Class 6 Maths Chapter 2 Notes Lines and Angles
Class 6 Maths Notes Chapter 2 – Class 6 Lines and Angles Notes
→ A point determines a location. It is denoted by a capital letter.
→ A line segment corresponds to the shortest distance between two points. The line segment joining points S and T is denoted by \(\overline{\mathrm{ST}}\).
→ A line is obtained when a line segment like \(\overline{\mathrm{ST}}\) is extended on both sides indefinitely; it is denoted by \(\stackrel{\leftrightarrow}{\mathrm{ST}}\) or sometimes by a single small letter like m.
→ A ray is a portion of a line starting at point D and going in one direction indefinitely. It is denoted by \(\overrightarrow{\mathrm{DP}}\) where P is another point on the ray.
→ An angle can be visualized as two rays starting from a common starting point. Two rays \(\overrightarrow{\mathrm{OP}}\) and \(\overrightarrow{\mathrm{OM}}\) form the angle ∠POM (also called ∠MOP); here, O is called the vertex of the angle, and the rays \(\overrightarrow{\mathrm{OP}}\) and \(\overrightarrow{\mathrm{OM}}\) are called the arms of the angle.
→ The size of an angle is the amount of rotation or turn needed about the vertex to rotate one ray of the angle onto the other ray of the angle.
→ The sizes of angles can be measured in degrees. One full rotation or turn is considered as 360 degrees and denoted as 360°.
→ Degree measures of angles can be measured using a protractor.
→ Angles can be straight (180°), right (90°), acute (more than 0° and less than 90°), obtuse (more than 90° and less than 180°), and reflex (more than 180° and less than 360°).
Point Class 6 Notes
Mark a dot on the paper with a sharp tip of a pencil. The sharper the tip, the thinner will be the dot. This tiny dot will give you an idea of a point. A point determines a precise location, but it has no length, breadth, or height. Some models for a point are given below.
If you mark three points on a piece of paper, you may be required to distinguish these three points. For this purpose, each of the three points may be denoted by a single capital letter such as Z, P, and T. These points are read as ‘Point Z’, ‘Point P’, and ‘Point T’. Of course, the dots represent precise locations and must be imagined to be invisibly thin.
Line Segment Class 6 Notes
Fold a piece of paper and unfold it. Do you see a crease? This gives the idea of a line segment. It has two end points, A and B. Mark any two points A and B on a sheet of paper. Try to connect A to B by various routes (Figure).
What is the shortest route from A to B?
This shortest path from Point A to Point B (including A and B) as shown here is called the line segment from A to B. It is denoted by either \(\overline{\mathrm{AB}}\) or \(\overline{\mathrm{BA}}\). The points A and B are called the endpoints of the line segment \(\overline{\mathrm{AB}}\).
Line Class 6 Notes
Imagine that the line segment from A to B (i.e., \(\overline{\mathrm{AB}}\)) is extended beyond A in one direction and beyond B in the other direction without any end (see Figure). This is a model for a line. Do you think you can draw a complete picture of a line? No. Why?
A line through two points A and B is written as \(\overleftrightarrow{\mathrm{AB}}\). It extends forever in both directions. Sometimes a line is denoted by a letter like l or m. Observe that any two points determine a unique line that passes through both of them.
Ray Class 6 Notes
A ray is a portion of a line that starts at one point (called the starting point or initial point of the ray) and goes on endlessly in a direction. The following are some models for a ray:
Look at the diagram (Figure) of a ray. Two points are marked on it. One is the starting point A and the other is a point P on the path of the ray. We then denote the ray by \(\overrightarrow{\mathrm{AP}}\).
Angle Class 6 Notes
An angle is formed by two rays having a common starting point. Here is an angle formed by rays \(\overrightarrow{\mathrm{BD}}\) and \(\overrightarrow{\mathrm{BE}}\) where B is the common starting point (Figure).
The point B is called the vertex of the angle, and the rays \(\overrightarrow{\mathrm{BD}}\) and \(\overrightarrow{\mathrm{BE}}\) are called the arms of the angle. How can we name this angle? We can simply use the vertex and say that it is the Angle B. To be clearer, we use a point on each of the arms together with the vertex to name the angle. In this case, we name the angle as Angle DBE or Angle EBD. The word angle can be replaced by the symbol ‘ ∠’, i.e., ∠DBE or ∠EBD. Note that in specifying the angle, the vertex is always written as the middle letter. To indicate an angle, we use a small curve at the vertex (refer to Figure). Vidya has just opened her book. Let us observe her opening the cover of the book in different scenarios.
