This MCQ module is based on: Chapter 5 — Figure it Out (Exercises)
TOPIC 19 OF 31
Chapter 5 — Figure it Out (Exercises)
🎓 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: Chapter 5 — Figure it Out (Exercises)
Targeting Class 7 level in Geometry, with Basic difficulty.
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Chapter 5 — Figure it Out (Exercises)
The following problems bring together everything we have learned: vertically opposite angles, linear pairs, perpendicular and parallel lines, and the properties of transversals (corresponding and alternate angles).
Q1. Find the angles marked a–j (Fig. 5.30, Page 123)
Find the values of \(a, b, c, d, e, f, g, h, i, j\) in Fig. 5.30.
\(a = 48^\circ\) (vertically opposite to 48°).
\(b = 52^\circ\) (alternate/corresponding to 52°, parallels).
\(c = 99^\circ\) or \(c = 81^\circ\) — follow linear-pair (\(180 - 81 = 99^\circ\)) giving \(c = 99^\circ\).
\(d = 99^\circ\) (corresponding to 99°).
\(e = 97^\circ\) or use linear pair: \(180 - 83 = 97^\circ\); here \(e = 83^\circ\) (co-interior / follow figure).
\(f = 48^\circ\) — found using \(180 - 132 = 48^\circ\).
\(g = 58^\circ\) (vertically opposite to 58°).
\(h = 60^\circ\) or via straight-line: \(180 - 120 - ... = 60^\circ\); here \(h = 75^\circ\) computed from the given 120° and 75° markings.
\(i = 54^\circ\) (vertically opposite to 54°).
\(j\) found using exterior-angle relation from 27°, 97°, 124°: \(j = 180 - 97 + 27 - 124 = \) compute step-wise.
Students should verify each using either corresponding angles, alternate angles, vertically opposite angles, or linear-pair relations.
\(b = 52^\circ\) (alternate/corresponding to 52°, parallels).
\(c = 99^\circ\) or \(c = 81^\circ\) — follow linear-pair (\(180 - 81 = 99^\circ\)) giving \(c = 99^\circ\).
\(d = 99^\circ\) (corresponding to 99°).
\(e = 97^\circ\) or use linear pair: \(180 - 83 = 97^\circ\); here \(e = 83^\circ\) (co-interior / follow figure).
\(f = 48^\circ\) — found using \(180 - 132 = 48^\circ\).
\(g = 58^\circ\) (vertically opposite to 58°).
\(h = 60^\circ\) or via straight-line: \(180 - 120 - ... = 60^\circ\); here \(h = 75^\circ\) computed from the given 120° and 75° markings.
\(i = 54^\circ\) (vertically opposite to 54°).
\(j\) found using exterior-angle relation from 27°, 97°, 124°: \(j = 180 - 97 + 27 - 124 = \) compute step-wise.
Students should verify each using either corresponding angles, alternate angles, vertically opposite angles, or linear-pair relations.
Q2. Find the angle represented by \(a\).
(i) Transversal makes 42° and 100° with two parallels; find \(a\). (ii) Double-arrow parallels with 62° given; find \(a\). (iii) Two parallels with 110° and 35°; find \(a\). (iv) Right angle and 67° given; find \(a\).
(i) \(a = 138^\circ\) — using alternate/corresponding angles and linear pair.
(ii) \(a = 118^\circ\) — since 62° and \(a\) are co-interior (add to 180° between the parallels).
(iii) \(a = 105^\circ\) — using transversal relations: \(a = 110^\circ - ...\) in fact \(a = 180 - 35 - 40 = 105^\circ\).
(iv) \(a = 23^\circ\) — since 67° + \(a\) = 90° at the right-angle mark (complementary to 67°).
(ii) \(a = 118^\circ\) — since 62° and \(a\) are co-interior (add to 180° between the parallels).
(iii) \(a = 105^\circ\) — using transversal relations: \(a = 110^\circ - ...\) in fact \(a = 180 - 35 - 40 = 105^\circ\).
(iv) \(a = 23^\circ\) — since 67° + \(a\) = 90° at the right-angle mark (complementary to 67°).
Q3. In the figures below, what angles do \(x\) and \(y\) stand for?
(i) A right-angle mark with 65° and angles \(x, y\) marked; find \(x\) and \(y\). (ii) Parallel-lines transversal with 53° and 78° given; find \(x\).
(i) \(x = 25^\circ\), \(y = 155^\circ\). \(x\) is complementary to 65° (\(90 - 65 = 25\)); \(y\) is supplementary to 25° (linear pair).
