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Lines and Angles — Class 6 Mathematics

Ganita Prakash · Chapter 2

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Chapter summary

This chapter builds the basic vocabulary of plane geometry. A point determines a precise location and has no length, breadth or height; it is named with a capital letter such as . A line segment is the shortest path between two end points and , written . Extending a segment endlessly in both directions gives a line, and extending it in only one direction gives a ray, which has one starting point. The chapter then introduces angles: an angle is formed by two rays sharing a common end point called the vertex, and the two rays are its arms. Angles are compared and measured in degrees using a protractor, where a full turn is . Special angles are named \u2014 a right angle is , a straight angle is , and a complete angle is . Angles smaller than are acute, those between and are obtuse, and those between and are reflex. Students learn to draw and measure angles of given sizes, building the foundation for constructing and analysing shapes in later chapters.

PointLine SegmentLine and RayAngle, Vertex and ArmsMeasuring and Drawing AnglesTypes of Angles

Key ideas & terms

Point: A precise location with no length, breadth or height, named by a capital letter such as .
Line segment: The shortest path between two end points and , denoted .
Ray: A part of a line that starts at one point and extends endlessly in one direction.
Angle: A figure formed by two rays sharing a common end point called the vertex.
Right angle: An angle that measures exactly ; a straight angle measures .
Protractor: An instrument marked in degrees used to measure and draw angles, with a full turn of .

Extra questions & answers

What is a point in geometry?

A point determines a precise location and has no length, breadth or height; it is shown as a tiny dot and named by a capital letter.

What is a line segment?

It is the shortest path between two end points and , written or .

How does a ray differ from a line?

A ray starts at one fixed point and goes endlessly in one direction, while a line extends endlessly in both directions.

What forms an angle?

An angle is formed by two rays (its arms) that share a common end point called the vertex.

What is the measure of a right angle?

A right angle measures exactly .

What is the measure of a straight angle?

A straight angle measures , which is half a complete turn.

What is an acute angle?

An angle whose measure is less than .

What is an obtuse angle?

An angle whose measure is greater than but less than .

What is a reflex angle?

An angle whose measure is greater than but less than .

What instrument is used to measure angles and what is its unit?

A protractor is used, and angles are measured in degrees , with a complete angle equal to .

Figure it Out

1. Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point?

Infinitely many lines can be drawn through a single point. If you rotate a ruler around that point you get a new line in every direction, so there is no limit.

1. Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points?

Only one unique line can pass through two distinct points. Any two points determine exactly one straight line.

2. Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?

The line segments are , , and . Points and lie on exactly one segment each. Points , and are the joining points where two segments meet, so they lie on two segments each.

3. Name the rays shown in Fig. 2.5. Is the starting point of each of these rays?

The rays are , and . There is also a ray starting at going through , i.e. . is the starting point of , and but is not the starting point of .

4. Draw a rough figure and write labels appropriately to illustrate each of the following: a. and meet at .

Draw two rays starting from the same point , one going through and the other through . The common vertex is .

4. Draw a rough figure and write labels appropriately to illustrate each of the following: b. and intersect at point .

Draw two lines crossing each other. Label one line and the other . Mark their intersection point as .

4. Draw a rough figure and write labels appropriately to illustrate each of the following: c. Line contains points and but not point .

Draw a straight line and label it . Mark points and on the line and mark point away from the line.

4. Draw a rough figure and write labels appropriately to illustrate each of the following: d. Point lies on .

Draw a line segment with endpoints and and mark a point somewhere between and on the segment.

5. In Fig. 2.6, name: a. Five points

Five points are , , , and .

5. In Fig. 2.6, name: b. A line

The line is which can also be written as , or .

5. In Fig. 2.6, name: c. Four rays

Four rays are , , and . The upward vertical ray from is also a ray.

5. In Fig. 2.6, name: d. Five line segments

Five line segments are , , , and . Any part between two named points counts as a segment.

6. Here is a ray (Fig. 2.7). It starts at and passes through the point . It also passes through the point . a. Can you also name it as ? Why?

Yes, and denote the same ray because both start at and go in the same direction, and lies on the path of .

6. Here is a ray (Fig. 2.7). It starts at and passes through the point . It also passes through the point . b. Can we write as ? Why or why not?

No, starts at and goes towards , while starts at and goes towards . They have opposite directions and different starting points, so they are different rays.

Figure it Out

1. Can you find the angles in the given pictures? Draw the rays forming any one of the angles and name the vertex of the angle.

Yes, every corner where two edges meet forms an angle. For example in the bicycle picture is formed by rays and with vertex at . The same idea can be applied to the other pictures.