Do you see angles being made in each of these cases? Can you mark their arms and vertex? Which angle is greater – the angle in Case 1 or the angle in Case 2?
Just as we talk about the size of a line based on its length, we also talk about the size of an angle based on its amount of rotation. So, the angle in Case 2 is greater as in this case, she needs to rotate the cover more. Similarly, the angle in Case 3 is even larger than that of Case 2, because there is even more rotation, and Cases 4, 5, and 6 are successively larger angles with more and more rotation.
Let’s look at some other examples where angles arise in real life by rotation or turn:
- In a compass or divider, we turn the arms to form an angle. The vertex is the point where the two arms are joined. Identify the arms and vertex of the angle.
- A pair of scissors has two blades. When we open them (or ‘turn them’) to cut something, the blades form an angle. Identify the arms and the vertex of the angle.
- Look at the pictures of spectacles, wallets, and other common objects. Identify the angles in them by marking out their arms and vertices.
- Do you see how these angles are formed by turning one arm over the other?
Comparing Angles Class 6 Notes
Look at these animals opening their mouths. Do you see any angles here? If yes, mark the arms and vertex of each one. Some mouths are open wider than others; the more the turning of the jaws, the larger the angle! Can you arrange the angles in this picture from smallest to largest?
Is it always easy to compare two angles?
Here are some angles. Label each of the angles. How will you compare them?
Draw a few more angles; label them and compare.
Comparing angles by superimposition
Any two angles can be compared by placing them one over the other, i.e., by superimposition. While superimposing, the vertices of the angles must overlap. After superimposition, it becomes clear which angle is smaller and which is larger.
The picture shows the two angles superimposed. It is now clear that ∠PQR is larger than ∠ABC.
Equal Angles:
Now consider ∠AOB and ∠XOY in the figure. Which is greater?
The corners of both of these angles match and the arms overlap with each other, i.e., OA ↔ OX and OB ↔ OY. So, the angles are equal in size. The reason these angles are considered to be equal in size is that when we visualize each of these angles as being formed out of rotation, we can see that there is an equal amount of rotation needed to move \(\overrightarrow{\mathrm{OB}}\) to \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OY}}\) to \(\overrightarrow{\mathrm{OX}}\). From the point of view of superimposition, when two angles are superimposed, and the common vertex and the two rays of both angles lie on top of each other, then the sizes of the angles are equal.
Comparing angles without superimposition
Two cranes are arguing about who can open their mouth wider, i.e., who is making a bigger angle. Let us first draw their angles. How do we know which one is bigger? As seen before, one could trace these angles, superimpose them, and then check. But can we do it without superimposition? Suppose we have a transparent circle that can be moved and placed on fiures. Can we use this for comparison?
Let us place the circular paper on the angle made by the first crane. The circle is placed in such a way that its center is on the vertex of the angle. Let us mark points A and B on the edge circle at the points where the arms of the angle pass through the circle.
Can we use this to find out if this angle is greater than, equal to, or smaller than the angle made by the second crane?
Let us place it on the angle made by the second crane so that the vertex coincides with the center of the circle and one of the arms passes through OA.
Can you now tell which angle is bigger?
Which crane was making the bigger angle?
If you can make a circular piece of transparent paper, try this method to compare the angles in the Figure with each other.
Making Rotating Arms Class 6 Notes
Let us make ‘rotating arms’ using two paper straws and a paper clip by following these steps:
1. Take two paper straws and a paper clip.
2. Insert the straws into the arms of the paper clip.
3. Your rotating arm is ready!
Make several ‘rotating arms’ with different angles between the arms. Arrange the angles you have made from smallest to largest by comparing and using superimposition.
Passing through a Slit:
Collect several rotating arms with different angles; do not rotate any of the rotating arms during this activity.
Take a cardboard and make an angle-shaped slit as shown below by tracing and cutting out the shape of one of the rotating arms.
Now, shuff and mix up all the rotating arms. Can you identify which of the rotating arms will pass through the slit? The correct one can be found by placing each of the rotating arms over the slit. Let us do this for some of the rotating arms:
Only the pair of rotating arms where the angle is equal to that of the slit passes through the slit. 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).
Special Types of Angles Class 6 Notes
Let us go back to Vidya’s notebook and observe her opening the cover of the book in different scenarios. She makes a full turn of the cover when she has to write while holding the book in her hand.
She makes a half turn off the cover when she has to open it on her table. In this case, observe the arms of the angle formed. They lie in a straight line. Such an angle is called a straight angle.
