This MCQ module is based on: 5.1 Across the Line
TOPIC 16 OF 31
5.1 Across the Line
🎓 Class 7
Mathematics
CBSE
Theory
Ch 5 — Parallel and Intersecting Lines
⏱ ~15 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: 5.1 Across the Line
Targeting Class 7 level in Geometry, with Basic difficulty.
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5.1 Across the Line
Take a piece of square paper and fold it in different ways. Now, observe the creases formed by the folds. Draw lines using a pencil and a scale. You will notice different lines on the paper. Take any pair of lines and observe their relationship with each other. Do they meet? If they do not meet within the paper, do you think they would meet if they were extended beyond the paper?
In this chapter, we will explore the relationship between lines on a plane surface?. The table top, your piece of paper, the blackboard, and the bulletin board are all examples of plane surfaces.
Let us observe a pair of lines that meet each other. You will notice that they meet at a point. When a pair of lines meet each other at a point on a plane surface, we say that the lines intersect each other. Let us observe what happens when two lines intersect.
How many angles do they form? In Fig. 5.2, where line \(l\) intersects line \(m\), we can see that four angles are formed around the point of intersection.
Can two straight lines intersect at more than one point? No. Two distinct straight lines can meet in at most one point.
Angle Patterns at an Intersection
Look carefully at Fig. 5.2. We notice two kinds of pairs:
Linear Pair
Two adjacent angles whose non-common arms form a straight line. In Fig. 5.2, \(\angle a\) and \(\angle b\) are a linear pair. Their measures add up to \(180^\circ\): \(\angle a + \angle b = 180^\circ\).
Vertically Opposite Angles
Angles that are opposite each other across the point of intersection. In Fig. 5.2, \(\angle a\) and \(\angle c\) are vertically opposite angles?; so are \(\angle b\) and \(\angle d\). Vertically opposite angles are always equal.
What patterns do you observe among these angles?
• \(\angle a = \angle c\) and \(\angle b = \angle d\) (vertically opposite).
• \(\angle a + \angle b = 180^\circ\) and \(\angle b + \angle c = 180^\circ\) (linear pairs).
Is this always true for any pair of intersecting lines? Yes.
• \(\angle a = \angle c\) and \(\angle b = \angle d\) (vertically opposite).
• \(\angle a + \angle b = 180^\circ\) and \(\angle b + \angle c = 180^\circ\) (linear pairs).
Is this always true for any pair of intersecting lines? Yes.
Figure it Out — Page 108
List all the linear pairs and the pairs of vertically opposite angles you observe in Fig. 5.3 (lines \(l\) and \(m\) intersecting with angles \(a, b, c, d\)).
Linear Pairs: \(\angle a\) & \(\angle b\), \(\angle b\) & \(\angle c\), \(\angle c\) & \(\angle d\), \(\angle d\) & \(\angle a\).
Vertically Opposite Angles: \(\angle b\) & \(\angle d\); \(\angle a\) & \(\angle c\).
Vertically Opposite Angles: \(\angle b\) & \(\angle d\); \(\angle a\) & \(\angle c\).
Why Vertically Opposite Angles are Equal — A Proof
In Fig. 5.2, notice that \(\angle a + \angle b = 180^\circ\) (linear pair along line \(l\)) and \(\angle b + \angle c = 180^\circ\) (linear pair along line \(m\)).
Therefore \(\angle a + \angle b = \angle b + \angle c\), which gives \(\angle a = \angle c\). Similarly, \(\angle b = \angle d\). This is a proof in mathematics — a logical justification that something is true.
Measurements and Geometry
You might have noticed that when you measure linear pairs, sometimes they may not add up to 180°, or measures of vertically opposite angles may be unequal by a degree or two. Why?
- Measurement errors because of improper use of measuring instruments — in this case, a protractor.
- Variation in the thickness of the lines drawn. The "ideal" line in geometry does not have any thickness! But it is not possible for us to draw it exactly on paper.
5.2 Perpendicular Lines
Can you draw a pair of intersecting lines such that all four angles formed are equal? You can figure out what will be the measure of each angle.
Let the four angles be \(\angle a, \angle b, \angle c, \angle d\). Since they are equal and together they make one full turn of \(360^\circ\), each angle equals \(\frac{360^\circ}{4} = 90^\circ\).
Definition — Perpendicular Lines
When two lines intersect so that all four angles formed are right angles (\(90^\circ\)), the lines are called perpendicular? to each other. We write \(l \perp m\) to mean "line \(l\) is perpendicular to line \(m\)."
Perpendiculars Around Us
Perpendicular lines and angles of \(90^\circ\) are everywhere: the corners of a book, the intersection of streets on a grid map, the vertical post of a goalpost meeting the crossbar, the legs of a chair meeting the floor, the edges of a window frame.