2. Draw and label an angle with arms and .

Draw a point as the vertex. From draw a ray through and another ray through . The angle formed is or with vertex .

3. Explain why cannot be labelled as .

At point there are three rays: , and . So there are three different angles at : , and . Writing just would not tell which one is meant, so we must use three letters.

4. Name the angles marked in the given figure.

The marked angles are and (the small curve shows and the larger curve shows which is the sum of the two).

5. Mark any three points on your paper that are not on one line. Label them , , . Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using , , ? Write them down, and mark each of them with a curve as in Fig. 2.9.

You get three lines: , and . Using , , you can name three angles: (or ), (or ) and (or ). They are the three interior angles of triangle .

6. Now mark any four points on your paper so that no three of them are on one line. Label them , , , . Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using , , , ? Write them all down, and mark each of them with a curve as in Fig. 2.9.

You get six lines: , , , , and . With four points many angles are formed. At each vertex there are three rays; for example at : , and . In total there are twelve small angles such as , , , , , , , , , , and .

Figure it Out

1. Fold a rectangular sheet of paper, then draw a line along the fold created. Name and compare the angles formed between the fold and the sides of the paper. Make different angles by folding a rectangular sheet of paper and compare the angles. Which is the largest and smallest angle you made?

When you fold a paper the crease makes four angles with the sides. If the fold is straight across, all four are right angles and equal. If the fold is slanted, two opposite angles are acute and two are obtuse, but each pair is equal. The largest is the obtuse one closest to and the smallest is the acute one closest to in your trials.

2. In each case, determine which angle is greater and why. a. or

is greater because is contained inside . , so it is larger.

2. In each case, determine which angle is greater and why. b. or

is greater because is a part of with the same vertex and one common arm.

2. In each case, determine which angle is greater and why. c. or

Neither is greater; they are equal if points and lie on the same ray from . In Fig. 2.10, and are the same ray, so .

3. Which angle is greater: or ? Give reasons.

By just looking you cannot be sure because the arms have different lengths and orientations. You need to superimpose the angles or measure with a protractor. After superimposition is found to be larger than in the given figure.

Figure it Out

1. How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?

A typical rectangular window has four right angles at its corners. A classroom usually has several windows, so multiply by the number of windows. Many other objects like doors, blackboards, books, desks and floor tiles also contain right angles.

2. Join to other grid points in the figure by a straight line to get a straight angle. What are all the different ways of doing it?

A straight angle is . Starting at you need to extend in the opposite direction to make a straight line . On the dot grid there is only one line through that continues straight to the opposite grid point .

3. Now join to other grid points in the figure by a straight line to get a right angle. What are all the different ways of doing it?

To make a right angle at you need a line through perpendicular to . On the grid the points directly above or below give a line that divides the straight angle into two angles.

4. Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease. a. How many right angles do you have now? Justify why the angles are exact right angles.

You now have four right angles. Folding so that the first crease falls exactly on itself makes the second crease divide the straight angle into two equal halves, i.e. . All four angles around the intersection are then .

4. Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease. b. Describe how you folded the paper so that any other person who doesn’t know the process can simply follow your description to get the right angle.

Make a slanted fold and keep it folded. Fold again so that the two parts of the first crease lie exactly on top of each other. Press firmly and unfold. The second crease is perpendicular to the first.

Figure it Out

1. Identify acute, right, obtuse and straight angles in the previous figures.

Acute angles are smaller than , right is exactly , obtuse is between and , straight is . In the fold examples the equal halves are right angles, slanted folds give one acute and one obtuse pair, and the unfolded edge is a straight angle.

2. Make a few acute angles and a few obtuse angles. Draw them in different orientations.

Draw an angle of about or for acute and about or for obtuse. Rotate the whole figure so that the vertex points up, down or sideways to see different orientations; the measure does not change.

3. Do you know what the words acute and obtuse mean? Acute means sharp and obtuse means blunt. Why do you think these words have been chosen?

An acute angle is narrow and looks sharp like a sharp pencil tip, while an obtuse angle is wide and looks blunt. The names describe how pointed the opening looks.

4. Find out the number of acute angles in each of the figures below. What will be the next figure and how many acute angles will it have? Do you notice any pattern in the numbers?

The first figure has 3 acute angles, the second has 12, the third has 21. The pattern increases by 9 each time. The next figure will have acute angles. In general the number is for figure .

Figure it Out

1. Write the measures of the following angles: a.

. Between and there are 30 unit parts of .