Let us consider a straight-angle ∠AOB. Observe that any ray \(\overrightarrow{\mathrm{OC}}\) divides it into two angles, ∠AOC and ∠COB.
Is it possible to draw OC such that the two angles are equal to each other in size?
Let’s Explore
We can try to solve this problem using a piece of paper. Recall that when a fold is made, it creates a straight crease. Take a rectangular piece of paper and on one of its sides, mark the straight angle AOB. By folding, try to get a line (crease) passing through O that divides ∠AOB into two equal angles. How can it be done?
Fold the paper such that OB overlaps with OA. Observe the crease and the two angles formed. Justify why the two angles are equal. Is there a way to superimpose and check? Can this superimposition be done by folding? Each of these equal angles formed is called the right angle. So, a straight angle contains two right angles.
If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?
Observe that a right angle resembles the shape of an ‘L’. An angle is a right angle only if it is exactly half of a straight angle. Two lines that meet at right angles are called perpendicular lines.
Classifying Angles
Angles are classified into three groups as shown below. Right angles are shown in the second group. What could be the common feature of the other two groups?
In the first group, all angles are less than a right angle or in other words, less than a quarter turn. Such angles are called acute angles. In the third group, all angles are greater than a right angle but less than a straight angle. The turning is more than a quarter turn and less than a half turn. Such angles are called obtuse angles.
Measuring Angles Class 6 Notes
We have seen how to compare two angles. But can we quantify how big an angle is using a number without having to compare it to another angle? We saw how various angles can be compared using a circle. Perhaps a circle could be used to assign measures for angles?
To assign precise measures to angles, mathematicians came up with an idea. They divided the angle in the center of the circle into 360 equal angles or parts. The angle measure of each of these unit parts is 1 degree, which is written as 1°. This unit part is used to assign a measure to any angle: the measure of an angle is the number of 1° unit parts it contains inside it. For example, see this figure:
It contains 30 units of 1° angle so we say that its angle measure is 30°. Measures of different angles: What is the measure of a full turn in degrees? As we have taken it to contain 360 degrees, its measure is 360°. What is the measure of a straight angle in degrees? A straight angle is half of a full turn. As a full turn is 360°, a half-turn is 180°. What is the measure of a right angle in degrees? Two right angles together form a straight angle. As a straight angle measures 180°, a right angle measures 90°.
A pinch of history
A full turn has been divided into 360°. Why 360? The reason why we use 360° today is not fully known. The division of a circle into 360 parts goes back to ancient times. The Rigveda, one of the very oldest texts of humanity going back thousands of years, speaks of a wheel with 360 spokes. Many ancient calendars, also going back over 3000 years—such as calendars of India, Persia, Babylonia, and Egypt—were based on having 360 days in a year. In addition, Babylonian mathematicians frequently used divisions of 60 and 360 due to their use of sexagesimal numbers and counting by 60s.
Perhaps the most important and practical answer for why mathematicians over the years have liked and continued to use 360 degrees is that 360 is the smallest number that can be evenly divided by all numbers up to 10, aside from 7. Thus, one can break up the circle into 1, 2, 3, 4, 5, 6, 8, 9, or 10 equal parts, and still have a whole number of degrees in each part! Note that 360 is also evenly divisible by 12, the number of months in a year, and by 24, the number of hours in a day. These facts all make the number 360 very useful. The circle has been divided into 1, 2, 3, 4, 5, 6, 8, 9 10, and 12 parts below. What are the degree measures of the resulting angles? Write the degree measures down near the indicated angles.
Degree Measures of Different Angles
How can we measure other angles in degrees? It is for this purpose that we have a tool called a protractor that is either a circle divided into 360 equal parts as shown in Figure, or a half circle divided into 180 equal parts.
Unlabelled Protractor
Here is a protractor. Do you see the straight angle at the center divided into 180 units of 1 degree? Only part of the lines dividing the straight angle are visible, though! Starting from the marking on the rightmost point of the base, there is a long mark for every 10°. From every such long mark, there is a medium-sized mark after 5°.
Labelled Protractor
This is a protractor that you find in your geometry box. It would appear similar to the protractor above except that there are numbers written on it. Will these make it easier to read the angles?
There are two sets of numbers on the protractor: one increasing from right to left and the other increasing from left to right. Why does it include two sets of numbers?
Name the different angles in the figure and write their measures.
Did you include angles such as ∠TOQ?
Which set of markings did you use – inner or outer?
What is the measure of ∠TOS?
Can you use the numbers marked to find the angle without counting the number of markings?
