degree measures of angles can be measured using a protractor.
A protractor is a tool used to measure and draw angles. It is semi-circular or circular in shape and marked with degree measurements, ranging from 0∘ to 180∘ (in a semi-circular protractor) or 0∘ to 360∘ (in a circular protractor).
Parts of a Protractor:
Baseline: The straight edge of the protractor used to align with one arm of the angle.
Center or Origin: The midpoint at the base of the protractor, marked with a small hole or dot. This is where the vertex of the angle is placed.
Degree Markings: The curved edge is marked with degrees from 0∘ to 180∘ (or 0∘ to 360∘ in a circular protractor).
Inner and Outer Scales:
The inner scale is used to measure angles starting from the right (clockwise).
The outer scale is used to measure angles starting from the left (counter-clockwise).
How to Use a Protractor to Measure an Angle
Step 1: Place the Protractor
Align the baseline of the protractor with one arm of the angle.
Place the centre of the protractor at the vertex of the angle.
Step 2: Read the Degree Markings
Locate the other arm of the angle on the protractor scale.
Check which degree marking it passes through. Use the correct scale (inner or outer) based on the angle's orientation.
Step 3: Note the Measurement
Record the degree where the arm crosses the protractor scale. This is the angle’s measure.
Steps to Draw an Angle Using a Protractor:
Step 1: Draw a Baseline
Use a ruler to draw a straight line. This will act as one arm of the angle.
Step 2: Place the Protractor
Align the baseline of the protractor with the line you just drew, and place the centre at one end of the line.
Step 3: Mark the Angle
Locate the desired angle measure on the protractor scale and make a small dot at that degree.
Step 4: Draw the Second Arm
Take away the protractor and use a ruler to connect the dot with the vertex of the angle. This forms the second arm.
Point: A position in space that indicates an exact location without any length, width, or depth, represented by a dot and named using a single capital letter.
Line Segment: The shortest straight path joining two distinct endpoints, possessing a definite and measurable length.
Line: A straight path formed when a line segment extends endlessly in both directions without any endpoints, denoted using arrowheads at both ends.
Ray: A part of a line that starts at a fixed initial point and extends infinitely in one single direction.
Opposite Rays: Two collinear rays that share a common initial endpoint and extend in completely opposite directions along the same line to form a straight line.
Collinear and Coplanar Elements: Points, segments, or rays are collinear if they lie along the exact same straight line, and coplanar if they exist within the same flat surface or plane.
Angle: A geometric figure formed when two distinct rays originate from the same common initial point called the vertex, with the rays forming the arms of the angle.
Degree: The standard unit of measurement used to quantify the amount of rotation or turning between two rays forming an angle.
Superimposition: A direct comparison method where one angle is placed directly over another so their vertices and base rays coincide, showing which angle is larger or if they are equal.
1.0Important Formulas: Lines and Angles
Geometric Symbols and Notations:
Line Segment AB Symbol: Written as AB or AB, representing a straight path bounded by endpoints A and B.
Line AB Symbol: Written as \(\overleftrightarrow{AB}\), AB with double arrowheads, or labeled using a small lowercase letter like m or l, extending infinitely in both directions.
Ray AB Symbol: Written as \(\overrightarrow{AB}\), representing a ray originating at endpoint A and passing endlessly through point B.
Angle AOB Notation: Written as AOB, where point O is the central vertex and rays \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) form the arms.
Perpendicular Lines Symbol (): Indicates two intersecting lines that meet at a 90 right angle, expressed as ABCD or PQXY.
Parallel Lines Symbol (): Indicates two straight lines lying in the same plane that never intersect and remain at an equal distance apart, expressed as ABCD.
Degree Measures and Rotation Rules:
Full Circle Revolution: One complete turn or revolution equals 360, which corresponds to 4 right angles.
Straight Angle Measure: A half turn of a ray equals 180, formed when the arms are opposite rays.
Right Angle Measure: A quarter turn of a ray equals 90, forming an L-shape.
Clock Angle Rule: A 12-hour clock face is divided into 12 equal parts, where each hour interval represents 112360=30. For example, at 3 o'clock, the angle between hands is 330=90, and at 7 o'clock, it is 530=150.