1. Write the measures of the following angles: b.

. It is more than .

1. Write the measures of the following angles: c.

. Counting from to gives 120 units.

Figure it Out

1. Find the degree measures of the following angles using your protractor.

Using a protractor: the first angle measures about , the second about , the third about . Place the centre on the vertex and count from .

2. Find the degree measures of different angles in your classroom using your protractor.

Corners of books, desks and windows measure . The angle between an open door and the wall can be e.g. or depending on how far it is opened.

3. Find the degree measures for the angles given below. Check if your paper protractor can be used here!

The small angle is about and the larger one about . The simple paper protractor with marks at every cannot measure these exactly, so a standard protractor is needed.

4. How can you find the degree measure of the angle given below using a protractor?

The marked reflex angle is greater than . Measure the smaller unmarked angle (about ) and subtract from : .

5. Measure and write the degree measures for each of the following angles: a.

- acute

5. Measure and write the degree measures for each of the following angles: b.

- obtuse

5. Measure and write the degree measures for each of the following angles: c.

- acute

5. Measure and write the degree measures for each of the following angles: d.

- obtuse

5. Measure and write the degree measures for each of the following angles: e.

- obtuse

5. Measure and write the degree measures for each of the following angles: f.

- acute

6. Find the degree measures of , , and .

, , and . Align the base with and read the scale.

7. Find the degree measures of , and .

, and .

8. Make the paper craft as per the given instructions. Then, unfold and open the paper fully. Draw lines on the creases made and measure the angles formed.

After the sequence of folds and unfolding, the creases meet at a point and form eight angles each of or multiples of . Measuring confirms they add up to .

9. Measure all three angles of the triangle shown in Fig. 2.21 (a), and write the measures down near the respective angles. Now add up the three measures. What do you get? Do the same for the triangles in Fig. 2.21 (b) and (c). Try it for other triangles as well, and then make a conjecture for what happens in general!

Each triangle's three interior angles add up to . For example a triangle might have or . The conjecture is that the sum of the interior angles of any triangle is .

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 and discuss how the reading could have been made and think how it can be corrected.

Incorrect: the vertex is not at the centre and one arm is not at . Correction: place centre on vertex and align one arm with , then read the scale where the other arm crosses.

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

Incorrect: the protractor is reversed and the base line is not along the arm. Correction: rotate protractor so that its straight edge coincides with one arm starting at .

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

Incorrect: the centre is away from the vertex. Correction: move protractor so its centre mark is exactly on vertex .

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

Incorrect: reading the inner scale instead of outer (or vice versa). Correction: note which arm is at and read the matching scale, giving about not if misread.

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

Incorrect: the arm does not pass through and centre is misaligned. Correction: align one arm to with centre on vertex, then read correct scale.

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

Incorrect: protractor is tilted and reading is off by -. Correction: keep base horizontal and read exactly at the mark, which should be when correctly placed.

Figure it Out

1. Angles in a clock: a. The hands of a clock make different angles at different times. At 1 o’clock, the angle between the hands is . Why?

A full circle is divided into 12 hours, so each hour mark is . At 1 o'clock the hands are one hour apart, so .

1. Angles in a clock: b. What will be the angle at 2 o’clock? And at 4 o’clock? 6 o’clock?

At 2 o'clock , at 4 o'clock , at 6 o'clock which is a straight angle.

1. Angles in a clock: c. Explore other angles made by the hands of a clock.

At 3 o'clock (right angle), at 9 o'clock reflex or the smaller side, at 5 o'clock etc. Every hour adds .

2. The angle of a door: Is it possible to express the amount by which a door is opened using an angle? What will be the vertex of the angle and what will be the arms of the angle?

Yes. The vertex is the hinge where the door joins the wall. One arm is along the wall and the other arm is along the edge of the door.

3. Vidya is enjoying her time on the swing. She notices that the greater the angle with which she starts the swinging, the greater is the speed she achieves on her swing. But where is the angle? Are you able to see any angle?

The angle is between the rope's starting position (hanging straight down) and its extreme lifted position. The vertex is at the swing's pivot at the top, with arms along the rope in the two positions.

4. Here is a toy with slanting slabs attached to its sides; the greater the angles or slopes of the slabs, the faster the balls roll. Can angles be used to describe the slopes of the slabs? What are the arms of each angle? Which arm is visible and which is not?

Yes, the steeper slab makes a larger acute angle with the horizontal. One arm is the horizontal table top (often invisible as a reference line) and the other arm is the edge of the sloping slab which is visible.

5. Observe the images below where there is an insect and its rotated version. Can angles be used to describe the amount of rotation? How? What will be the arms of the angle and the vertex?

Yes. Fix a reference horizontal line through the insect's centre. The angle between the original orientation line and the rotated image's line gives the rotation amount. The vertex is the centre of rotation.

Figure it Out

1. In Fig. 2.23, list all the angles possible. Did you find them all? Now, guess the measures of all the angles. Then, measure the angles with a protractor. Record all your numbers in a table. See how close your guesses are to the actual measures.

Possible angles include , , , , , , , etc. Measure each with a protractor and compare with your guess; most guesses will be within - after practice.

2. Use a protractor to draw angles having the following degree measures: a.

Draw a base ray, place protractor centre on the vertex, mark from and join. This is an obtuse angle.

2. Use a protractor to draw angles having the following degree measures: b.

Mark from the base line; this is an acute angle.

2. Use a protractor to draw angles having the following degree measures: c.

Mark from the base line; acute.

2. Use a protractor to draw angles having the following degree measures: d.

Mark from the base line; obtuse.

2. Use a protractor to draw angles having the following degree measures: e.

Mark from the base line; obtuse.

3. Draw an angle whose degree measure is the same as the angle given below: Also, write down the steps you followed to draw the angle.

Measure with a protractor (say it is about ). Then on a new point draw a base ray, place protractor centre on the vertex, mark the same number of degrees and join to get a congruent copy.

Figure it Out

1. In each of the below grids, join to other grid points in the figure by a straight line to get: a. An acute angle

Join to a grid point close to so that the angle between and the new line is less than , e.g. or .

1. In each of the below grids, join to other grid points in the figure by a straight line to get: b. An obtuse angle

Join to a grid point on the opposite side so the angle exceeds but is less than , e.g. .

1. In each of the below grids, join to other grid points in the figure by a straight line to get: c. A reflex angle

The larger side of the same lines forms a reflex angle; choose the greater rotation around (more than and less than ), e.g. .

2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. a.

, so it is acute.

2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. b.

, acute.

2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. c.

, obtuse.

2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. d.

is the reflex angle at going the other way, about (since ), so reflex.

Let’s Explore

In this figure, . What is the measure of ? What is the measure of ?

is a straight angle . So . is complementary to if is ; then (or ).

Figure it Out

1. Draw angles with the following degree measures: a.

Draw a base ray, align protractor at , mark and join; obtuse.

1. Draw angles with the following degree measures: b.

Mark ; acute.

1. Draw angles with the following degree measures: c.

Mark the smaller on the other side (since ) or mark going the long way; reflex.

1. Draw angles with the following degree measures: d.

Mark ; acute.

1. Draw angles with the following degree measures: e.

Mark ; acute.

2. Estimate the size of each angle and then measure it with a protractor: a.

Estimated about , measured about , acute.

2. Estimate the size of each angle and then measure it with a protractor: b.

Estimated about , measured about , obtuse (just over right).

2. Estimate the size of each angle and then measure it with a protractor: c.

Estimated about , measured about , obtuse.

2. Estimate the size of each angle and then measure it with a protractor: d.

Estimated about , measured about , acute.

2. Estimate the size of each angle and then measure it with a protractor: e.

Estimated about , measured about , obtuse.

2. Estimate the size of each angle and then measure it with a protractor: f.

Estimated about , measured about , reflex.

3. Make any figure with three acute angles, one right angle and two obtuse angles.

Draw an irregular hexagon or a house shape where three corners are acute (e.g. ), one corner is and two corners are each. Label each angle.

4. Draw the letter ‘M’ such that the angles on the sides are each and the angle in the middle is .

Draw a letter M with two outer legs slanting outward making with the vertical and the middle V making at its bottom point.

5. Draw the letter ‘Y’ such that the three angles formed are , and .

Draw a Y with a vertical stem. The two upper arms form between them, and each arm makes with the stem on its side (since around the junction).

6. The Ashoka Chakra has 24 spokes. What is the degree measure of the angle between two spokes next to each other? What is the largest acute angle formed between two spokes?

Full circle divided by 24 gives between adjacent spokes. Multiples of are possible; the largest acute less than is (6 spokes would be which is not acute).

7. Puzzle: I am an acute angle. If you double my measure, you get an acute angle. If you triple my measure, you will get an acute angle again. If you quadruple (four times) my measure, you will get an acute angle yet again! But if you multiply my measure by 5, you will get an obtuse angle measure. What are the possibilities for my measure?

Let the angle be with , , , , and and . Combining gives . So integer possibilities are , , and .