This MCQ module is based on: 5.5 Transversals
TOPIC 18 OF 31
5.5 Transversals
🎓 Class 7
Mathematics
CBSE
Theory
Ch 5 — Parallel and Intersecting Lines
⏱ ~16 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: 5.5 Transversals
Targeting Class 7 level in Geometry, with Basic difficulty.
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5.5 Transversals
We saw what happens when two lines intersect in different ways. Let us explore what happens when one line intersects two different lines.
Definition — Transversal
A line that intersects two (or more) distinct lines at two distinct points is called a transversal?. In Fig. 5.14 above, line \(t\) crosses lines \(l\) and \(m\), so \(t\) is a transversal. Eight angles are formed where the transversal crosses a pair of lines.
Is it possible for all the eight angles to have different measurements? Why or why not?
No. Because \(\angle 1 = \angle 3\), \(\angle 2 = \angle 4\), \(\angle 5 = \angle 7\), and \(\angle 6 = \angle 8\) (vertically opposite angles are equal at each intersection).
No. Because \(\angle 1 = \angle 3\), \(\angle 2 = \angle 4\), \(\angle 5 = \angle 7\), and \(\angle 6 = \angle 8\) (vertically opposite angles are equal at each intersection).
What about five different angles — 6, 5, 4 and 2?
No. Because \(\angle 2 = \angle 4\). Similarly, the maximum number of distinct measurements possible among the eight angles is four: at each of the two intersections, two pairs of vertically opposite angles give only two distinct values; across the two intersections these may or may not coincide.
No. Because \(\angle 2 = \angle 4\). Similarly, the maximum number of distinct measurements possible among the eight angles is four: at each of the two intersections, two pairs of vertically opposite angles give only two distinct values; across the two intersections these may or may not coincide.
5.6 Corresponding Angles
In Fig. 5.14, we notice that the transversal \(t\) forms two sets of angles — one with line \(l\) and another with line \(m\). There are angles in the first set that correspond to angles in the second set based on their position.
For example, \(\angle 1\) and \(\angle 5\) both lie above their respective line and to the left of the transversal. They are in the same position. Similarly \(\angle 2\) and \(\angle 6\), \(\angle 4\) and \(\angle 8\), \(\angle 3\) and \(\angle 7\) are the corresponding angle pairs formed by transversal \(t\) intersecting lines \(l\) and \(m\).
Definition — Corresponding Angles
Two angles, one at each intersection of a transversal with two lines, that sit in the same relative position (same side of the transversal, same side of the line) are called corresponding angles?.
Activity 3 — Drawing a Transversal
Activity 3: Corresponding Angles at a Non-Parallel Pair
L3 ApplyMaterials: Ruler, protractor, pencil.
Predict: If two lines are not parallel, will corresponding angles still be equal?
- Draw a pair of lines \(l\) and \(m\) such that they form two distinct angles (i.e., are not parallel).
- Draw a transversal \(t\) that cuts both lines.
- Measure each of the eight angles with a protractor.
- Check each corresponding pair: are the measurements equal?
Observation: When \(l\) and \(m\) are not parallel, corresponding angles are not equal. So equality of corresponding angles is tied to the lines being parallel.
Activity 4 — Corresponding Angles at a Parallel Pair
Activity 4: Tracing Parallel Corresponding Angles (Fig. 5.19)
L3 ApplyMaterials: Ruler, tracing paper, pencil.
- Draw two parallel lines \(l\) and \(m\) and a transversal \(t\).
- Observe the corresponding angles \(\angle a\) and \(\angle b\) at the two intersections.
- Place a tracing paper over \(\angle a\) and trace the angle.
- Now slide the tracing paper along \(t\) until the traced \(\angle a\) lies over \(\angle b\). Do the angles align exactly?
Observation: They align exactly — meaning \(\angle a = \angle b\). When a transversal crosses parallel lines, corresponding angles are always equal.
Corresponding-Angles Property
When a transversal intersects a pair of parallel lines, each pair of corresponding angles is equal. The converse also holds: if corresponding angles at a pair of lines cut by a transversal are equal, the two lines are parallel.
5.7 Alternate Angles
Why are lines \(l\) and \(m\) parallel to each other? Because the corresponding angles they make with a transversal are equal. Let's discover another pair-property that parallel lines share.
5.8 Alternate Angles
In Fig. 5.25, \(\angle d\) is called the alternate angle of \(\angle f\), and \(\angle c\) is the alternate angle of \(\angle e\).
Definition — Alternate Angles
Between two parallel lines cut by a transversal, two angles on opposite sides of the transversal and between the two parallel lines are called alternate (interior) angles?.
You can find the alternate angle of a given angle, say \(\angle f\), by first finding the corresponding angle of \(\angle f\), which is \(\angle b\), and then finding the vertically opposite angle of \(\angle b\), which is \(\angle d\).
Activity 5 — Alternate Angles are Equal
Activity 5: Equal Alternate Angles
L3 Apply- In Fig. 5.25, if \(\angle f\) is 120°, what is the measure of its alternate angle \(\angle d\)?
- We find the measure of \(\angle d\) if we know \(\angle b\), because vertically opposite angles are equal.
- What is the measure of \(\angle b\)? \(\angle b\) is 120° because it is the corresponding angle of \(\angle f\).
- So, \(\angle d\) also measures 120°.
In fact, \(\angle f = \angle d\) irrespective of the measure of \(\angle f\). Why? Because \(\angle f\) is the corresponding angle of \(\angle b\) (so \(\angle f = \angle b\)). And \(\angle b\) is vertically opposite to \(\angle d\) (so \(\angle b = \angle d\)). Therefore \(\angle f = \angle d\).
Similarly, \(\angle c = \angle e\), because \(\angle c\) is the corresponding angle of \(\angle g\) and \(\angle g\) is vertically opposite \(\angle e\). So \(\angle c = \angle e\) is the case that \(\angle c = \angle e\). Using our understanding of corresponding angles, we have justified that alternate angles are always equal to each other.
Alternate-Angles Property
Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.
Example 1 — Using Alternate/Corresponding Angles
In Fig. 5.26, parallel lines \(l\) and \(m\) are intersected by the transversal \(t\). If \(\angle 6 = 135^\circ\), what are the measures of the other angles?
Using the three facts — linear pair sums to 180°, vertically opposite angles are equal, corresponding angles between parallels are equal:
\(\angle 6 = 135^\circ\).
\(\angle 5 = 180^\circ - 135^\circ = 45^\circ\) (linear pair with \(\angle 6\)).
\(\angle 7 = \angle 5 = 45^\circ\) (vertically opposite).
\(\angle 8 = \angle 6 = 135^\circ\) (vertically opposite).
\(\angle 2 = \angle 6 = 135^\circ\) (corresponding).
\(\angle 1 = 45^\circ, \angle 3 = 45^\circ, \angle 4 = 135^\circ\) by the same reasoning at the upper intersection.
\(\angle 6 = 135^\circ\).
\(\angle 5 = 180^\circ - 135^\circ = 45^\circ\) (linear pair with \(\angle 6\)).
\(\angle 7 = \angle 5 = 45^\circ\) (vertically opposite).
\(\angle 8 = \angle 6 = 135^\circ\) (vertically opposite).
\(\angle 2 = \angle 6 = 135^\circ\) (corresponding).
\(\angle 1 = 45^\circ, \angle 3 = 45^\circ, \angle 4 = 135^\circ\) by the same reasoning at the upper intersection.
5.9 Parallel Illusions
Consider Fig. 5.35 (the Café Wall illusion, the Hering illusion, and the Ehrenstein illusion). In these famous optical illusions, lines that are parallel appear to bend or tilt — yet a ruler confirms they are perfectly straight and perfectly parallel.
What causes these illusions? Our visual system tries to interpret small repeated contrasts along the edge of a line and nudges our perception — the brain "bends" straight lines to reconcile the pattern. A simple ruler test confirms that the lines are in fact parallel.
Competency-Based Questions
Scenario: A civil engineer is marking lane lines on a straight highway. Two chalk lines \(l\) and \(m\) are supposed to be parallel lanes; a third line \(t\) crosses both at an angle representing a lane-change trajectory. At the first crossing, \(t\) makes a 75° angle with lane \(l\) measured to the right-and-above.
Q1. If \(l \parallel m\), what is the corresponding angle at the second crossing?
L3 Apply(a) 75°. Corresponding angles between parallel lines cut by a transversal are equal.
Q2. After the measurements, the inspector notices the corresponding angle at the second crossing reads 78°, not 75°. Analyse what this tells you about the lanes.
L4 AnalyseIf the corresponding angles are unequal (75° vs 78°), then \(l\) and \(m\) are not parallel. The lanes are diverging slightly — by 3° — which would cause the road to widen or narrow along the highway. The lines must be re-drawn to make them parallel.
Q3. A student argues: "Alternate interior angles and corresponding angles are basically the same property." Evaluate.
L5 EvaluatePartly right, partly wrong. They are different pairs of angles (alternate angles are inside the parallels, on opposite sides of the transversal; corresponding angles are in matching positions on the same side). But both properties follow from each other via vertically opposite angles, so proving one proves the other. So they are linked but not identical.
Q4. Create a real-world problem in which knowing one angle and the parallelism of two lines lets you find another angle. Pose the problem and give its solution.
L6 CreateSample: A ladder leans against two parallel horizontal ledges. It makes a 65° angle with the lower ledge. Find the angle it makes with the upper ledge. Solution: Since the ledges are parallel and the ladder is a transversal, the corresponding angle at the upper ledge is also 65°. Many variations are possible.
Assertion–Reason Questions
Assertion (A): When two parallel lines are cut by a transversal, the alternate interior angles are equal.
Reason (R): Corresponding angles are equal and vertically opposite angles are equal, which together force alternate angles to be equal.
Reason (R): Corresponding angles are equal and vertically opposite angles are equal, which together force alternate angles to be equal.
Answer: (a). R is exactly the chain of reasoning that proves A.
Assertion (A): At most four distinct angle measures appear among the 8 angles formed by a transversal crossing two lines.
Reason (R): At each of the two intersections, vertically opposite angles are equal.
Reason (R): At each of the two intersections, vertically opposite angles are equal.
Answer: (a). Two pairs of vertically opposite angles at each crossing give at most two distinct measures per crossing; across two crossings, at most 4.
Assertion (A): If the alternate angles between two lines cut by a transversal are unequal, the lines cannot be parallel.
Reason (R): Alternate angles between parallel lines must be equal.
Reason (R): Alternate angles between parallel lines must be equal.
Answer: (a). The contrapositive of R gives A directly.
Frequently Asked Questions — Parallel and Intersecting Lines
What is Part 3 — Transversals, Corresponding & Alternate Angles | Class 7 Maths Ch 5 | MyAiSchool in NCERT Class 7 Mathematics?
Part 3 — Transversals, Corresponding & Alternate 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 3 — Transversals, Corresponding & Alternate Angles | Class 7 Maths Ch 5 | MyAiSchool step by step?
To solve problems on Part 3 — Transversals, Corresponding & Alternate 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 3 — Transversals, Corresponding & Alternate Angles | Class 7 Maths Ch 5 | MyAiSchool important for the Class 7 board exam?
Part 3 — Transversals, Corresponding & Alternate 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 3 — Transversals, Corresponding & Alternate Angles | Class 7 Maths Ch 5 | MyAiSchool?
Common mistakes in Part 3 — Transversals, Corresponding & Alternate 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 3 — Transversals, Corresponding & Alternate Angles | Class 7 Maths Ch 5 | MyAiSchool?
End-of-chapter NCERT exercises for Part 3 — Transversals, Corresponding & Alternate 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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