Bicycle Wheel Spoke Rule: When a circular wheel is divided into N equal parts by spokes, the angle between adjacent spokes equals 360N. For a wheel with 24 spokes, the adjacent angle is 36024=15.
Unmarked/Reflex Angle Rule: The measure of a marked angle plus its unmarked exterior angle equals 360. Therefore, Unmarked Angle =360-MarkedAngle.
Angle Bisector Formula: An angle bisector divides an angle into two equal halves. If a ray bisects a 90 right angle, each resulting angle measures 902=45.
Angle Classifications:
Zero Angle: Measures exactly 0, formed when the rotating ray has not moved from its initial position.
Acute Angle: Measures strictly greater than 0 and less than 90 (0<x<90).
Right Angle: Measures exactly 90, formed by perpendicular rays making a quarter turn.
Obtuse Angle: Measures strictly greater than 90 and less than 180 (90<x<180).
Straight Angle: Measures exactly 180, formed by opposite collinear rays making a half turn.
Reflex Angle: Measures strictly greater than 180 and less than 360 (180<x<360).
Complete Angle: Measures exactly 360, formed when a ray completes one full revolution back to its starting position.
2.0Learning Guide for Class 6 Maths Lines and Angles – Chapter 2
Making a Handmade Protractor Activity: Fold a semicircular paper disc in half to locate the 90 center crease. Fold the quarter circle in half again to obtain 45 and 135 creases. Fold once more in half to derive 22.5, 67.5, 112.5, and 157.5 mark lines, demonstrating how a protractor scale is constructed through repeated angle bisection.
Paper Folding Angle Bisector Activity: Draw an angle AOB on paper and fold the sheet through vertex O so that ray \(\overrightarrow{OA}\) coincides with ray \(\overrightarrow{OB}\). Unfolding reveals crease \(\overrightarrow{OX}\), which acts as the line of symmetry and angle bisector where AOX=BOX.
Measuring Angles with a Labelled Protractor: Place the protractor center directly over vertex A with its base line aligned along arm \(\overrightarrow{AC}\). Read the degree mark where arm \(\overrightarrow{AB}\) crosses the curved scale, choosing the inner scale (0 to 180 right to left) or outer scale (0 to 180 left to right) based on which baseline aligns with 0.
Constructing an Angle Step-by-Step: To construct XYZ=65, draw base ray \(\overrightarrow{YZ}\), place the protractor center on point Y with the baseline along \(\overrightarrow{YZ}\), mark point X at 65 on the scale, remove the protractor, and draw ray \(\overrightarrow{YX}\).
Constructing a Reflex Angle: To construct a 220 reflex angle, calculate the interior angle as 360-220=140. Draw a 140 angle with a protractor, then mark the outer rotation arc to show the 220 reflex angle.
Identifying Perpendicular Models in Real Life: Adjacent edges of a tabletop and line segments forming the letter 'L' intersect at 90 angles, serving as real-life models of perpendicular lines.
Identifying Parallel Models in Real Life: Railway tracks and opposite edges of a ruler maintain equal spacing and never meet, serving as real-life models of parallel lines.
Exam Revision Tip on Rotation: Remember that the length of the rays does not affect the size of an angle; angle magnitude depends entirely on the amount of rotation between the rays.
6.0Benefits of CBSE Notes for Class 6 Maths Chapter 2 - Lines and Angles
Focus on Key Points:CBSE Notes for Class 6 highlight the most crucial information and formulas, ensuring that students don't miss out on essential details.
Easy Accessibility: Digital notes can be accessed anytime and anywhere, making learning more convenient.
Self-Paced Learning: Students can refer to the CBSE Notes at their own pace, revisiting topics they find challenging.
Reduced Textbook Burden: Notes can act as a handy reference, reducing the need to constantly flip through the entire NCERT Solutions textbook.
Preparation for Higher Classes: A strong foundation in the concepts of lines and angles in CBSE Class 6 is crucial for understanding more advanced geometry in higher classes.