Activity 1 — Folding Perpendiculars
Activity 1: Creating a Perpendicular by Folding
L3 ApplyMaterials: A rectangular sheet of paper, ruler, pencil.
Predict: Can a single fold create two perpendicular lines? How?
- Draw a straight line \(l\) across the paper.
- Choose any point \(P\) on the line.
- Fold the paper so that the line falls exactly on itself, with the crease passing through \(P\).
- Open the paper and mark the crease as line \(m\).
- Using a protractor or a set-square, measure the angle between \(l\) and \(m\).
Observation: The crease makes an angle of \(90^\circ\) with the line — so \(m \perp l\).
Explanation: When the line falls on itself after folding, the two parts of the line on each side of the fold are mirror images. The angles on either side of the fold are equal. Since they form a linear pair adding to \(180^\circ\), each must equal \(90^\circ\).
History Corner
The word perpendicular comes from the Latin perpendiculum — meaning a plumb line, a string with a weight used by masons to check that a wall is truly vertical. The vertical plumb line is always perpendicular to the horizontal ground.
Competency-Based Questions
Scenario: Anika is folding a paper aeroplane. On her square sheet she makes a diagonal fold and then a horizontal fold. She draws along both creases and measures the angles at their point of intersection. Three angles measure \(42^\circ, 138^\circ, 42^\circ\). The fourth has been smudged.
Q1. What is the measure of the smudged fourth angle, and how do you know?
L3 Apply138°. The four angles around the intersection are in two pairs of vertically opposite angles. The three known values (42°, 138°, 42°) tell us that 42° repeats as a vertically opposite pair, so the missing angle must be vertically opposite to 138°, hence 138°. Check: 42 + 138 + 42 + 138 = 360° ✓
Q2. Anika now claims the two crease-lines are perpendicular. Analyse whether her claim is correct.
L4 AnalyseIncorrect. For perpendicular lines all four angles must be 90°. Here we have 42° and 138°, so the lines intersect but are not perpendicular.
Q3. Evaluate the argument: "If one angle at an intersection is 90°, then the two lines are perpendicular." Is this always, sometimes, or never true?
L5 EvaluateAlways true. If one angle is 90°, its linear-pair partner is \(180^\circ - 90^\circ = 90^\circ\). The remaining two angles are vertically opposite to these, so they also equal 90°. All four are 90°, hence the lines are perpendicular.
Q4. Create a real-life design (e.g. a tile pattern, logo, or classroom object) that uses two intersecting lines where neither pair of opposite angles is 90°. Describe the angles.
L6 CreateSample: An "X" logo with arms at 60° and 120° — like scissor blades half open. Vertically opposite angles: 60°, 60°, 120°, 120°. Check: 60+120+60+120=360° ✓. Many valid answers possible.
Assertion–Reason Questions
Assertion (A): Vertically opposite angles formed by two intersecting lines are always equal.
Reason (R): Each vertically opposite pair shares a common linear-pair partner that sums to 180°.
Reason (R): Each vertically opposite pair shares a common linear-pair partner that sums to 180°.
Answer: (a). If \(\angle a + \angle b = 180°\) and \(\angle b + \angle c = 180°\), then \(\angle a = \angle c\). R correctly explains A.
Assertion (A): Two lines can intersect at two different points.
Reason (R): Through any two distinct points there passes exactly one straight line.
Reason (R): Through any two distinct points there passes exactly one straight line.
Answer: (d). A is false (two distinct lines intersect in at most one point). R is true and is exactly the reason A is false.
Assertion (A): If one angle formed by two intersecting lines is 90°, all four angles are 90°.
Reason (R): Linear-pair angles sum to 180° and vertically opposite angles are equal.
Reason (R): Linear-pair angles sum to 180° and vertically opposite angles are equal.
Answer: (a). Both true; R directly implies A.
Frequently Asked Questions — Parallel and Intersecting Lines
What is Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool in NCERT Class 7 Mathematics?
Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool is a key concept covered in NCERT Class 7 Mathematics, Chapter 5: Parallel and Intersecting Lines. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool step by step?
To solve problems on Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 7 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 5: Parallel and Intersecting Lines?
The essential formulas of Chapter 5 (Parallel and Intersecting Lines) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool important for the Class 7 board exam?
Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool is part of the NCERT Class 7 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool?
Common mistakes in Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool?
End-of-chapter NCERT exercises for Part 1 — Intersecting Lines & Vertically Opposite Angles | Class 7 Maths Ch 5 | MyAiSchool cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 5, and solve at least one previous-year board paper to consolidate your understanding.
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