NCERT Solutions for Class 6 Maths Chapter 02 Lines and Angles

Page 15

Question 1. Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point?
Answer: An unlimited number of lines can be drawn through a single point. Since a line extends infinitely in both directions and there is no restriction on the direction a line can take from a given point, countless lines are possible.
In simple words: Through one point, you can draw as many lines as you want - there is no limit.

Exam Tip: Remember that a single point does not determine a unique line - you need two points to fix one line uniquely.

Question 2. Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points?
Answer: Only one line can be drawn through two given points. Two distinct points determine a unique line passing through both of them.
In simple words: If you have two points, there is only one straight line that goes through both.

Exam Tip: This is a fundamental axiom in geometry - always use this principle when identifying unique lines.

Question 3. Name the five marked points in the figure. Name the line segments. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?
Answer: The five marked points are L, M, P, Q, and R. The line segments in the figure are LM, MP, PQ, and QR. Points L and R are on exactly one line segment each - L is an endpoint of segment LM, and R is an endpoint of segment QR. Points M, P, and Q lie on two line segments each - M is where LM and MP meet, P is where MP and PQ meet, and Q is where PQ and QR meet.
In simple words: Points at the ends are on one segment. Points in the middle where two segments meet are on two segments.

Exam Tip: Carefully distinguish between endpoints (on one segment) and intersection points (on two segments).

Question 4. Name the rays shown in figure. Is T the starting point of each of these rays?
Answer: The two rays shown are ray TA and ray TB. Both rays start at point T, which is their common starting point. T is the initial point from which both rays extend infinitely.
In simple words: Both rays begin at T and go in different directions from there.

Exam Tip: A ray always starts at a specific point (the vertex) and extends infinitely in one direction only.

Question 5. Draw a rough figure and write labels appropriately to illustrate each of the following:
(a) Ray OP and ray OQ meet at O.
(b) Line XY and line PQ intersect at point M.
(c) Line l contains points E and F but not point D.
(d) Point P lies on AB.
Answer: For each case, draw the appropriate geometric figures: (a) Draw two rays starting from the same point O, one passing through P and another through Q. (b) Draw two lines crossing each other at point M, with one line passing through X and Y, and the other through P and Q. (c) Draw a line l passing through points E and F, with point D positioned away from the line. (d) Draw a line segment or line AB with point P marked on it between A and B or on the segment itself.
In simple words: Draw the shapes as described, making sure all points and lines are clearly labeled.

Exam Tip: Use clear labels and ensure all geometric relationships mentioned are visually represented in your figure.

Question 6. In figure, name:
(a) Five points
(b) A line
(c) Four rays
(d) Five line segments
Answer: (a) Five points are D, E, O, C, and B. (b) A line shown is line DC (or the line extending through D and C). (c) Four rays are ray OD, ray OE, ray OC, and ray OB. (d) Five line segments are DE, EO, OC, BO, and DB.
In simple words: Identify each geometric object carefully - points are locations, a line goes forever in both directions, rays start at one point and go forever one way, and segments have two endpoints.

Exam Tip: Always name geometric objects using the correct notation - points as single letters, lines with two points or a line name, rays with starting point first, and segments with both endpoints.

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Question. Do you see angles being made in each of these cases? Can you mark their arms and vertex?
Answer: Yes, angles are visible in all six cases shown. In each case, the arms of the angle are the two line segments or rays forming the angle, and the vertex is the point where these two arms meet. When marking the angles, the arms should be highlighted in green and the vertices should be marked with red dots to clearly identify them in each case.
In simple words: An angle has two arms (sides) that meet at a point called the vertex. You can see angles in all the pictures.

Exam Tip: Always identify the vertex first, then mark the two rays or line segments forming the angle arms.

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Question. 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.
Answer: Angles can be identified in various real-world objects shown in the pictures - the bicycle has angles at the joints, the checkered pattern shows angles formed by intersecting lines, the ladder has angles between its steps and side rails, and the bridge structure shows multiple angles. For any one of these angles, draw the two rays (sides) that form the angle, and label the point where they meet as the vertex. The rays should start from the vertex and extend in the directions that form the observed angle.
In simple words: Look for places where two lines or edges meet - that point is the vertex, and the two edges form the angle.

Exam Tip: Practice identifying angles in everyday objects to develop visual recognition of geometric concepts.

Question 2. Draw and label an angle with arms ST and SR.
Answer: Draw point S. From S, draw one ray towards T and another ray towards R. The angle formed is labeled \( \angle TSR \) or \( \angle RST \), with S as the vertex and ST and SR as the arms.
In simple words: Start at point S and draw two rays going toward T and R.

Exam Tip: When labeling angles, the vertex is always written in the middle - for example, \( \angle TSR \) has S (the vertex) in the center.

Question 3. Explain why \( \angle APC \) cannot be labelled as \( \angle P \).
Answer: \( \angle APC \) cannot be simply labelled as \( \angle P \) because when multiple angles share the same vertex P, using only the vertex letter creates ambiguity. There are several different angles at P - \( \angle APB \), \( \angle APC \), and \( \angle BPC \) - so naming an angle with just the vertex letter does not specify which of these angles is meant. The three-letter notation clearly identifies the angle by including the two rays forming it along with the vertex, removing all confusion.
In simple words: If many angles meet at point P, saying just "angle P" is confusing. You must name the two rays to show which angle you mean.

Exam Tip: Use three-letter notation for angles whenever there are multiple angles at the same vertex - the first and third letters are on the rays, and the middle letter is always the vertex.

Question 4. Name the angles marked in the given figure.
Answer: The angles marked in the figure are \( \angle PTQ \) and \( \angle QTR \).
In simple words: Look at where the angle marks (curves) are drawn - each curve shows one angle, and you name it using the vertex and the two rays.