(ii) \(x = 25^\circ\). Using exterior-angle relation: \(78^\circ = 53^\circ + x\) gives \(x = 25^\circ\).
(ii) \(x = 25^\circ\). Using exterior-angle relation: \(78^\circ = 53^\circ + x\) gives \(x = 25^\circ\).
Q4. In the Figure, \(\angle ABC = 45^\circ\) and \(\angle IKJ = 78^\circ\). Find \(\angle GEH, \angle HEF, \angle FED\).
Find \(\angle GEH\), \(\angle HEF\), and \(\angle FED\).
\(\angle GEH = 45^\circ\) (corresponding to \(\angle ABC = 45^\circ\), since CD is the transversal and AB ∥ ... or alternate depending on the figure).
\(\angle HEF = 57^\circ\) (remaining angle at E on the line: \(180° - 45° - 78° = 57°\)).
\(\angle FED = 78^\circ\) (corresponding to \(\angle IKJ = 78^\circ\)).
\(\angle HEF = 57^\circ\) (remaining angle at E on the line: \(180° - 45° - 78° = 57°\)).
\(\angle FED = 78^\circ\) (corresponding to \(\angle IKJ = 78^\circ\)).
Q5. AB ∥ CD and CD ∥ EF; EA ⊥ AB. If ∠BEF = 55°, find x and y.
In Fig. 5.34, AB is parallel to CD and CD is parallel to EF. Also, EA is perpendicular to AB. If \(\angle BEF = 55^\circ\), find the values of \(x\) and \(y\).
Since AB ∥ CD ∥ EF, and the transversal makes \(\angle BEF = 55°\), the co-interior angles between EF and CD add to 180°. The line from E through D making angle \(y\) is found from the transversal relations.
\(\angle x = \angle y = 125°\) (both corresponding angles computed from the 55° and the perpendicular, giving \(180° - 55° = 125°\)).
\(\angle x = \angle y = 125°\) (both corresponding angles computed from the 55° and the perpendicular, giving \(180° - 55° = 125°\)).
Q6. What is \(\angle NOP\)?
Given L↑, M (at 40°), a zigzag with N (96°), bend at O (\(a^\circ\)), and P (52°), Q↑. LM is parallel to PQ. Find \(\angle NOP\). [Hint: Draw lines parallel to LM and PQ through points N and O.]
\(\angle NOP = 108°\).
Draw a line through O parallel to LM (and PQ). This line splits \(\angle NOP\) into two parts.
The upper part equals \(96° - 40° = 56°\) using alternate angles at N.
The lower part equals \(52°\) using alternate angles at P.
So \(\angle NOP = 56° + 52° = 108°\).
Draw a line through O parallel to LM (and PQ). This line splits \(\angle NOP\) into two parts.
The upper part equals \(96° - 40° = 56°\) using alternate angles at N.
The lower part equals \(52°\) using alternate angles at P.
So \(\angle NOP = 56° + 52° = 108°\).
Q. Figure it Out (Page 119) — Drawing a Parallel Line
Can you draw a line parallel to line \(l\) that goes through point A? How would you do it with the tools from your geometry box? Describe your method.
Tools: Ruler, set-square (right-angled triangle), pencil, eraser.
Steps:
Steps:
- Place the set-square so that one side lies along line \(l\).
- Hold the ruler against the other side of the set-square so that the ruler won't move.
- Slide the set-square along the ruler until one side reaches point A.
- Draw a line along the edge of the set-square through A. This new line is parallel to \(l\).
Construction Activity
Activity: Drawing Parallel Lines with a Ruler & Set-Square
L3 ApplyMaterials: Ruler, set-square, pencil, paper.
Predict: Can you draw three parallel lines that all pass through three given points not lying on the same line?
- Mark three points A, B, C not on a line.
- Draw a line through A in any direction — call it \(\ell_A\).
- Using ruler + set-square sliding, draw \(\ell_B\) through B parallel to \(\ell_A\).
- Repeat for C to get \(\ell_C\).
- Verify: measure the angle each line makes with a fixed transversal; all should be equal.
The three lines are all parallel because each was drawn using the same fixed-angle sliding technique. Any transversal cuts the three at equal corresponding angles.
Chapter 5 — Summary (Key Ideas)
- Intersecting lines: Two straight lines in a plane meet at exactly one point, forming four angles.
- Linear pair: Two adjacent angles on a straight line; they sum to \(180^\circ\).
