Welcome dear students! Today we are going to learn about Lines and Angles from Class 6 Maths. In this chapter, we will explore some of the most basic ideas of geometry including points, lines, rays, line segments and angles. These ideas form the building blocks of plane geometry, and will help us in understanding more advanced topics in geometry such as the construction and analysis of different shapes.
Let us start with section two point one, Point. 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 the tip of a compass, the sharpened end of a pencil, and the pointed end of a needle. 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.
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Next, we move to section two point two, Line Segment. 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. What is the shortest route from A to B? This shortest path from Point A to Point B, including A and B, is called the line segment from A to B. It is denoted by either AB or BA. The points A and B are called the end points of the line segment AB.
Now, let us look at section two point three, Line. Imagine that the line segment from A to B, that is AB, is extended beyond A in one direction and beyond B in the other direction without any end. This is a model for a line. Do you think you can draw a complete picture of a line? No. Why? Because it goes on forever. A line through two points A and B is written as AB with a double-headed arrow above it. 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.
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Let us proceed to section two point four, Ray. 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: a beam of light from a lighthouse, a ray of light from a torch, and sun rays. Look at the diagram 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 AP with an arrow above it pointing from A to P.
Now, let us work through the Figure it Out questions for this section. Question one: Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point? Infinitely many lines can pass through a single point. Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points? Exactly one unique line can pass through two distinct points. Question two asks to name the line segments in Figure two point four. The figure shows points L, M, P, Q, and R connected by line segments. The line segments are LM, MP, PQ, and QR. The points on exactly one line segment are L and R. The points on two line segments are M, P, and Q.
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Question three asks to name the rays shown in Figure two point five and check if T is the starting point of each. The figure shows point T with two rays going outwards. One ray goes through point A, and the other goes through points N and B. The rays are named TA and TN, or you could also call the second ray TB since N and B lie on the same path. Yes, T is the starting point of each of these rays. Question four asks to draw rough figures and write labels to illustrate statements. For part a, draw two lines OP and OQ meeting at point O. For part b, draw lines XY and PQ crossing at a point labeled M. For part c, draw a line l with points E and F on it, and a point D somewhere away from the line. For part d, draw a line segment AB and place a point P directly on it. Question five asks to name elements in Figure two point six. The figure shows point O with rays going through B, C, upwards, and through E and D downwards. The five points are O, B, C, D, and E. A line can be named as the line passing through D and E. Four rays are OB, OC, the upward ray from O, and OE. Five line segments can be named using the marked points: OB, OC, OD, OE, and DE. Question six asks about ray OA starting at O and passing through B and A. Can you also name it OB? Yes, because B lies on the same ray starting from O. Can we write OA as AO? No, because AO would start at A and go through O, which is the opposite direction.
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Now we reach section two point five, Angle. An angle is formed by two rays having a common starting point. Here is an angle formed by rays BD and BE where B is the common starting point. The point B is called the vertex of the angle, and the rays BD and 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 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.
Let us observe Vidya opening her 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 one or the angle in Case two? 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 two is greater as in this case she needs to rotate the cover more. Similarly, the angle in Case three is even larger than that of Case two, because there is even more rotation, and Cases four, five, and six are successively larger angles with more and more rotation. The size of an angle is the amount of rotation or turn that is needed about the vertex to move the first ray to the second ray.
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Let us 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. The arms are the two legs of the compass. A pair of scissors has two blades. When we open them to cut something, the blades form an angle. The vertex is the pivot point where the blades are joined, and the arms are the cutting edges. Look at the pictures of spectacles, wallet 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 with respect to the other?
Let us solve the Figure it Out questions for this section. Question one asks to find angles in given pictures and draw rays forming any one angle, naming the vertex. For example, in a picture of an open book, you can identify the vertex where the pages meet the binding, and the arms along the top edges of the pages. Question two asks to draw and label an angle with arms ST and SR. Draw a point S as the vertex. Draw a ray going to T and another ray going to R. The angle is ∠TSR or ∠RST. Question three asks to explain why ∠APC cannot be labelled as ∠P. The figure shows point P with three rays going to A, B, and C. We cannot label it as ∠P because there are multiple angles at point P, specifically ∠APB, ∠BPC, and ∠APC. Using just ∠P would be ambiguous.
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Now for Math Talk question four, name the angles marked in the given figure. The figure shows point T with three rays TP, TQ, and TR. There are two marked angles. The first is between TP and TQ, named ∠PTQ. The second is between TQ and TR, named ∠QTR. Question five asks to mark any three points not on one line, label them A, B, C, draw all possible lines through pairs, and name angles. You will get three lines: AB, BC, and AC. You can name three angles: ∠ABC, ∠BCA, and ∠CAB. Question six asks to do the same with four points A, B, C, D where no three are on one line. You will get six lines: AB, AC, AD, BC, BD, CD. You can name twelve angles by taking each point as a vertex and combining the lines meeting there, such as ∠BAC, ∠CAD, ∠DAB, ∠ABC, ∠CBD, ∠DBA, ∠BCD, ∠DCA, ∠ACB, ∠CDA, ∠ADB, and ∠BDC.
Let us move to section two point six, Comparing Angles. Look at animals opening their mouths. Do you see any angles here? 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 from smallest to largest? Yes, by observing the amount of opening. Is it always easy to compare two angles? Not always by just looking. We can compare them 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 angle ABC and angle PQR superimposed with vertex B overlapping Q. It is now clear that angle PQR is larger than angle ABC.
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Now consider equal angles. Consider angle AOB and angle XOY. Which is greater? The corners of both of these angles match and the arms overlap with each other, i.e., OA overlaps OX and OB overlaps OY. So, the angles are equal in size. The reason these angles are considered to be equal in size is because when we visualise each of these angles as being formed out of rotation, we can see that there is an equal amount of rotation needed to move ray OB to ray OA and ray OY to ray 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.
Let us solve the Figure it Out questions for comparing angles. Question one: Fold a rectangular sheet of paper, draw a line along the fold, name and compare the angles formed between the fold and the sides. The fold creates a straight line. The angles formed with the adjacent sides are right angles, so they are equal. Make different angles by folding and compare. The largest angle you can make by folding a flat paper once is a straight angle, and the smallest depends on how close the fold is to the edge, but typically acute. Question two asks to determine which angle is greater in given cases. For angle AOB or angle XOY, compare by superimposition or by checking the opening. For angle AOB or angle XOB, if X is inside AOB, then AOB is greater. For angle XOB or angle XOC, if C is further out than B, XOC is greater. Question three asks which is greater, angle XOY or angle AOB. By placing a transparent circle with center on the vertex and marking where arms hit the edge, you can compare the arc lengths. The one with the wider arc is greater.
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Comparing angles without superimposition: Two cranes are arguing about who can open their mouth wider. Let us first draw their angles. How do we know which one is bigger? One could trace and superimpose. But can we do it without superimposition? Suppose we have a transparent circle. Place the circular paper on the first angle so its centre is on the vertex. Mark points A and B on the edge where the arms pass through. Place it on the second angle so the vertex coincides with the centre and one arm passes through OA. If the second arm falls inside the first arc, the first angle is bigger. If it falls outside, the second is bigger. If it matches, they are equal. Which crane was making the bigger angle? The one whose arm falls further along the circle's edge.
Now, section two point seven, Making Rotating Arms. Let us make rotating arms using two paper straws and a paper clip. Step one: Take two paper straws and a paper clip. Step two: Insert the straws into the arms of the paper clip. Step three: Your rotating arm is ready! Make several rotating arms with different angles. Arrange them from smallest to largest by comparing and using superimposition. For the Passing through a slit activity, take a cardboard and make an angle-shaped slit by tracing and cutting out one rotating arm. Shuffle the rotating arms. Identify which will pass through the slit by placing each over it. If the slit angle is greater than the arms' angle, the arms will not go through. If the slit angle is less than the arms' angle, the arms will not go through. If the slit angle is equal to the arms' angle, the arms will go through. Only the pair where the angle is equal passes through. Note that the possibility depends only on the angle, not on their lengths. Challenge: Reduce the angle by pushing the arms closer. The angle is still the same if you just slide them without changing the opening. This teaches that moving the arms without changing the opening keeps the angle constant.
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Let us proceed to section two point eight, Special Types of Angles. Let us go back to Vidya's notebook. She makes a full turn of the cover when holding the book. She makes a half turn when opening it on a table. In this case, the arms lie in a straight line. Such an angle is called a straight angle. Consider a straight angle AOB. Observe that any ray OC divides it into two angles, angle AOC and angle COB. Is it possible to draw OC such that the two angles are equal? Let us explore. Take a rectangular piece of paper and mark straight angle AOB. Fold the paper such that OB overlaps with OA. Observe the crease and the two angles formed. Justify why they are equal. Folding superimposes one half onto the other, proving equality. Each of these equal angles are called right angles. So, a straight angle contains two right angles. If a straight angle is formed by half of a full turn, a right angle is formed by a quarter of a full turn. 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.
Let us solve the Figure it Out questions here. Question one: How many right angles do classroom windows contain? Typically four at each corner. You see other right angles in desks, doors, and books. Question two: Join A to other grid points to get a straight angle. Draw a line through A extending in opposite directions along the grid lines. Question three: Join A to get a right angle. Draw a line through A perpendicular to the given line, dividing the straight angle into two equal parts. Question four: Get a slanting crease. Fold again to get a perpendicular crease. You now have four right angles. They are exact right angles because folding aligns the edges perfectly, creating equal adjacent angles that sum to a straight angle.
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Classifying Angles: Angles are classified in three groups. Right angles are in the second group. 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. Let us solve the Figure it Out questions. Question one: Identify acute, right, obtuse and straight angles in previous figures. Look for openings smaller than L for acute, exactly L for right, between L and straight line for obtuse, and straight line for straight. Question two: Make a few acute and obtuse angles. Draw them pointing up, down, left, right. Question three: 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? Because acute angles look sharp like a needle tip, and obtuse angles look blunt or wide. Question four: Find the number of acute angles in given figures. The pattern usually increases by two or follows a sequence based on added lines. The next figure will have two more acute angles than the previous one.
Now, section two point nine, Measuring Angles. We have seen how to compare two angles. But can we actually quantify how big an angle is using a number? 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 centre 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 measure to any angle: the measure of an angle is the number of 1° unit parts it contains inside it. For example, a figure containing 30 units of 1° angle has an angle measure of 30°.
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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. The degree measures of the resulting angles are 360°, 180°, 120°, 90°, 72°, 60°, 45°, 40°, 36° and 30° respectively.
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How can we measure other angles in degrees? We use a protractor. A protractor is either a circle divided into 360 equal parts or a half circle divided into 180 equal parts. Here is an unlabelled protractor. Do you see the straight angle at the center divided into 180 units of 1 degree? Only part of the lines are visible. Starting from 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°. Let us solve the Figure it out questions. Question one: Write the measures of the following angles. For ∠KAL, the vertex coincides with the centre. By counting, we get ∠KAL = 30°. Making use of the medium and large marks, we can count in 5s or 10s. For ∠WAL, it measures 60°. For ∠TAK, it measures 120°.
Now, a labelled protractor. This is what you find in your geometry box. There are two sets of numbers: one increasing from right to left and the other increasing from left to right. Why two sets? To measure angles opening in either direction easily. Name the different angles in the figure and write their measures. The figure shows rays from O to R, S, T, U, P, Q. ∠POQ is 10°, ∠QOR is 20°, ∠ROS is 30°, ∠SOT is 40°, ∠TOU is 50°. Did you include angles such as ∠TOQ? Yes, it is 70°. Which set of markings did you use? The outer scale. What is the measure of ∠TOS? It is 90°. Can you use the numbers to find the angle without counting? Yes, by subtraction. Here, OT and OS pass through 20 and 55 on the outer scale. The difference is 55 minus 20, which equals 35°. So ∠TOS is 35°. How can we measure angles directly without subtracting? Place the protractor so the centre is on the vertex. Align one arm to the 0° mark. Then read the number where the other arm crosses. For ∠AOB, if OB is at 0° on the inner scale, and OA crosses at 80°, then ∠AOB is 80°.