Exam Tip: The curved arc marking an angle helps you identify which angle is being referred to.

Question 5. 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 figure.
Answer: When you mark three non-collinear points A, B, C and draw all possible lines through pairs of points, you get three lines: AB, BC, and CA. These three lines form a triangle. Using the three points, you can form three angles: \( \angle ABC \) (the angle at vertex B), \( \angle BCA \) (the angle at vertex C), and \( \angle CAB \) (the angle at vertex A). Each angle should be marked with a small curve at its vertex to clearly identify it.
In simple words: Three points that are not on one line make a triangle. You can draw three sides and name three angles - one at each point.

Exam Tip: Non-collinear points always form a triangle, and a triangle always has exactly three angles.

Question 6. 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 figure.
Answer: When you mark four points A, B, C, D such that no three are collinear, you can draw six lines through all possible pairs of points: AB, AC, AD, BC, BD, and CD. These four points form six angles - \( \angle ABC \), \( \angle ABD \), \( \angle ACD \), \( \angle BAC \), \( \angle CAD \), \( \angle BAD \), \( \angle BCA \), \( \angle BCD \), \( \angle ACD \), \( \angle ADB \), \( \angle ADC \), and \( \angle BDC \). In total, using the four points, you can name twelve distinct angles. Each angle should be marked with a curve at its vertex.
In simple words: Four points not on one line give you six different straight lines connecting them and twelve different angles you can name.

Exam Tip: When you have n non-collinear points, the number of lines is \( \binom{n}{2} \) and you can identify many angles by choosing any three points at a time.

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Question. Is it always easy to compare two angles?
Answer: It is not always easy to compare two angles by just looking at them. The visual appearance can be misleading, especially when angles are drawn in different orientations or positions. To accurately determine which angle is larger or smaller, we need to use systematic methods such as superimposing one angle over the other by tracing or using transparent paper, or by measuring them with appropriate tools. Simply observing the angles without using these techniques may lead to incorrect conclusions about their relative sizes.
In simple words: You cannot always tell which angle is bigger just by looking. You need to measure or trace them to compare properly.

Exam Tip: Always use proper comparison methods like superimposition or measurement rather than relying on visual estimation alone.

Question. Where else do we use superimposition to compare?
Answer: Superimposition is used in many real-world situations to compare objects. Common examples include comparing angles formed by scissors, compasses, or other tools that involve angles. We can place one pair of scissors over another to see which has a wider angle between its blades. Similarly, for other tools with angles - like the opening of a compass or the angle between the steps of a ladder - superimposition helps determine which angle is larger or if they are equal.
In simple words: Superimposition means placing one thing on top of another to compare. We use this method with scissors, compasses, and many other objects that have angles.

Exam Tip: Think of everyday objects with angles and practice identifying situations where superimposition would be the practical method to compare them.

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Question 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?
Answer: When you fold a rectangular sheet of paper diagonally and draw a line along the fold, several angles are created. By folding the paper in different ways, you create angles of varying sizes. Through this process, you will discover that when the fold passes through opposite corners, you create the largest angles, whereas when the fold is nearly parallel to one side, you create smaller angles. By comparing all the angles made through different folding patterns, the largest angle will be the one closest to the diagonal of the rectangle, and the smallest angles will be those formed when the fold is nearly aligned with the sides. Through systematic folding and comparison, you can identify which configurations produce the maximum and minimum angles.
In simple words: Fold paper in different ways and compare the angles. The widest fold angle is the biggest, and the narrowest is the smallest.

Exam Tip: When comparing angles from folds, mark each angle clearly and use superimposition to determine which is larger.

Question 2. 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 \( \angle XOC \)
Answer: (a) \( \angle AOB > \angle XOY \), because \( \angle XOY \) is contained within \( \angle AOB \). When one angle is part of another larger angle, the containing angle is always greater. (b) \( \angle AOB > \angle XOB \), because \( \angle XOB \) is a part of \( \angle AOB \). The point X lies on ray OA, so \( \angle XOB \) is contained in \( \angle AOB \). (c) \( \angle XOB = \angle XOC \), because both angles are formed with the same two rays OX and OB and OX and OC respectively. Actually, upon closer examination, these angles are equal because they are formed with the same rays.
In simple words: When one angle is inside another angle, the bigger one is the one that contains it.

Exam Tip: Always check if one angle is contained within another - if it is, the containing angle is the larger one.

Question 3. Which angle is greater: \( \angle XOY \) or \( \angle AOB \)? Give reasons.
Answer: \( \angle XOB > \angle AOB \), because \( \angle AOB \) is contained within \( \angle XOB \). This means that \( \angle AOB \) forms a part of the larger angle \( \angle XOB \), making the latter the greater angle.
In simple words: The angle that contains the other angle inside it is the bigger one.

Exam Tip: Always visualize which angle is the larger container - that will always be the greater angle.

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Question. Is it possible to draw OC such that the two angles are equal to each other in size?
Answer: Yes, it is possible to draw ray OC such that it divides the straight angle AOB into two equal angles. If ray OC becomes perpendicular to line AB, then the two angles formed - \( \angle AOC \) and \( \angle COB \) - will be equal to each other, both measuring 90 degrees.
In simple words: If you draw a ray from O that is perpendicular to line AB, it splits the straight angle into two equal parts.

Exam Tip: A perpendicular ray divides a straight angle into two equal right angles.

Question. If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?
Answer: A straight angle is formed by half of a full turn, measuring 180 degrees. Since a right angle is half of a straight angle, a right angle is formed by one - quarter (1/4) of a full turn. This means a right angle measures 90 degrees, which is one - quarter of the 360 degrees that make up a complete full turn.
In simple words: A full turn is 360 degrees. Half of that is a straight angle (180 degrees). Half of that is a right angle (90 degrees).

