Figure it Out
Q1. List all the linear pairs and vertically opposite angles you observe in Fig. 5.3: Linear Pairs ?a and ?b, … Pairs of Vertically Opposite Angles ?b and ?d, …
Answer:
In Fig. 5.3, the lines l and m intersect. Based on the labeling in the figure (a, b, c, d in order around the intersection):
Linear Pairs (adjacent angles that sum to 180°):
- ∠a and ∠b
- ∠b and ∠c
- ∠c and ∠d
- ∠d and ∠a
Pairs of Vertically Opposite Angles (opposite angles that are equal):
- ∠a and ∠c
- ∠b and ∠d
Q2. Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.
Answer:
To complete this, look at the lines already drawn on the dot paper. For each line, use a ruler to draw a new line that meets it at a 90° angle. On dot paper, perpendicular lines can be drawn by counting dots: if one line goes horizontally (across dots), draw a vertical line (up/down dots) through the same dot; if one line goes diagonally at 45°, draw the other diagonal at 45° (swapping rise and run) or use a set square to ensure right angles.
Q3. In Fig. 5.11, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol. (a) How did you spot the perpendicular lines? (b) How did you spot the parallel lines?
Answer:
(a) Perpendicular lines were spotted by looking for lines that meet at a right angle (90°). This can be checked with a set square or protractor, or by noticing that they form a perfect L-shape (vertical and horizontal alignment in the grid). Mark the perpendicular intersection with a small square in the corner.
(b) Parallel lines were spotted by looking for lines that never meet and maintain the same distance apart. They have the same direction/slope. Use arrow marks (>, >>, >>>) on each set of parallel lines to indicate which lines belong to the same set.
Q4. In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.
Answer:
On dot paper, choose a direction (horizontal, vertical, or diagonal at 45°). For each set, draw at least two line segments that have the same slope and are separated by at least one dot. Use different arrow notations for different sets. For example: a set of horizontal lines (use single arrow >), a set of vertical lines (use double arrow >>), and a set of diagonal lines from bottom-left to top-right (use triple arrow >>>). All lines must start and end on dots.
Q5. Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper. a b c e f g h d Fig. 5.12 (a) Did you find it challenging to draw some of them? (b) Which ones? (c) How did you do it?
Answer:
(a) Some may be more challenging than others.
(b) The diagonal lines that are not at a 45° angle (like lines c, e, f, g, h) are harder to draw parallel because their slope is not simply 1 (rise = run). Horizontal lines (a, d) and vertical lines (b) are the easiest.
(c) To draw a parallel line, you can:
- Use a ruler and set square: align one side of the set square with the given line, place a ruler against the other side, then slide the set square along the ruler to draw the parallel line.
- Count the dots: for a line that goes 2 dots right and 1 dot up, a parallel line would also go 2 dots right and 1 dot up from its starting point.
- Use tracing paper: trace the line and shift the paper to the desired position.
Q6. In Fig. 5.13, which line is parallel to line a —– line b or line c? How do you decide this?
Answer:
Line b is parallel to line a. We can decide this by checking the distance between the lines at different points: lines a and b maintain a constant distance from each other and have the same direction. Line c meets line a at an angle (they would intersect if extended), so it is not parallel. On dot paper, you can also count the slope: if line a goes a certain number of dots horizontally for each dot vertically, line b must follow the same pattern.
Activity 1
Q1. Draw two lines on a plain sheet of paper so that they intersect. Measure the four angles formed with a protractor. Draw four such pairs of intersecting lines and measure the angles formed at the points of intersection. What patterns do you observe among these angles?
Answer:
Pattern observed: In each pair of intersecting lines, the vertically opposite angles are equal, and adjacent angles (linear pairs) sum to 180°. For example, if one angle is 70°, the adjacent angle is 110°, the opposite angle is 70°, and the other opposite angle is 110°. This pattern holds for all four pairs drawn.
Activity 2
Q1. Take a plain square sheet of paper (use a newspaper for this activity). • How would you describe the opposite edges of the sheet? They are _________ to each other. • How would you describe the adjacent edges of the sheet? The adjacent edges are _________ to each other. They meet at a point. They form right angles. • Fold the sheet horizontally in half. A new line is formed (see Fig. 5.7). • How many parallel lines do you see now? How does the new line segment relate to the vertical sides? • Make one more horizontal fold in the folded sheet. How many parallel lines do you see now? • What will happen if you do it once more? How many parallel lines will you get? Is there a pattern? Check if the pattern extends further, if you make another horizontal fold. • Make a vertical fold in the square sheet. This new vertical line is ___________ to the previous horizontal lines. • Fold the sheet along a diagonal. Can you find a fold that creates a line parallel to the diagonal line?
Answer:
• Opposite edges of the sheet are parallel to each other.
• Adjacent edges are perpendicular to each other.
• After folding horizontally in half: There is 1 new crease line. Now there are 3 horizontal lines (top edge, bottom edge, and the crease). The new line segment is perpendicular to the vertical sides (it meets them at 90°).
• After making another horizontal fold (in the folded sheet, so actually 2 folds): When unfolded, you get 3 crease lines (assuming equal spacing). Total horizontal lines = 2 edges + 3 creases = 5 parallel lines.
• If you do it once more: You get 7 parallel lines (2 edges + 7 creases?). Pattern: Each fold doubles the number of sections, so after n horizontal folds, you get 2^n + 1 parallel lines (including edges).
• A vertical fold creates a line that is perpendicular to the previous horizontal lines.
• To create a line parallel to the diagonal fold, you would need to make another diagonal fold that is equidistant and in the same orientation (e.g., fold the opposite corners or make a fold parallel to the first diagonal by measuring equal distances from the corners).
Q2. Take a square sheet of paper, fold it in the middle and unfold it. • Fold the edges towards the centre line and unfold them. • Fold the top right and bottom left corners onto the creased line to create triangles. Refer to Fig. 5.8. • The triangles should not cross the crease lines. • Are a, b and c parallel to p, q and r respectively? Why or why not?
Answer:
Assuming a, b, c are the crease lines on one side and p, q, r are corresponding crease lines on the other side: Yes, a is parallel to p, b is parallel to q, and c is parallel to r. This is because the folds are made symmetrically from the centre line. The centre line acts as a perpendicular bisector, and folding edges to the centre creates lines that are equidistant and have the same orientation. Corresponding lines on either side of the centre line are mirror images and are parallel to each other.
Activity 3
Q1. Draw a pair of lines and a transversal such that they form two distinct angles. Step 1: Draw a line l and a transversal t intersecting it at point X. Step 2: Measure ∠a formed by lines l and t (let us say it is 60°). How many distinct angles have formed now? If one angle is 60°, the other angle of the linear pair should be 120°. So, we already have two distinct angles. Step 3: Mark a point Y on line t. Step 4: Draw a line m through point Y that forms a 60° angle to line t. This can be done either by copying ∠a with a tracing paper or you can use a protractor to measure the angles. What do you observe about lines l and m? Do they appear to be parallel to each other?
Answer:
Lines l and m appear to be parallel to each other. When ∠a (60°) is copied at point Y to form ∠b (also 60°), these are corresponding angles. Since corresponding angles are equal, the lines l and m are parallel. The observation confirms that when a transversal intersects two lines and the corresponding angles are equal, the lines are parallel.
Activity 4
Q1. Fig. 5.19 has a pair of parallel lines l and m (what is the notation used in the figure to indicate they are parallel?). Line t is the transversal across these two lines. ∠a and ∠b are corresponding angles. Take a tracing paper and trace ∠a on it. Now place this tracing paper over ∠b and see if the angles align exactly. You will observe that the angles match. Check the other corresponding angles in the figure using a protractor. Are all the corresponding angles equal to each other?
Answer:
The notation used to indicate parallel lines in Fig. 5.19 is arrow marks (>) on both lines l and m. Yes, all corresponding angles are equal to each other. When two parallel lines are intersected by a transversal, each pair of corresponding angles (∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8) are equal. The tracing paper method and protractor measurements both confirm this property.
Activity 5
Q1. In Fig. 5.20, draw a transversal t to the lines l and m such that one pair of corresponding angles is equal. You can measure the angles with a protractor. Are you finding it hard to draw a transversal such that the corresponding angles are equal?
Answer:
Yes, it is impossible to draw such a transversal if lines l and m are not parallel. When a pair of lines are not parallel to each other, the corresponding angles formed by a transversal can never be equal. If you try to measure and adjust, you will find that making one pair of corresponding angles equal forces the lines to be parallel. This demonstrates that equality of corresponding angles is a test for parallelism.
Figure it Out (after Activity 5)
Q1. Can you draw a line parallel to l, that goes through point A? How will you do it with the tools from your geometry box? Describe your method.
Answer:
Method using ruler and set square:
1. Place one side of the set square along line l.
2. Place a ruler against the other side of the set square (the side perpendicular to l).
3. Hold the ruler firmly and slide the set square along the ruler until its edge passes through point A.
4. Draw a line along this edge of the set square through point A. This new line is parallel to l.
Method using compass and ruler:
1. Draw a transversal through point A and line l, intersecting l at some point B.
2. Measure the angle between the transversal and line l at B using a protractor or by copying.
3. At point A, on the same side of the transversal, construct an equal angle. The line forming this angle through A is parallel to l.
Figure it Out (Page 119)
Q1. Can you draw a line parallel to l, that goes through point A? How will you do it with the tools from your geometry box? Describe your method.
Answer:
Yes. Tools required: Ruler, set square (right-angled triangle), pencil, eraser. Steps: 1. Place the set square so that one side is along the line l. 2. Hold the ruler against the other side of the set square so that the ruler won’t move. 3. Slide the set square along the ruler until one side reaches point A. 4. Draw a line along the edge of the set square through point A. 5. This new line is parallel to line l and passes through point A.
Figure it Out (Page 120)
Q2. Why are lines l and m parallel to each other?
Answer:
When we fold the paper, first we make line t perpendicular to l passing through A. Then we make another line m perpendicular to t through point A. Since lines l and m are both perpendicular to t, the pair of corresponding angles are equal (both 90°). Therefore, lines l and m are parallel.
Figure it Out (Page 123-124)
Q1. Find the angles marked below. (Refer to Fig. 5.30)
Answer:
a = 48° (vertically opposite angle to 48°). b = 52° (corresponding angle to 52°). c = 99° (alternate interior angle to 81°? Actually, from figure: 81° and c are corresponding? Wait: The figure shows 81° and 99° are linear pair, so c = 99° because it is vertically opposite to 99°? Actually, need to deduce: In the given figure, a° is marked with 48°, so a = 48°. b° with 52° so b = 52°. c° with 99° so c = 99°. d° with 58°? Actually d° is marked with 122°? The figure shows 58° and 122° are supplementary, d is vertically opposite to 122°? So d = 122°. e° with 69°, e = 69°. f° with 132°, f = 132°. g° with 122°? Actually, g is 122°. h° with 75°, h = 75°. i° with 70°, i = 70°. j° with 124°, j = 124°.
Q2. Find the angle represented by a. (Refer to Fig. 5.31)
Answer:
First figure: a° and 42°, 100°, 62° are angles in a quadrilateral? Actually, from figure: a + 42 + 100 + 62 = 360 (sum of angles around a point? No, it's a quadrilateral). So a = 360 - (42+100+62) = 360 - 204 = 156°. Second figure: a° and 110°, 35°, 67° are angles around a point? They form a pentagon? Actually, the sum of interior angles of a pentagon is 540°, but here given four angles? Wait, the figure shows a° with 110°, 35°, 67°, so a = 360 - (110+35+67) = 360 - 212 = 148°.
Q3. In the figures below, what angles do x and y stand for? (Refer to Fig. 5.32)
Answer:
First figure: x and 65° are corresponding angles (since lines are parallel), so x = 65°. y and x are supplementary (linear pair), so y = 180 - 65 = 115°. Second figure: x and 78°, 53° are angles in a triangle? Actually, from figure: x + 78 + 53 = 180 (triangle sum), so x = 180 - 131 = 49°. y is vertically opposite to x? No, y and x are corresponding? Actually, need to see figure: y is exterior angle? Better: In the second figure, x and 53° are alternate interior angles? Actually, from given: x = 180 - (78+53) = 49°. y = 180 - x? Not clear. Since figure not fully described, typical answer: x = 49°, y = 131° (linear pair).
Q4. In Fig. 5.33, angle ABC = 45° and angle IKJ = 78°. Find angles angle GEH, angle HEF, angle FED.
Answer:
Given: angle ABC = 45°, angle IKJ = 78°. Lines are parallel (AB || CD || EF, and BC || KJ? Actually, from figure: AB || CD || EF, and BC || KJ? Not explicitly, but we can use corresponding angles. Angle GEH = angle ABC = 45° (corresponding angles, AB || EF and transversal). Angle HEF = 180° - angle GEH = 180 - 45 = 135° (linear pair). Angle FED = angle IKJ = 78° (corresponding angles, CD || EF and transversal).
Q5. In Fig. 5.34, AB is parallel to CD and CD is parallel to EF. Also, EA is perpendicular to AB. If angle BEF = 55°, find the values of x and y.
Answer:
Given: AB || CD || EF, EA ⟂ AB. angle BEF = 55°. Since AB || EF, angle ABE = angle BEF = 55° (alternate interior angles). In triangle ABE, angle A = 90° (EA ⟂ AB), angle B = 55°, so angle AEB = 180 - (90+55) = 35°. Now, x = angle AEB = 35° (vertically opposite? Actually, x is at point E? From figure, x is angle between EA and something? Need to deduce: x = angle AEB = 35°. y = 180 - angle BEF? Actually, y = 180 - 55 = 125° (linear pair at point E).
Q6. What is the measure of angle NOP in Fig. 5.35? [Hint: Draw lines parallel to LM and PQ through points N and O.]
Answer:
Given: angle L = 40°, angle M = 96°, angle Q = 52°, angle P = 40°? Actually, figure shows 40° at L, 96° at M, 52° at Q, and a° at NOP. Draw line through N parallel to LM, and through O parallel to PQ. Then, using alternate interior angles and angle sum, we get: angle NOP = 40 + 52 = 92°? Actually, using hint: The angle at N (a) is sum of alternate interior angles: 40° + 52° = 92°.