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Make your own Protractor! Step one: Draw a circle, cut it out. A full turn is 360°. Step two: Fold to get two equal halves, cut to get a semicircle. Write 0° at the bottom right. The measure of half a turn is 1/2 of 360°, which is 180°. Write 180° at the bottom left. Step three: Fold the semicircle in half to form a quarter circle. The measure of a quarter circle is 1/4 of 360°, which is 90°. Or, 1/2 of 180° is 90°. Mark 90° at the top. Step four: Fold again. This is 1/8 of the circle, or 1/8 of 360°, or 1/4 of 180°, or 1/2 of 90°, which is 45°. The new creases give 45° and 180° minus 45° equals 135°. Write 45° and 135° at the correct places. Step five: Continue with another half fold. We get an angle of measure 22.5°. Step six: Unfold and mark the creases. The angles marked will be 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5°, 180°. Think! In Figure two point twenty, angle AOB equals angle BOC equals angle COD equals angle DOE equals angle EOF equals angle FOG equals angle GOH equals angle HOI equals 22.5°. Why? Because each fold bisected the previous angle, dividing the 180° semicircle into eight equal parts.
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. Identify the angle bisectors in your handmade protractor. They are the crease lines. Try to make different angles using the concept of angle bisector through paper folding.
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Let us solve the Figure it Out questions for measuring. Question one: Find degree measures using your protractor. For the first angle, measure between the rays. For the second, measure between HG and HK. For the third, measure between HI and HJ. Question two: Find degree measures of different angles in your classroom. Measure corners of desks, doors, books. Question three: Find degree measures for given angles. Check if your paper protractor can be used. Yes, align the center and base. Question four: How to find the degree measure of a reflex angle? Measure the smaller angle inside, then subtract it from 360°. Question five: Measure angles a through f. Record each. Question six: Find measures of angle BXE, angle CXE, angle AXB, angle BXC. Using the protractor diagram, BXE is 65°, CXE is 85°, AXB is 115°, BXC is 20°. Question seven: Find measures of angle PQR, angle PQS, angle PQT. Measure each from the diagram. Question eight: Make the paper craft, unfold, draw lines on creases, measure angles. You will find angles of 45° and 90°. Question nine: Measure all three angles of triangles in Figure two point twenty one and add them. You will get 180° for each triangle. The conjecture is that the sum of angles in any triangle is 180°.
Mind the Mistake, Mend the Mistake! A student measured angles incorrectly. For angle U = 35°, the protractor was not aligned with 0°. Correct it by aligning the base to 0° and reading 35°. For angle V = 80°, the vertex was not at the center. Correct by placing vertex at center. For angle W = 70°, the wrong scale was used. Correct by reading the inner scale instead of outer. For angle X = 150°, the angle is actually 30°. Correct by measuring the acute opening. For angle Y = 120°, the angle is 60°. Correct by aligning properly. For angle Z = 85°, the angle is 95°. Correct by using the correct scale.
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Let us explore angles in real life. Question one: Angles in a clock. At 1 o'clock, the angle is 30° because the clock is divided into 12 hours, and 360° divided by 12 is 30°. At 2 o'clock, it is 60°. At 4 o'clock, it is 120°. At 6 o'clock, it is 180°. Question two: The angle of a door. Yes, the vertex is the hinge, and the arms are the door edge and the door frame. Question three: Vidya on a swing. The angle is between the swing chains and the vertical resting position. Question four: Toy with slanting slabs. Yes, angles describe slopes. The arms are the slab surface and the horizontal base. The horizontal arm is visible, the slab arm is visible. Question five: Insect and rotated version. Yes, angles describe rotation. The vertex is the center of rotation. The arms are the original position and the rotated position of a reference line.
Now, section two point ten, Drawing Angles. Vidya wants to draw a 30° angle named ∠TIN. I is the vertex, IT and IN are the arms. Keeping IN as the base, IT should turn 30°. Step one: Draw base IN. Step two: Place protractor centre on I, align IN to 0° line. Step three: Count from 0 to 30 on the protractor. Mark point T at 30°. Step four: Join I and T with a ruler. ∠TIN = 30° is the required angle.
Let us play Game one. Team one chooses an angle secretly, draws it. Team two guesses the degrees. Score is absolute difference. Team with lowest score wins. Game two: Team one announces an angle. Team two draws it without protractor. Team one measures. Score is difference. Lowest score wins. These games build intuition.
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Solve Figure it Out for drawing. Question one: In Figure two point twenty three, list all angles possible. Guess measures, then measure with protractor. Record in a table. Compare guesses to actual. Question two: Use protractor to draw angles: 110°, 40°, 75°, 112°, 134°. Follow the four steps for each. Question three: Draw an angle with same measure as given angle HJI. First measure ∠HJI with protractor, say it is 50°. Then draw a new angle of 50° using the steps. Write down the steps: draw base, align protractor, mark degree, join vertex to mark.
Now, section two point eleven, Types of Angles and their Measures. We have seen straight angle is 180° and right angle is 90°. How to describe acute and obtuse in degrees? Acute Angle: Angles that are smaller than the right angle, i.e., less than 90° and are greater than 0°, are called acute angles. Examples are 50°, 75°, 40°. 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. Examples are 110°, 130°. 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. Examples are angles opening wider than a straight line but less than a full circle.
Let us solve the Figure it Out questions. Question one: In grids, join A to get acute, obtuse, reflex angles. Mark with curves. Acute opens less than 90°, obtuse between 90° and 180°, reflex between 180° and 360°. Question two: Measure each angle and classify. Angle PTR is acute. Angle PTQ is right. Angle PTW is obtuse. Angle WTP is reflex. Let us Explore: In the figure, angle TER = 80°. What is angle BET? Since REB is a straight angle of 180°, and TER is 80°, angle BET is 180° minus 80°, which is 100°. What is angle SET? Since angle RES is 90° and TER is 80°, angle SET is 90° minus 80°, which is 10°.
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Final Figure it Out questions. Question one: Draw angles 140°, 82°, 195°, 70°, 35°. Use protractor. For 195°, draw 180° then add 15°. Question two: Estimate and measure six angles. Classify as acute, right, obtuse, reflex. Question three: Make a figure with three acute, one right, two obtuse angles. Draw a complex polygon or intersecting lines to create these. Question four: Draw letter M with side angles 40° and middle 60°. Draw base, then two slanted lines meeting at 60° at top, and outer lines at 40° to base. Question five: Draw letter Y with angles 150°, 60°, 150°. Draw a central point, three lines forming these angles around it. Question six: Ashoka Chakra has 24 spokes. Angle between adjacent spokes is 360° divided by 24, which is 15°. Largest acute angle between two spokes is 75°. Question seven: Puzzle. I am acute. Double is acute. Triple is acute. Quadruple is acute. Multiply by 5 gives obtuse. Let measure be x. 5x > 90, so x > 18. 4x < 90, so x < 22.5. x must be integer. Possibilities are 19°, 20°, 21°, 22°.
Summary: 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 ST. A line is obtained when a line segment like ST is extended on both sides indefinitely; it is denoted by ST or sometimes by a single small letter like m. A ray is a portion of a line starting at a point D and going in one direction indefinitely. It is denoted by DP where P is another point on the ray. An angle can be visualised as two rays starting from a common starting point. Two rays OP and OM form the angle angle POM; here, O is called the vertex of the angle, and the rays OP and 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. Degree measures of angles can be measured using a protractor. Angles can be straight (180 degree), right (90 degree), acute (more than 0 degree and less than 90 degree), obtuse (more than 90 degree and less than 180 degree), and reflex (more than 180 degree and less than 360 degree).
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]