Exam Tip: Remember these key relationships: full turn = 360°, straight angle = 180°, right angle = 90°.

Page 29

Question 1. How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?
Answer: The number of right angles in classroom windows depends on the specific design, but typically rectangular windows contain four right angles - one at each corner. Beyond windows, many other objects in a classroom contain right angles. The corners where walls meet, the corners of doors, the edges where the blackboard or whiteboard meets the wall, the corners of desks and tables, and the corners of tiles on the floor or walls all form right angles. Nearly every rectangular or square object in a classroom demonstrates right angles because these shapes have 90 - degree corners by definition.
In simple words: Windows and doors have corners that are right angles. Desks, tables, and walls also have right angles where they meet.

Exam Tip: Right angles are found everywhere in man-made structures because rectangular and square shapes are very common.

Question 2. 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?
Answer: To form a straight angle using point A on a grid, you need to join A to another grid point such that the line passes through A and extends infinitely in opposite directions to form a 180 - degree angle. All the different ways of joining A to other grid points in a straight line include: joining A horizontally to any point on the same horizontal line, joining A vertically to any point on the same vertical line, and joining A diagonally to any point that maintains a constant slope. Each of these directions, when extended in both directions from A, forms a straight angle.
In simple words: Draw a line through point A in any direction - horizontal, vertical, or diagonal. Each line represents a straight angle when extended both ways.

Exam Tip: A straight angle is any line - it goes straight through, creating a 180 - degree angle.

Question 3. 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?
Answer: To form a right angle using point A on a grid, you need to identify two perpendicular lines - that is, two lines that meet at point A at exactly 90 degrees. On a rectangular grid, the horizontal and vertical lines are naturally perpendicular to each other. Therefore, all the different ways to form a right angle at A include: drawing a horizontal line through A and a vertical line through A (these form four right angles where they intersect), or drawing any two lines through A that are perpendicular to each other. By carefully rotating one line while keeping it perpendicular to another, you can create multiple right angles at A.
In simple words: Draw one line horizontally through A and another vertically through A - they make a right angle. You can also draw other pairs of lines that cross at 90 degrees.

Exam Tip: Right angles are always 90 degrees - look for lines that cross perpendicularly.

Question 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.
(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.
Answer: (a) After creating a slanting crease and folding to make a second crease perpendicular to it, you will have four right angles formed at the point where the two creases intersect. These are exact right angles because when two lines (or creases) are perpendicular to each other, they meet at exactly 90 degrees by the definition of perpendicularity. Each of the four angles formed at the intersection point measures 90 degrees. (b) First, fold a rectangular or square piece of paper diagonally (corner to opposite corner) to create a slanting crease. Unfold the paper to see the diagonal line. Next, take one edge of the paper and fold it so that the edge aligns exactly with the slanting crease at its center point. Make sure the edge is flush with the crease line. When you unfold, the new crease will be perpendicular to the slanting crease, creating four perfect right angles at their intersection. By following these precise folding steps, any person can replicate the right angles without prior knowledge.
In simple words: Fold the paper to make two creases that cross each other. If they cross at 90 degrees, the four corners formed are right angles.

Exam Tip: Perpendicular lines always create four right angles (90 degrees each) at their point of intersection.

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Question 1. Identify acute, right, obtuse and straight angles in the figures.
Answer: Acute angles are angles that measure less than 90 degrees and appear sharp or narrow. Right angles measure exactly 90 degrees and are formed by perpendicular lines. Obtuse angles measure between 90 and 180 degrees and appear wider or more open. Straight angles measure exactly 180 degrees and form a straight line. In the given figures, you should identify and label each angle according to these categories based on their visual appearance and measurement.
In simple words: Acute angles are sharp, right angles are square corners, obtuse angles are wide, and straight angles are flat lines.

Exam Tip: Use the 90 - degree mark as your reference - angles smaller than this are acute, equal to this are right, larger are obtuse, and exactly 180 degrees are straight.

Question 2. Make a few acute angles and a few obtuse angles. Draw them in different orientations.
Answer: Draw several examples of acute angles by creating angles that open less than 90 degrees - such as 30 degrees, 45 degrees, and 60 degrees. Draw these same angles in various orientations on the page - some pointing upward, some to the side, some tilted at different angles. Similarly, draw obtuse angles of varying measures between 90 and 180 degrees - such as 120 degrees, 135 degrees, and 150 degrees - also in multiple orientations to show that the orientation of an angle does not change its classification. This practice helps demonstrate that an angle's size is independent of how it is drawn or oriented.
In simple words: Draw sharp angles (acute) and wide angles (obtuse) pointing in different directions on your paper.

Exam Tip: Remember that an angle's type (acute, obtuse, etc.) depends only on its measure, not on its position or orientation.

Question 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?
Answer: The words acute and obtuse have been chosen because they relate to the everyday meaning of these words and match the visual appearance of the angles. Acute comes from the idea of sharpness - an acute angle measures less than 90 degrees and visually appears pointed or sharp, much like the tip of a knife or a needle. Obtuse comes from the idea of bluntness - an obtuse angle measures more than 90 degrees and appears wider and less sharp, similar to a blunt or dull object. These word choices make it easy to remember: when you see a sharp - looking angle, it is acute; when you see a wide or blunt - looking angle, it is obtuse. The mathematical terms are therefore connected to their everyday English meanings.
In simple words: Acute angles look sharp like a point. Obtuse angles look blunt or dull like a flat edge.

Exam Tip: Connect the mathematical definitions to their everyday meanings - this makes them easier to remember and apply.

Question 4. Find out the number of acute angles in each of the figures below.
Answer: To count the acute angles in each figure, identify all angles that measure less than 90 degrees. In the first triangle shown, there are three acute angles (all three angles of the triangle). In the second figure with subdivided triangles, count each acute angle formed by the divisions. In the third figure with further subdivisions, systematically identify every angle less than 90 degrees. The exact count depends on the specific subdivisions and geometric arrangements shown in each figure. Work through each figure carefully to avoid missing any acute angles, including those formed by the intersecting lines within the shapes.
In simple words: Look at each shape and count all the sharp - looking angles (those less than 90 degrees).

Exam Tip: When counting angles in subdivided figures, examine all possible angles formed by line intersections, not just the obvious ones.

Question. What will be the next figure and how many acute angles will it have? Do you notice any pattern in the numbers?
Answer: Following the pattern of subdividing triangles, the next figure would have the fourth triangle subdivided into smaller triangles. Looking at the sequence of acute angles, there is a clear pattern: the first triangle has 3 acute angles, the second has 9 acute angles, and the third has 27 acute angles. This follows a pattern where each subsequent figure has three times as many acute angles as the previous one - each triangle is subdivided into 3 smaller triangles, and each smaller triangle contributes three acute angles. Therefore, the fourth figure would have 81 acute angles (27 × 3 = 81). The pattern is a geometric progression with a common ratio of 3, following the formula: number of acute angles = \( 3^n \), where n is the figure number.
In simple words: Each new figure has three times more acute angles than the last one. The pattern grows by multiplying by 3 each time.

Exam Tip: Look for patterns in sequences by checking if numbers are being added, multiplied, or follow some other rule - in this case it is multiplication by 3.

Question. 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?
Answer: A straight angle measures 180° and a right angle measures 90°.
In simple words: A straight line makes a 180° angle. An angle that is exactly a quarter of a full turn (or half of a straight angle) is 90°.

Exam Tip: These two angle measures (180° and 90°) are fundamental reference points - memorize them as they appear in many geometry problems.

Question. 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.
Answer: The degree measures are shown in the circles as follows:
360°, 180°, 120°, 90°, 72°
60°, 45°, 40°, 36°, 30°
In simple words: When you divide a full circle (360°) into equal parts, each piece has a measure of 360 divided by the number of parts. For example, 6 equal pieces means each piece is 360 ÷ 6 = 60°.

Exam Tip: Always use the formula: angle measure = 360° ÷ number of parts. This method works for dividing circles into any number of equal sections.

Figure it Out

Question. Write the measures of the following angles:
(a) \( \angle KAL \)
(b) \( \angle WAL \)
(c) \( \angle TAK \)
Answer:
(a) \( \angle KAL = 30° \)
(b) \( \angle WAL = 50° \)
(c) \( \angle TAK = 40° \)
In simple words: Using a protractor or the angle markings in the figure, measure each angle by finding where its two rays intersect on the degree scale.

Exam Tip: When reading angles from a protractor diagram, always check which scale (inner or outer) you should use based on the angle's position.

Question. Name the different angles in the figure and write their measures.
Answer: The angles are:
\( \angle UOT = 20° \), \( \angle UOS = 55° \), \( \angle UOR = 85° \), \( \angle UOQ = 145° \), \( \angle UOP = 180° \),
\( \angle TOS = 35° \), \( \angle TOR = 65° \), \( \angle TOQ = 125° \), \( \angle TOP = 160° \),
\( \angle SOR = 30° \), \( \angle SOQ = 90° \), \( \angle SOP = 125° \),
\( \angle ROQ = 60° \), \( \angle ROP = 95° \),
\( \angle QOP = 35° \)

Yes, I have included angles such as \( \angle TOQ \). I have used both set of markings - inner and outer, depending on angles. The measure of \( \angle TOS = 35° \).
In simple words: To find all angles in a figure with several rays from one point, measure each pair of rays by reading the protractor scale carefully, using either the inner or outer markings as needed.

Exam Tip: When naming angles, always include all possible combinations of rays. Check that your measurements are consistent by verifying that adjacent angles add up correctly.

Think!

Question. In Figure, we have \( \angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOG = \angle GOH = \angle HOI = \)____. Why?
Answer: \( \angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOG = \angle GOH = \angle HOI = 22.5° \). A straight angle equals 180°. When divided into 8 equal parts, the measure of each angle is \( 180° / 8 = 22.5° \).
In simple words: All eight angles are the same size because the straight line is split into 8 equal pieces, so each piece gets 180° divided by 8, which equals 22.5°.

Exam Tip: When angles are marked as equal in a diagram, check if they all fit within a straight angle or full rotation - this tells you how many equal parts the total angle is divided into.

Question. 1. Find the degree measures of the following angles using your protractor.
Answer: Use a protractor to measure each angle shown in the figure. Place the center point of the protractor at the angle's vertex and align the baseline with one ray of the angle. Read the degree measure where the second ray crosses the protractor scale.
In simple words: To measure an angle, put the protractor's center point on the angle's corner and line up one ray with the zero mark. The other ray tells you the angle's size in degrees.

Exam Tip: Always check that the protractor's center is exactly at the vertex and one ray aligns with the baseline - small shifts cause large reading errors.

Question. 2. Find the degree measures of different angles in your classroom using your protractor.
Answer: Measure various angles in your classroom environment using a protractor. Examples might include the angle between a wall and floor (90°), the corner of a window frame (90°), the slant of a poster on a wall (varies), how far a book opens (varies), how slightly a door opens (small acute angle), and the angle where a chair backrest meets the seat (varies). Record all measurements in degrees.
In simple words: Walk around your classroom and find different angles - between objects, on walls, on furniture - and measure each one with your protractor.

Exam Tip: When measuring angles in real objects, make sure your protractor is positioned flat against the surfaces that form the angle and is not tilted.

Question. 3. Find the degree measures for the angles given below. Check if your paper protractor can be used here!
Answer: Measure the angles in the figure using a protractor. Check whether your paper protractor can be placed correctly on all the angles shown, or if the angle's position makes it impossible to use a paper protractor. Some angles may require a circular (full 360°) protractor rather than a semicircular one.
In simple words: Try to measure each angle with your protractor. If an angle is too wide or in a tricky position, a paper protractor might not work - you may need a different type of protractor.

Exam Tip: Semicircular protractors work best for angles up to 180°. For reflex angles (greater than 180°) or awkwardly positioned angles, you need a full circular protractor.

Question. 4. How can you find the degree measure of the angle given below using a protractor?
Answer: To measure a reflex angle (an angle greater than 180°) using a protractor, measure the smaller angle on the opposite side (the non-reflex angle). Then subtract this measurement from 360° to find the reflex angle measure. For example, if the non-reflex angle measures 120°, the reflex angle measures 360° - 120° = 240°.
In simple words: When an angle is too wide for your protractor, measure the angle on the other side instead, then take that away from 360° to find the big angle.

Exam Tip: Reflex angles are always larger than 180°. Always use the 360° - smaller angle method, as most protractors cannot directly measure them.

Question. 5. Measure and write the degree measures for each of the following angles:
(a) An acute angle
(b) An obtuse angle
(c) An acute angle
(d) An obtuse angle
(e) An acute angle
(f) An acute angle
Answer: Use your protractor to measure each angle. Your answers will depend on the specific angles shown in your worksheet. Record each measurement as a whole number of degrees. All acute angles should be between 0° and 90°, while all obtuse angles should be between 90° and 180°.
In simple words: Measure each angle carefully with your protractor. Remember - acute angles are less than 90° and obtuse angles are more than 90° but less than 180°.

Exam Tip: Always double-check that you're reading the correct scale on the protractor - use the scale that starts from the baseline you aligned with the angle's arm.

Question. 6. Find the degree measures of \( \angle BXE, \angle CXE, \angle AXB \) and \( \angle BXC \).
Answer: Using the protractor shown in the figure:
\( \angle BXE = 115° \)
\( \angle CXE = 85° \)
\( \angle AXB = 65° \)
\( \angle BXC = 30° \)
In simple words: Place your protractor with its center at point X and read where each ray meets the degree scale to find each angle's measure.

Exam Tip: When multiple angles share the same vertex, verify your work by checking that adjacent angles add up to angles that contain them (e.g., \( \angle AXC \) should equal \( \angle AXB + \angle BXC \)).

Question. 7. Find the degree measures of \( \angle PQR, \angle PQS \) and \( \angle PQT \).
Answer: Using the protractor in the figure:
\( \angle PQR = 45° \)
\( \angle PQS = 100° \)
\( \angle PQT = 152° \)
In simple words: Measure each angle from ray QP to the other rays (R, S, and T) by reading where each ray falls on the protractor scale.

Exam Tip: List your angle measures in order from smallest to largest to make sure they increase logically and that no reading errors have been made.

Question. 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.
Answer:
Step 1: When we fold the square paper diagonally, we create two right angles (90°) at the bottom corners and a 45° angle at the top.
Step 2: After folding, the base of the triangle is flat, forming 90° angles at the bottom corners, while the top vertex remains 45°.
Step 3: When we fold the base upwards, we create a small right angle (90°) at the corner where the base meets the triangle.
Step 4: Folding in the sides to form the ears creates acute angles near the top, likely around 30° to 45°.

In the complete bunny face, we will have multiple angles - the ears form acute angles (around 30° to 45°), the face near the chin forms an obtuse angle, close to 120°, and the sides of the face are around 90° to 120° depending on the precision of the folds.

When we unfold and examine the creases:
The sum of three angles of the triangle in (a): 45° + 65° + 70° = 180°
The sum of three angles of the triangle in (b): 56° + 62° + 62° = 180°
The sum of three angles of the triangle in (c): 30° + 55° + 95° = 180°
In simple words: When you fold and unfold paper, the creases show where angles were formed. Every triangle's three angles always add up to exactly 180°, no matter what shape or size the triangle is.

Exam Tip: This hands-on activity teaches the angle sum property of triangles - the fact that the three interior angles always total 180° is a key property to remember for all geometry problems.

Question. 9. Measure all three angles of the triangle shown in Figure (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 Figure (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.
Answer:
1. The degree measures of the angles are as follows:
\( \angle HIJ = 48° \)
\( \angle HIJ = 23° \)
\( \angle HIJ = 109° \)

2. The degree measures of different angles in my classroom:
Corner of a Window Frame - 90°
Edge of a Desk - 90°
Poster Slant on a Wall - 45°
Book opened halfway - 180°
Door slightly opened - 30°
Chair Backrest - 120°

3. The degree measures for the angles using paper protractor are given below:
\( \angle HIJ = 41° \)
\( \angle HIJ = 125° \)

In simple words: Measure the three angles in each triangle and add them together. You will always get 180°, even though the triangles may look very different from each other.

Exam Tip: The triangle angle sum property (all interior angles add to 180°) is one of the most important facts in geometry and appears in many problems - verify it yourself by measuring actual triangles to make it memorable.

Question. 4. The degree measure of the angle is as follows:
Answer: \( \angle HIJ = 102° \)
In simple words: Using a protractor, place its center point at the angle's vertex and read the degree measure where the angle's ray falls on the scale.

Exam Tip: When an angle appears to be larger than 90°, always check that you are reading the outer scale of the protractor, not the inner scale.

Question. 5. The degree measures for the following angles are given below:
(a) 78°
(b) 120°
(c) 58°
(d) 130°
(e) 127°
(f) 60°
Answer: The protractor readings for each angle are shown above.
In simple words: Each protractor diagram shows where to read the degree measure - look where the angle's ray aligns with the number scale on the protractor.

Exam Tip: When reading protractor measurements, place your finger on the baseline of the angle and trace along the protractor to the second ray - this ensures you're reading the correct measurement.

Question. 6. The degree measures of \( \angle BXE, \angle CXE, \angle AXB \) and \( \angle BXC \) are as follows:
\( \angle BXE = 115° \)
\( \angle CXE = 85° \)
\( \angle AXB = 65° \)
\( \angle BXC = 30° \)
Answer: These measurements are read from the protractor diagram by placing the center at point X and reading where each ray intersects the degree scale.
In simple words: All four angles share the same vertex X, so you read each one by starting from ray XA and moving to rays XB, XC, and XE on the protractor scale.

Exam Tip: Add adjacent angles to verify: 65° + 30° should equal 95°, which is \( \angle AXC \). This check confirms your protractor readings are accurate.

Question. 7. The degree measures of \( \angle PQR, \angle PQS \) and \( \angle PQT \) are as follows:
\( \angle PQR = 45° \)
\( \angle PQS = 100° \)
\( \angle PQT = 152° \)
Answer: Using the protractor shown in the figure, measure from ray QP to each of the other rays (QR, QS, and QT). The measures increase in order because each successive ray is further from ray QP.
In simple words: Starting from ray QP, ray QR is closest (45°), then QS is further (100°), and finally QT is furthest (152°).

Exam Tip: When multiple rays emanate from one point, arrange them in order by angle measure to double-check that your measurements make logical sense.

Question. 8. Step 1: When we fold the square paper diagonally, we create two right angles (90°) at the bottom corners and a 45° angle at the top. Step 2: After folding, the base of the triangle is flat, forming 90° angles at the bottom corners, while the top vertex remains 45°. Step 3: When we fold the base upwards, we create a small right angle (90°) at the corner where the base meets the triangle. Step 4: Folding in the sides to form the ears creates acute angles near the top, likely around 30° to 45°.
Answer: When you fold square paper to make this craft:

The creases show various angles created during each fold. After unfolding the paper completely, you can draw lines along all creases and measure each angle formed. The specific measurements depend on how precisely you made each fold, but typically:

- The original diagonal folds create 45° angles
- Folding the sides inward creates acute angles in the range of 30° to 45°
- The ears form angles close to 30° to 45°
- The chin area shows an obtuse angle near 120°
- The face sides show angles between 90° and 120°

After unfolding, the pattern of creases reveals that within any triangle formed by the creases, the three interior angles always sum to 180°.
In simple words: Each time you fold the paper, you create angles. When you unfold and measure, you discover that the three angles in every triangle always add to 180°.

Exam Tip: This activity demonstrates the triangle angle sum property practically - the fact that interior angles of a triangle always total 180° is universal and applies to all triangles, regardless of their shape or size.

Question. General Conjecture
Answer: For any triangle, the sum of the three interior angles will always equal 180 degrees. This is a fundamental property of triangles in Euclidean geometry. We can test this with different types of triangles (isosceles, scalene, equilateral, etc.) and the result will always be the same - the sum of the interior angles is 180°. This property is key in geometry and forms the basis for understanding more complex shapes and relationships in future studies.
In simple words: No matter what kind of triangle you draw or measure - whether it is tall, flat, wide, or narrow - if you add up its three inside angles, you always get 180°.

Exam Tip: Memorize the triangle angle sum property - this rule appears constantly in geometry problems, from basic triangle classification to proofs involving complex angle relationships.

Page 45

Figure it Out

Question. Where are the angles?
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 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.
Answer: 1. (a) There are 12 numbers on a clock representing for 12 hours. The total angle covered in 12 hours is 360°. So, angle between two successive numbers = 360°/12 = 30°. That is why at 1 o'clock, the angle between the hands is 30°.
(b) The angle at 2 o'clock = 60°
The angle at 4 o'clock = 120°
The angle at 6 o'clock = 180°
(c) The other angles make by the hands of a clock:
The angle at 3 o'clock = 90°
The angle at 9 o'clock = 270°
The angle at 6 o'clock = 150°
The angle at 11 o'clock = 330°
In simple words: A clock face is divided into 12 equal spaces. Each space is 30° wide (since 360° ÷ 12 = 30°). At different times, the hour and minute hands form different angles based on how many spaces apart they are.

Exam Tip: Always remember the clock angle formula - each hour mark represents 30°, and you can quickly calculate any angle by counting the hour spaces between the hands.

Question. 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?
Answer: Yes, it is possible to measure the angle of door, when it is opened. The vertex of the angle is B and its arms are AB and BC.
In simple words: Yes, the door's opening can be described as an angle. The corner of the door frame is the angle's vertex, and the two sides (one wall and the open door) form the angle's arms.

Exam Tip: When applying angles to real-world situations, always identify the vertex (the corner or pivot point) and the two arms (the rays forming the angle).

Question. 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?
Answer: The angle is made between the rope of swing and vertical line. It is shown in the picture.
In simple words: The angle forms between the rope (when pulled back) and a vertical hanging position. A bigger angle means the rope is pulled further back, making the swing go faster.

Exam Tip: In real-world applications, angles are often measured from a reference line (like vertical or horizontal) - identify this reference line first before measuring or describing the angle.

Question. 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?
Answer: Yes, angles can be used to describe the slopes of the slabs in this toy. Each slab forms an inclined plane and the angle of inclination (the angle between the slab and the horizontal surface) determines how fast the ball rolls down. The greater the angle of the slab's slope, the steeper the incline, which causes the ball to roll faster. The smaller the angle, the gentler the slope and the ball rolls more slowly.

One arm is the horizontal surface (the base of the toy or any horizontal plane the slab is attached to). The other arm is the slanting slab itself. The visible arm is the slanting slab that we can see in the picture. The invisible arm is the horizontal base, which is directly visible in the image but can be imagined as the surface parallel to the ground.
In simple words: The steeper the tilt of a slab, the faster a ball rolls down it. One arm of the angle is the flat ground (or base), and the other arm is the tilted slab itself.

Exam Tip: In slope problems, the angle of inclination is always measured from the horizontal. The steeper the angle, the faster objects slide - this principle applies to ramps, slides, and inclined planes in physics.

Question. 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? Hint: Observe the horizontal line touching the insects.
Answer: Yes, angles can describe the rotation between an insect and its rotated version by measuring the turn around a central point. The vertex of the angle is the insect's centre, while the arms are lines from the centre to a reference point on the body, before and after rotation. For instance, if using the tip of a snail's head or a ladybug's antenna as reference points, the angle between these lines from the centre before and after rotation quantifies the rotation degree. This angle quantifies the rotation, like 180 degrees for an upside-down position.
In simple words: Draw a line from the center of the insect to a point (like its head tip or antenna). When the insect rotates, the angle between the old line and the new line shows how much it turned.

Exam Tip: Rotation angles are always measured from a fixed center point. The vertex of the rotation angle is always at the center, and you measure the angle between corresponding points before and after the turn.

Question. Page 49 Figure it Out
1. In Figure, 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.
Answer: 1. List of angles: \( \angle PAC, \angle APD, \angle DPS, \angle LPR, \angle PLS, \angle ARP, \angle PRS, \angle RSL \) and \( \angle ALC \).

Now, let's guess the measurements in degrees:
\( \angle PAC = 120° \)
\( \angle APD = 90° \)
\( \angle DPS = 60° \)
\( \angle LPR = 110° \)
\( \angle PLS = 90° \)
\( \angle ARP = 360° \)
\( \angle PRS = 110° \)
\( \angle RSL = 80° \)
\( \angle ALC = 45° \)
In simple words: Find every angle in the figure by listing all pairs of rays that share a common vertex. Then estimate each angle's measure by looking at its size, and later measure with a protractor to check your estimates.

Exam Tip: When finding all angles in a figure, be systematic - start at each vertex and list all possible angle combinations to ensure you don't miss any.

Question. 2. Use a protractor to draw angles having the following degree measures:
(a) 110°
(b) 40°
(c) 75°
(d) 112°
(e) 134°
Answer: Using a protractor, draw each angle as follows:

For each angle, draw one ray (the baseline), place the protractor's center at one end of the ray, align the baseline with the protractor's zero mark, mark a point at the required degree measure, and draw another ray through this marked point.

(a) 110° - an obtuse angle
(b) 40° - an acute angle
(c) 75° - an acute angle
(d) 112° - an obtuse angle
(e) 134° - an obtuse angle
In simple words: To draw any angle, start with a straight line, place your protractor at one end, find the degree number you want, mark a dot there, and draw a line from the starting point to your dot.

Exam Tip: Always double-check your drawn angles by measuring them afterward with a protractor to ensure accuracy - this confirms your construction was correct.

Question. 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.
Answer:
Steps for construction:
- Draw a ray HJ which will serve as one arm of the angle.
- Mark the vertex a point H on the ray HJ. This point will be the vertex of the angle.
- Place the protractor with its centre point at H and align the baseline of the protractor with ray HJ.
- Using the protractor, mark a point at the required degree measure for the angle (as per the given figure).
- Remove the protractor and draw another ray HI from the vertex H through the marked point. This ray will form the second arm of the angle.
- Label the new ray's endpoint as I, completing the construction of angle \( \angle I H J \).

This completes the construction of the angle as shown in the image.
In simple words: Draw one ray, place your protractor at the starting point, mark the degree you want, then draw another ray through that mark to complete your angle.

Exam Tip: When constructing angles with a protractor, always ensure the vertex point is exactly at the protractor's center mark and one ray aligns perfectly with the baseline for accurate results.

Exam Tip: Recording your guesses and actual measurements helps you develop better angle estimation skills - over time, you will find your guesses get closer to the actual values.

Question. Page 51 Figure it Out
1. 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
Answer: Using the grid points, draw lines from point A to create:
(a) An acute angle - by connecting A to a point that forms an angle less than 90°
(b) An obtuse angle - by connecting A to a point that forms an angle between 90° and 180°
(c) A reflex angle - by connecting A to a point that forms an angle greater than 180°
In simple words: From point A, draw lines to other dots. An acute angle is sharp (less than 90°), an obtuse angle is wide (more than 90° but less than 180°), and a reflex angle is very wide (more than 180°).

Exam Tip: When classifying angles, remember these ranges: acute (0° to 90°), obtuse (90° to 180°), and reflex (180° to 360°) - visualizing these ranges helps you quickly identify any angle type.

Question. 2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex.
(a) \( \angle PTR \)
(b) \( \angle PTQ \)
(c) \( \angle PTW \)
(d) \( \angle WTP \)
Answer: Measure each angle using a protractor and classify it. The classifications depend on the specific measurements in your figure, but use these guidelines:
- Acute angles: less than 90°
- Right angles: exactly 90°
- Obtuse angles: between 90° and 180°
- Reflex angles: between 180° and 360°
In simple words: Measure each angle with your protractor and then name it - if it's less than 90° it's acute, if it's 90° it's right, if it's between 90° and 180° it's obtuse, and if it's more than 180° it's reflex.

Exam Tip: Always classify angles after measuring them, as this classification is often required in exam answers - just giving a measurement without naming the angle type is incomplete.

Page 53

Question. Let's Explore: In this figure, ∠TER = 80°. What is the measure of ∠BET? What is the measure of ∠SET?
Answer: ∠BET = 100°
∠SET = 10°
In simple words: When you know one angle is 80°, you can find the others by using the fact that angles on a straight line add up to 180°, and angles around a point add up to 360°.

Exam Tip: Always identify whether angles are on a straight line or around a point - this tells you what they must add up to.

Page 53 - Figure it Out

Question 1. Draw angles with the following degree measures:
(a) 140°
(b) 82°
(c) 195°
(d) 70°
(e) 35°
Answer: Use a protractor to draw each angle. Place the protractor's center point at the vertex, align the baseline with one ray, and mark the degree measure on the protractor's scale. Draw the second ray through the marked degree point.
In simple words: To draw an angle, use a protractor. Put its center at a point, line it up with one ray, and mark the degrees you need on the curved scale.

Exam Tip: Make sure the protractor's center is exactly at the vertex and the baseline is perfectly aligned with one ray for an accurate angle.

Question 2. Estimate the size of each angle and then measure it with a protractor:
Answer: First, look at each angle carefully and write down your best guess about how many degrees it is. Then use a protractor to measure the actual angle. The protractor should be placed with its center point at the vertex of the angle and one ray aligned with the baseline. Read the degree measure where the second ray crosses the protractor's scale.
In simple words: Make a guess about the angle size. Then measure it with a protractor to see how close you were.

Exam Tip: Estimating first helps train your angle sense - over time, you'll get better at predicting angles before measuring them.

Question. Classify these angles as acute, right, obtuse or reflex angles.
Answer: An acute angle is between 0° and 90°. A right angle is exactly 90°. An obtuse angle is between 90° and 180°. A reflex angle is between 180° and 360°. To classify each angle shown, measure it with a protractor and compare the result to these ranges.
In simple words: Check each angle's measure. If it's less than 90°, it's acute. If it's 90°, it's a right angle. If it's between 90° and 180°, it's obtuse. If it's more than 180°, it's reflex.

Exam Tip: Memorize these four angle types and their degree ranges - they appear on almost every geometry test.

Question 3. Make any figure with three acute angles, one right angle and two obtuse angles.
Answer: Draw a six-sided shape (hexagon). Make three of the interior angles less than 90° (acute), one angle exactly 90° (right), and two angles between 90° and 180° (obtuse). The sum of all interior angles in a hexagon is 720°, so adjust each angle measure to fit this total while meeting the stated requirements.
In simple words: Create a shape with six corners. Three corners should be pointed and sharp (acute), one corner should be a square corner (right), and two corners should be wide and open (obtuse).

Exam Tip: Draw the angles with a protractor to make sure you have the correct types - don't just estimate.

Question 4. Draw the letter 'M' such that the angles on the sides are 40° each and the angle in the middle is 60°.
Answer: Draw a vertical line. From the top, draw a line at 40° to the left and another at 40° to the right. These lines should meet at a point below, forming a 60° angle between them. Continue from this point with a line going down. The result looks like the letter M with the specified angles at the sides and 60° in the middle.
In simple words: To make an M shape, start with two lines going down at 40° angles from a top point. They meet at a middle point where the angle between them is 60°.

Exam Tip: Use a protractor at each vertex - this ensures your angles are exact and the M has the right shape.

Question 5. Draw the letter 'Y' such that the three angles formed are 150°, 60° and 150°.
Answer: Draw a vertical line pointing downward. From a point on this line, draw two lines going upward at angles that create 150° on the left side, 60° at the top center, and 150° on the right side. The angles are arranged so the two upper lines diverge away from the vertical line with the 60° angle between them.
In simple words: Draw a Y shape where the top two lines form a 60° angle between them, and each of the outer angles is 150°.

Exam Tip: The three angles at the center point must add up to 360° since they go all the way around - check this as a way to verify your Y is drawn correctly.

Question 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?
Answer: To find the angle between two adjacent spokes, divide the total degrees in a circle (360°) by the number of spokes (24). This gives 360° ÷ 24 = 15°. The largest acute angle formed between two spokes is the angle between two spokes that are closest together, which is 15°. This is the largest acute angle because adding any more spokes would give an angle of 30° or more, which would still be acute, but 15° is the smallest angle we can make with adjacent spokes.
In simple words: Divide 360° by 24 spokes to get 15° between each pair of next-door spokes. The biggest sharp angle you can make between any two spokes is 15°.

Exam Tip: When a shape is divided equally (like the spokes of a wheel), always divide 360° by the number of equal parts to find the angle between adjacent parts.

Question 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?
Answer: Let the angle be θ. For doubling the angle to stay acute: 2θ must be less than 90°, so θ must be less than 45°. For tripling to stay acute: 3θ must be less than 90°, so θ must be less than 30°. For quadrupling to stay acute: 4θ must be less than 90°, so θ must be less than 22.5°. For multiplying by 5 to give an obtuse angle: 5θ must be greater than 90°, so θ must be greater than 18°. Combining these conditions: 18° < θ < 22.5°. Therefore, possible values for the angle are any value in this range, such as 19°, 20°, 21°, or 22°.
In simple words: The angle must be bigger than 18° but smaller than 22.5°. So it could be 19°, 20°, 21°, or 22°.

Exam Tip: For "puzzle" questions like this, write down what each condition tells you as an inequality, then find where all the inequalities overlap.