- Vertically opposite angles: Always equal. This is proven via linear pairs.
- Perpendicular lines (\(l \perp m\)): Intersect so that all four angles are \(90^\circ\).
- Parallel lines (\(l \parallel m\)): Lie in the same plane and never meet when extended. Marked with matching arrowheads.
- Transversal: A line cutting two (or more) lines at distinct points. It produces 8 angles.
- Corresponding angles: In matching positions at the two intersections. Between parallel lines they are equal.
- Alternate (interior) angles: On opposite sides of the transversal, between the two lines. Between parallel lines they are equal.
- Co-interior angles: On the same side of the transversal, between the parallels. They sum to \(180^\circ\).
- Parallel illusions: Famous visual illusions where the brain wrongly perceives parallel lines as non-parallel — only a ruler test confirms.
Key Terms
linear pair?, vertically opposite angles?, perpendicular?, parallel?, transversal?, corresponding angles, alternate angles, co-interior angles, plane surface.
Competency-Based Questions (Chapter Review)
Scenario: A window grill has a pair of long horizontal bars (\(l\) and \(m\)) that are supposed to be parallel. A diagonal brace \(t\) runs across both bars. The welder has left one angle measured — the angle between \(t\) and \(l\) (top-right of their intersection) is 72°.
Q1. If the grill is correctly made (\(l \parallel m\)), what is the value of the co-interior angle on the right side between \(t\) and \(m\)?
L3 Apply(b) 108°. Co-interior angles between parallels add to 180°, so \(180° - 72° = 108°\).
Q2. Suppose the welder later measures the co-interior angle and finds 112° (not 108°). Analyse the nature and size of the error in the parallelism.
L4 AnalyseThe bars are not parallel. Expected sum 180°; actual 72°+112° = 184°. The excess of 4° means one bar tilts by 4° relative to the other, so the grill's bars diverge along their length. The welder must re-weld with a 4° correction.
Q3. Evaluate: "Two angles that look alternate are automatically equal." Is this always, sometimes, or never true?
L5 EvaluateSometimes true. Alternate angles are equal only when the two lines crossed by the transversal are parallel. If the lines are not parallel, alternate angles differ.
Q4. Design a "parallel-line tester" — a simple handheld instrument or method that workers on a construction site can use to check if two long painted lines on a floor are truly parallel. Describe the method using ideas from this chapter.
L6 CreateSample design: Stretch a taut string across both lines (acting as a transversal). Use a protractor to measure the angle at each intersection on the same corresponding position. If both angles are equal (within ~1° tolerance), lines are parallel. Alternative: measure the perpendicular distance between the two lines at three different places; if distances are all equal, the lines are parallel.
Assertion–Reason Questions (Chapter Review)
Assertion (A): If a transversal crosses two lines such that co-interior angles sum to 180°, the two lines are parallel.
Reason (R): Co-interior angles between parallel lines always sum to 180°.
Reason (R): Co-interior angles between parallel lines always sum to 180°.
Answer: (a). The converse of R is exactly A; both true and linked.
Assertion (A): If two lines are each perpendicular to the same third line, they are parallel to each other.
Reason (R): Two lines in the same plane that do not meet are parallel.
Reason (R): Two lines in the same plane that do not meet are parallel.
Answer: (b). Both statements are true. R is the definition of parallel but doesn't specifically explain A — A follows from equal corresponding angles (both 90°) rather than the definition alone.
Assertion (A): In the Café Wall illusion, the lines are actually curved.
Reason (R): Contrasting black and white stripes cause our brain to misinterpret straight lines as curved.
Reason (R): Contrasting black and white stripes cause our brain to misinterpret straight lines as curved.
Answer: (d). The lines are actually straight (A is false). R correctly explains the perceived curvature — it is an optical illusion, not a real curve.
Frequently Asked Questions — Parallel and Intersecting Lines
What is Part 4 — Exercises & Summary | Parallel and Intersecting Lines | Class 7 Maths Ch 5 | MyAiSchool in NCERT Class 7 Mathematics?
Part 4 — Exercises & Summary | Parallel and Intersecting Lines | 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 4 — Exercises & Summary | Parallel and Intersecting Lines | Class 7 Maths Ch 5 | MyAiSchool step by step?
To solve problems on Part 4 — Exercises & Summary | Parallel and Intersecting Lines | 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 4 — Exercises & Summary | Parallel and Intersecting Lines | Class 7 Maths Ch 5 | MyAiSchool important for the Class 7 board exam?
Part 4 — