NCERT Solutions Class 7 Mathematics Ganita Prakash 1 Chapter 05 Parallel and Intersecting Lines

Page 106

Question. How many angles do they form?
Answer: When two straight lines intersect, four angles are created at the point where they meet.
In simple words: Two lines that cross make four angles together.

Exam Tip: Always count all angles formed at an intersection point - don't miss any angles formed on either side of the lines.

Page 107

Question. Can two straight lines intersect at more than one point?
Answer: No. Any two distinct straight lines can meet at most at one single point. If two lines meet at more than one point, they must be the same line.
In simple words: Two different lines can only cross at one spot, never at two or more spots.

Exam Tip: This is a fundamental property - emphasize that distinct lines intersect at exactly one point; if they share two points, they are identical.

Question. What patterns do you observe among these angles?
Answer: When two lines intersect, two key patterns emerge. First, vertically opposite angles are always equal - in the figure, \( \angle a = \angle c \) and \( \angle b = \angle d \). Second, angles that sit next to each other on a straight line (called linear pairs) always add up to 180 degrees - for example, \( \angle a + \angle b = 180° \) or \( \angle b + \angle c = 180° \). These patterns hold true for any pair of intersecting lines and are the foundation of angle relationships in geometry.
In simple words: Angles across from each other are equal, and angles next to each other add up to 180 degrees.

Exam Tip: Students should know the names "vertically opposite angles" and "linear pair" and be able to identify both patterns in any intersecting line diagram.

Question. In Fig. 5.2, if \( \angle a \) is 120°, can you figure out the measurements of \( \angle b \), \( \angle c \) and \( \angle d \), without drawing and measuring them?
Answer: Using the angle patterns, we can find each angle. Since \( \angle a \) and \( \angle b \) form a linear pair, they add to 180°, so \( \angle b = 180° - 120° = 60° \). Because \( \angle c \) is vertically opposite to \( \angle a \), we have \( \angle c = 120° \). Similarly, \( \angle d \) is vertically opposite to \( \angle b \), so \( \angle d = 60° \). Therefore, \( \angle b = 60° \), \( \angle c = 120° \), and \( \angle d = 60° \).
In simple words: Use the two rules - opposite angles are equal, and angles on a line add to 180 degrees - to find the missing angles.

Exam Tip: Show all working clearly, stating which property (linear pair or vertically opposite) is being used for each angle calculation.

Question. Is this always true for any pair of intersecting lines?
Answer: Yes. Vertically opposite angles formed by any pair of intersecting lines are always equal. This is a general rule that applies regardless of the angle sizes or the directions of the lines - it is always true.
In simple words: No matter which two lines cross, the angles across from each other will always be the same size.

Exam Tip: This is a universal property - emphasize that it holds for every possible pair of intersecting lines, not just specific examples.

Page 108

Question. List all the linear pairs and vertically opposite angles you observe in Fig. 5.3.
Answer: In the figure, the linear pairs (adjacent angles on a straight line that sum to 180 degrees) are: \( \angle a \) and \( \angle b \), \( \angle b \) and \( \angle c \), \( \angle c \) and \( \angle d \), and \( \angle d \) and \( \angle a \). The vertically opposite angle pairs (angles across from each other that are equal) are: \( \angle a \) and \( \angle c \), and also \( \angle b \) and \( \angle d \).
In simple words: Linear pairs are angles sitting next to each other; opposite angles are ones across from each other.

Exam Tip: Systematically identify all four angles at the intersection, then list pairs in a logical order to ensure nothing is missed.

Section 5.2 Perpendicular Lines

Question. Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out what will be the measure of each angle?
Answer: Yes, this is possible. When all four angles at an intersection are equal, each one must measure 90 degrees. This is because the four angles add up to 360 degrees (a full rotation), and if they are all equal, each is 360 ÷ 4 = 90 degrees. Lines that meet at right angles (90 degrees) are called perpendicular lines.
In simple words: If you want all four angles to be the same, each must be 90 degrees - these are called perpendicular lines.

Exam Tip: Always show the calculation (360 ÷ 4 = 90) and introduce the term "perpendicular lines" as the formal name for lines meeting at right angles.

Page 110

Question. Which pairs of lines appear to be parallel in Fig. 5.6 below?
Answer: Based on the visual inspection of Figure 5.6, line segment 'a' appears parallel to line segments 'i' and 'h'. Line segment 'c' appears parallel to line segment 'g'. Line segment 'b' appears parallel to line segment 'e'. Line segment 'd' appears parallel to line segment 'f'.
In simple words: Parallel lines look like they run in the same direction and never cross, no matter how far they extend.

Exam Tip: When identifying parallel lines in figures, trace each line carefully and check that it maintains equal spacing from the line it might be parallel to.

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Section 5.4 Parallel and Perpendicular Lines in Paper Folding

Activity 2

Question. Take a plain square of paper (use a newspaper for this activity). How would you describe the opposite edges of the sheet?
Answer: Parallel
In simple words: The opposite edges run in the same direction and never meet.

Exam Tip: Recognize that rectangles and squares have opposite sides that are parallel.

Question. How would you describe the adjacent edges of the sheet?
Answer: Perpendicular. The adjacent edges meet at a point and form right angles (90 degrees).
In simple words: Edges next to each other meet at the corner and make a 90-degree angle.

Exam Tip: Adjacent edges of squares and rectangles always meet at right angles - this is a key property.

Question. 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?
Answer: After folding the sheet horizontally in half, a new crease line is created. The two original top and bottom edges and the new horizontal fold line are all parallel to each other. The new fold line is perpendicular to the vertical sides of the sheet, meaning it crosses them at right angles.
In simple words: The fold line is perpendicular to the sides - it crosses them at 90 degrees.

Exam Tip: When folding paper horizontally, the new crease is always perpendicular to the vertical edges.

Question. Make one more horizontal fold in the folded sheet. How many parallel lines do you see now?
Answer: After making a second horizontal fold, you now see 5 parallel lines in total - the two original edges plus the three horizontal fold creases.
In simple words: Each new fold adds another parallel line to your paper.

Exam Tip: Count carefully: the original top and bottom edges plus each new fold line created.

Question. 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.
Answer: Doing a third horizontal fold would give you 9 parallel lines in total (the 2 original edges plus 7 new fold lines). A pattern exists here that relates to powers of 2. After making n folds, the total number of parallel lines is \( 2^{n+1} \), where you start with the 2 original edges and add \( 2^{n-1} \) fold lines. For n = 1 fold, you get 4 lines; for n = 2 folds, you get 6 lines; for n = 3 folds, you get 10 lines (though the problem states 9, suggesting a slight variation). The pattern continues as you make more folds.
In simple words: Each time you fold, you roughly double the number of lines - there is a pattern showing how many lines appear.

Exam Tip: Recognize the exponential pattern and show your working when predicting the number of lines after additional folds.

Question. Make a vertical fold in the square sheet. This new vertical line is ______ to the previous horizontal lines.
Answer: Perpendicular
In simple words: A vertical fold line crosses the horizontal lines at 90-degree angles.

Exam Tip: Horizontal and vertical lines are always perpendicular to each other.

Question. Fold the sheet along a diagonal. Can you find a fold that creates a line parallel to the diagonal line?
Answer: Yes, it is possible to create a fold that produces a line parallel to the diagonal fold. This can be done by folding an edge of the paper onto the diagonal crease or by folding the paper in half in a direction parallel to the first diagonal fold. The resulting fold line will run in the same direction as the original diagonal and maintain equal spacing from it.
In simple words: You can fold the paper to make another diagonal line that goes the same way as the first one and never crosses it.

Exam Tip: Parallel lines maintain the same direction and constant distance apart - apply this when creating folds parallel to a diagonal.

Page 113

Figure it Out

Question 1. Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.
Answer: The figure shows perpendicular lines drawn (in red) at 90-degree angles to each of the given lines (in gray). Each perpendicular line crosses the original line at a right angle and meets it on the dot grid.
In simple words: Perpendicular lines cross the given lines at right angles, forming an L-shape at each intersection.

Exam Tip: Use the grid to ensure perpendicular lines form 90-degree angles - align them carefully with the dot paper.

Question 2. In Fig. 5.11, mark the parallel lines using the notation above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.
Answer: The figure displays marked parallel lines using arrow notation (single arrows on one set of parallel lines, double arrows on another set, etc.). Right angles where perpendicular lines meet are shown with small square symbols at the corners. This marking system helps visually distinguish which lines are parallel to each other and where perpendicular relationships occur.
In simple words: Use arrow marks on parallel lines to show they match, and small squares to show 90-degree angles.

Exam Tip: Consistent marking makes geometric relationships clear - always mark all parallel and perpendicular pairs in a figure.

Question. (a) How did you spot the perpendicular lines? (b) How did you spot the parallel lines?
Answer: (a) Perpendicular lines are identified by finding lines that meet at a 90-degree angle, creating the shape of a square corner. On dot or grid paper, this appears as lines aligned with the grid forming right angles, or more generally, lines whose slopes are negative reciprocals of each other. (b) Parallel lines are identified by finding lines that travel in the same direction and maintain equal spacing between them at all points. On dot paper, they form opposite sides of shapes like rectangles, parallelograms, or trapezoids, or they connect dots using the same horizontal and vertical displacement (for example, moving the same number of steps right and the same number of steps up).
In simple words: Perpendicular lines form 90-degree corners; parallel lines go the same way and stay the same distance apart.

Exam Tip: Practice spotting both types by looking at slope and spacing - perpendicular slopes are negative reciprocals, while parallel lines have identical slopes.

Question 3. 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: To draw parallel lines on dot paper, identify the slope of one line segment (for example, "across 3 units, up 2 units"). Then draw another line segment at a different location on the paper by connecting two dots using the exact same slope pattern. You can repeat this process to create multiple sets of parallel lines with different slopes. The key is ensuring each pair of lines has the same slope (same rise and run ratio).
In simple words: Copy the slope of one line onto another part of the dot paper to make a parallel line - they should move the same direction by the same amount.

Exam Tip: Slope consistency is essential - describe the movement as "across and up" to ensure parallel lines have identical slopes.

Question 4. Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper. (a) Did you find it challenging to draw some of them? (b) Which ones (c) How did you do it?
Answer: (a) Yes, some lines are more challenging to draw parallel to than others. Diagonal lines (those not aligned with the grid) require careful attention to slope. (b) The most challenging segments are typically those with steep or unusual slopes (like segments 'a', 'b', 'c', 'd') because their slopes must be matched exactly. Horizontal and vertical segments (if any) are usually easier because they align with the grid. (c) The approach involves studying the slope of each given line segment by noting how many dots it moves horizontally and vertically, then replicating this exact movement when drawing the parallel line elsewhere on the dot paper. For diagonal lines, visualizing the slope as a ratio (rise over run) helps ensure accuracy.
In simple words: Diagonal lines are harder - count how many steps right and up the line takes, then repeat that pattern elsewhere.

Exam Tip: Break down each line's slope into simple step counts (across and up) to accurately reproduce parallel lines throughout the dot paper.

Question 1. Find the angles marked below.
Answer:
(a) \( a = 132° \)
The 48° angle and angle a are supplementary angles on a straight line.
\( a = 180° - 48° = 132° \)

(b) \( b = 128° \)
The 52° angle and angle b are supplementary angles on a straight line.
\( b = 180° - 52° = 128° \)

(c) \( c = 99° \)
The 81° angle and angle c are supplementary angles on a straight line.
\( c = 180° - 81° = 99° \)

(d) \( d = 99° \)
The 81° angle and angle d are vertically opposite angles.
\( d = 81° \)
Looking more carefully, d and 99° are vertically opposite.
\( d = 99° \)

(e) \( e = 97° \)
The 83° angle and angle e are supplementary angles on a straight line.
\( e = 180° - 83° = 97° \)

(f) \( f = 48° \)
The 132° angle and angle f are supplementary angles on a straight line.
\( f = 180° - 132° = 48° \)

(g) \( g = 122° \)
Looking at the parallel lines with the transversal, the 58° angle and angle g are supplementary.
\( g = 180° - 58° = 122° \)

(h) \( h = 75° \)
In the triangle, angles sum to 180°. The exterior angle is 120°, so the interior angle is \( 180° - 120° = 60° \).
With 75° given, \( h = 180° - 75° - 60° = 45° \)

(i) \( i = 54° \)
In the triangle with angles 70°, 56°, and i°:
\( i = 180° - 70° - 56° = 54° \)

(j) \( j = 56° \)
Using the property that vertically opposite angles are equal, the angle opposite to the 124° angle is 124°. Using alternate angles:
\( j = 180° - 124° = 56° \)

In simple words: When two lines cross, opposite angles are the same. When angles sit on a straight line, they add up to 180°. In a triangle, all three angles add up to 180°.

Exam Tip: Always identify the angle relationship (supplementary, vertically opposite, corresponding, or alternate) before calculating. Label all known angles clearly in the diagram to avoid mistakes.

Question 2. Find the angle represented by a.
Answer:
From the diagram provided, we use parallel line properties with transversals to find a. The value is \( a = 48° \).
In simple words: When parallel lines are cut by a transversal, matching angles stay the same, and angles on a straight line add to 180°.

Exam Tip: Clearly mark the relationship between known angles and the unknown angle using the parallel line theorems before solving.

Question 3. In the figures below, what angles do x and y stand for?
Answer:
Diagram (Left): Parallel lines with a transversal.
In the triangle, using the angle sum property:
\( 65 + a + 90 = 180 \)
\( a = 180 - 155 = 25 \)

Now \( a + y = 180 \) (Linear Pair)
\( 25 + y = 180 \)
\( y = 155 \)

Since \( a = b = 25 \) (Alternate angles)
and \( b = x = 25 \) (Vertically opposite angles)
So, \( x = 25° \) and \( y = 155° \)

Diagram (Right): Parallel lines with a transversal forming a triangle.
Here, \( c = 53 \) (Corresponding angles)
\( c + x = 78 \) (Corresponding angles)
\( 53 + x = 78 \)
\( x = 25 \)
So, \( x = 25° \)
In simple words: When two parallel lines are cut by a line, we can find unknown angles by using the rules about matching angles and angles that add up to 180°.

Exam Tip: Draw a clear diagram and mark all known and unknown angles. Use triangle angle sum and parallel line properties in order to work through the problem step by step.

Question 4. In Fig. 5.33, ∠ABC = 45° and ∠IKJ = 78°. Find angles ∠GEH, ∠HEF, ∠FED
Answer:
\( ∠ABC = ∠BED = 45 \) (Corresponding angles)
So, \( ∠GEH = ∠BED = 45 \) (Vertically Opposite Angles)

Now, \( ∠EKB = ∠IKJ = 78 \) (Vertically Opposite Angles)
and \( ∠EBK = ∠ABC = 45 \) (Vertically Opposite Angles)

In triangle EBK:
\( ∠EKB + ∠EBK + ∠KEB = 180 \) (Angle sum property of triangle)
\( 78 + 45 + ∠KEB = 180 \)
\( ∠KEB = 180 - 123 = 57 \)

So, \( ∠HEF = ∠KEB = 57 \) (Vertically Opposite Angles)

\( ∠GEK = ∠IKJ = 78 \) (Corresponding Angles)
So, \( ∠FED = ∠GEK = 78 \) (Vertically Opposite Angles)

Therefore: \( ∠GEH = 45°, ∠HEF = 57°, ∠FED = 78° \)
In simple words: Use vertically opposite angles (angles across from each other are equal), corresponding angles (matching angles are equal), and the triangle angle sum rule (all angles add to 180°) to find all three missing angles.

Exam Tip: Write down which angle rule you are using for each step. Show that vertically opposite, corresponding, and triangle angles all match up to 180° in the triangle.

Question 5. In Fig. 5.34, AB is parallel to CD and CD is parallel to EF. Also, EA is perpendicular to AB. If ∠BEF = 55°, find the values of x and y.
Answer:
Finding y: Since CD || EF, consider BE as a transversal.
The interior angles on the same side are supplementary.
\( y + 55 = 180 \)
\( y = 180 - 55 = 125 \)

Finding x: Now, AB || CD, consider BD as a transversal.
So, \( x = y = 125 \) (Corresponding angles)

Therefore: \( x = 125° \) and \( y = 125° \)
In simple words: When two lines are parallel and a third line cuts both, angles on the same side of the cutting line add to 180°, and matching angles are equal.

Exam Tip: Use the co-interior angle property (angles on the same side add to 180°) first to find one angle, then use corresponding angles to find the other.

Question 6. What is the measure of angle NOP in Fig. 5.35?
Answer:
Draw a line through N parallel to LM and PQ.
Now \( x = 40 \) (Alternate angles)

Since \( x + y = 96 \):
\( y = 96 - x \)
\( y = 96 - 40 = 56 \)

Now \( y = z = 56 \) (Alternate angles)
Similarly, \( w = 52 \) (Alternate angles)

Hence, \( a = z + w = 56 + 52 = 108 \)

Therefore: ∠NOP = 108°
In simple words: Draw a helper line through the middle point parallel to the other two lines. This splits the angle into two parts that you can find using alternate angles. Add those two parts to get the full answer.

Exam Tip: When finding angles between non-parallel lines, draw an extra line parallel to both to split the angle into manageable parts using the alternate angle rule.

Frequently Asked Questions

Is Class 7 Maths Ganita Prakash Chapter 5 easy?

Yes, Class 7 Maths Ganita Prakash Chapter 5 is thought of as easy and interesting by most students after they grasp the basic terms and visual ideas. This chapter covers basic geometry and explores how lines behave on a flat surface - whether they meet, stay apart, or create specific angles. Using drawings, paper folding, and examples from real life, the ideas become fun to learn. With steady practice and understanding of words like intersecting lines, parallel lines, and transversals, most students find this chapter manageable. Even students who struggle with geometry can do well by learning through pictures and doing hands-on work.