Here, OT and OS pass through the numbers 20 and 55 on the outer scale. How many units of 1 degree are contained between these two arms?
Can subtraction be used here?
How can we measure angles directly without having to subtract?
Place the protractor so the center is on the vertex of the angle. Align the protractor so that one of the arms passes through the 0º mark as in the picture below.
What is the degree measure of ∠AOB?
Make your Protractor!
You may have wondered how the different equally spaced markings are made on a protractor. We will now see how we can make some of them!
1. Draw a circle of a convenient radius on a sheet of paper. Cut out the circle (Figure). A circle or one full turn is 360°.
2. Fold the circle to get two halves and cut it through the crease to get a semicircle. Write ‘0°’ in the bottom right corner of the semi-circle.
3. Fold the semi-circular sheet in half as shown in Figure to form a quarter circle.
4. Fold the sheet again as shown in Figures:
When folded, this is \(\frac {1}{8}\) of the circle, or \(\frac {1}{8}\) of a turn, or \(\frac {1}{8}\) of 360°, or \(\frac {1}{4}\) of 180° or \(\frac {1}{2}\) of 90° = _________.
The new creases formed give us measures of 45° and 180° – 45° = 135° as shown. Write 45° and 135° at the correct places on the new creases along the edge of the semicircle.
5. Continuing with another half fold as shown in Figure, we get an angle of measure ______.
6. Unfold and mark the creases as OB, OC, …, etc., as shown in Figures.
Angle Bisector
At each step, we folded in halves. This process of getting half of a given angle is called bisecting the angle. The line that bisects a given angle is called the angle bisector of the angle.
Mind the Mistake, Mend the Mistake!
A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor discuss how the reading could have been made, and think about how it can be corrected.
Drawing Angles Class 6 Notes
Vidya wants to draw a 30° angle and name it ∠TIN using a protractor. In ∠TIN will be the vertex, and IT and IN will be the arms of the angle. Keeping one arm, say IN, as the reference (base), the other arm IT should take a turn of 30°.
Step 1: We begin with the base and draw \(\overrightarrow{\mathrm{IN}}\):
Step 2: We will place the center point of the protractor on I and align IN to the 0 lines.
Step 3: Now, starting from 0, count your degrees (0, 10, 30) up to 30 on the protractor. Mark point T at the label 30°.
Step 4: Using a ruler join the point I and T.
∠TIN = 30° is the required angle.
Let’s Play a Game #1
This is an angle-guessing game! Play this game with your classmates by making two teams, Team 1 and Team 2. Here are the instructions and rules for the game:
- Team 1 secretly chooses an angle measure, for example, 49°, and makes an angle with that measure using a protractor without Team 2 being able to see it.
- Team 2 now gets to look at the angle. They have to quickly discuss and guess the number of degrees in the angle (without using a protractor!).
- Team 1 now demonstrates the true measure of the angle with a protractor.
- Team 2 scores the number of points, which is the absolute difference in degrees between their guess and the correct measure.
- For example, if Team 2 guesses 39°, they score 10 points (49° – 39°).
- Each team gets five turns. The winner is the team with the lowest score!
Let’s Play a Game #2
We now change the rules of the game a bit. Play this game with your classmates by again making two teams, Team 1 and Team 2. Here are the instructions and rules:
- Team 1 announces to all, an angle measure, e.g., 34°.
- A player from Team 2 must draw that angle on the board without using a protractor. Other members of Team 2 can help the player by speaking words like ‘Make it bigger!’ or ‘Make it smaller!’.
- A player from Team 1 measures the angle with a protractor for all to see.
- Team 2 scores the number of points which is the absolute difference in degrees between Team 2’s angle size and the intended angle size.
- For example, if the player’s angle from Team 2 is measured to be 25°, then Team 2 scores 9 points (34° – 25°).
- Each team gets five turns. The winner is again the team with the lowest score.
Types of Angles and their Measures
We have read about different types of angles in this chapter. We have seen that a straight angle is 180° and a right angle is 90°. How can other types of angles — acute and obtuse — be described in terms of their degree measures?
Acute Angle:
Angles that are smaller than the right angle, i.e., less than 90° and greater than 0°, are called acute angles.
Obtuse Angle:
Angles that are greater than the right angle and less than the straight angle, i.e., greater than 90° and less than 180°, are called obtuse angles.
Have we covered all the possible measures that an angle can take? Here is another type of angle.
Reflex Angle:
Angles that are greater than the straight angle and less than the whole angle, i.e., greater than 180° and less than 360°, are called reflex angles.
