Lines and Angles

Fold a piece of paper and unfold it. Do you see a crease?

Mark any two points A and B on a sheet of paper. Try to connect A to B by various routes (Fig. 2.1). What is the shortest route from A to B?

2.7 Making Rotating Arms

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.

Now, shuffle and mix up all the rotating arms. Can you identify which of the rotating arms will pass through the slit?

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 crease which is straight.

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 $\angle AOB$ 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.

Fold the semi-circular sheet in half as shown in Fig. 2.15 to form a quarter circle.

Fold the sheet again as shown in Figs. 2.16 and 2.17:

When folded, this is $\frac{1}{8}$ of the circle, or $\frac{1}{8}$ of a turn, or $\frac{1}{8}$ of $360^\circ$, or $\frac{1}{4}$ of $180^\circ$ or $\frac{1}{2}$ of $90^\circ$ = ________.

The new creases formed give us measures of $45^\circ$ and $180^\circ - 45^\circ = 135^\circ$ as shown. Write $45^\circ$ and $135^\circ$ at the correct places on the new creases along the edge of the semicircle.

Continuing with another half fold as shown in Fig. 2.18, we get an angle of measure ________

Unfold and mark the creases as OB, OC, ..., etc., as shown in Fig. 2.19 and Fig. 2.20.

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.

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.

• $\angle U = 35^\circ$
• $\angle V = 80^\circ$
• $\angle W = 70^\circ$
• $\angle X = 150^\circ$
• $\angle Y = 120^\circ$
• $\angle Z = 85^\circ$

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 choose 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 that is the absolute difference in degrees between their guess and the correct measure. For example, if Team 2 guesses 39°, then 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 that is the absolute difference in degrees between Team 2’s angle size and the intended angle size. For example, if 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.

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

Can you help Rihan and Sheetal find their answers?

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?

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

Draw a rough figure and write labels appropriately to illustrate each of the following:

a. $\overleftrightarrow{\text{OP}}$ and $\overleftrightarrow{\text{OQ}}$ meet at O.

b. $\overleftrightarrow{\text{XY}}$ and $\overleftrightarrow{\text{PQ}}$ intersect at point M.

c. Line $l$ contains points E and F but not point D.

d. Point P lies on $\overline{\text{AB}}$.

In Fig. 2.6, name:

a. Five points

b. A line

c. Four rays

d. Five line segments

Here is a ray $\overrightarrow{OA}$ (Fig. 2.7). It starts at O and passes through the point A. It also passes through the point B.

a. Can you also name it as $\overrightarrow{OB}$? Why?

b. Can we write $\overrightarrow{OA}$ as $\overrightarrow{AO}$? Why or why not?

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.

Draw and label an angle with arms $ST$ and $SR$.

Explain why $\angle APB$ cannot be labelled as $\angle P$.

Name the angles marked in the given figure.

Mark any three points on your paper that are not on one line. Label them A, B, C. 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 A, B, C? Write them down, and mark each of them with a curve as in Fig. 2.9.

Now mark any four points on your paper so that no three of them are on one line. Label them A, B, C, D. 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 A, B, C, D? Write them all down, and mark each of them with a curve as in Fig. 2.9.

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?

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

a. $\angle AOB$ or $\angle XOY$ b. $\angle AOB$ or $\angle XOB$ c. $\angle XOB$ or $\angle XOC$

Discuss with your friends on how you decided which one is greater.

Which angle is greater: $\angle XOY$ or $\angle AOB$? Give reasons.

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

Join A 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?

Now join A 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?

Hint: Extend the line further as shown in the figure below. To get a right angle at A, we need to draw a line through it that divides the straight angle CAB into two equal parts.

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.

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.

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

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

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?

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?

Write the measures of the following angles:

(a) $\angle\text{KAL}$

Notice that the vertex of this angle coincides with the centre of the protractor. So the number of units of 1 degree angle between KA and AL gives the measure of $\angle\text{KAL}$. By counting, we get: $\angle\text{KAL} = 30^\circ$

Making use of the medium-sized and large-sized marks, is it possible to count the number of units in 5s or 10s?

(b) $\angle\text{WAL}$

(c) $\angle\text{TAK}$

Name the different angles in the figure and write their measures.

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

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

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

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

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

Find the degree measures of $\angle BXE$, $\angle CXE$, $\angle AXB$ and $\angle BXC$.

Find the degree measures of $\angle PQR$, $\angle PQS$ and $\angle PQT$.

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 30°. Why?

b. What will be the angle at 2 o'clock? And at 4 o'clock? 6 o'clock?

c. Explore other angles made by the hands of a clock.

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?

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?

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?

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?

Hint: Observe the horizontal line touching the insects.

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.

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

a. 110° b. 40° c. 75° d. 112° e. 134°

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.

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

a. An acute angle

b. An obtuse angle

c. A reflex angle

Mark the intended angles with curves to specify the angles. One has been done for you.

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

a. $\angle\text{PTR}$ b. $\angle\text{PTQ}$ c. $\angle\text{PTW}$ d. $\angle\text{WTP}$

Draw angles with the following degree measures: a. $140^\circ$ b. $82^\circ$ c. $195^\circ$ d. $70^\circ$ e. $35^\circ$

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

a. b. c. d. e. f.

Classify these angles as acute, right, obtuse or reflex angles.

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

Draw the letter 'M' such that the angles on the sides are $40^\circ$ each and the angle in the middle is $60^\circ$.

Draw the letter 'Y' such that the three angles formed are $150^\circ$, $60^\circ$ and $150^\circ$.

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?

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?

Do you think you can draw a complete picture of a line? No. Why?

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?

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 vertex of the angle.

Look at the pictures of spectacles, wallet and other common objects. Identify the angles in them by marking out their arms and vertices.

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.

Suppose we have a transparent circle which can be moved and placed on figures. Let us place the circular paper on the angle made by the first crane. The circle is placed in such a way that its centre is on the vertex of the angle. Let us mark the points A and B on the edge circle at the points where the arms of the angle pass through the circle.

Q. Can we use this to find out if this angle is greater than, or equal to or smaller than the angle made by the second crane?

Let us place the circle on the angle made by the second crane so that the vertex coincides with the centre of the circle and one of the arms passes through OA.

Q. 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 Fig. 2.10 with each other.

Is it possible to draw $\overrightarrow{OC}$ such that the two angles are equal to each other in size?

If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?

Angles are classified in three groups as shown below. Right angles are shown in the second group. What could be the common feature of the other two groups?

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°.

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.

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?

What is the degree measure of $\angle$AOB?

In Fig. 2.19, we have $\angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOG = \angle GOH = \angle HOI = \_\_\_\_\_\_$. Why?

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! We will come back to why this happens in a later year.

Let's Explore

In this figure, $\angle\text{TER} = 80^\circ$. What is the measure of $\angle\text{BET}$? What is the measure of $\angle\text{SET}$?

Hint: Observe that $\angle\text{REB}$ is a straight angle. Hence, the degree measure of $\angle\text{REB} = 180^\circ$ of which $80^\circ$ is covered by $\angle\text{TER}$. A similar argument can be applied to find the measure of $\angle\text{SET}$.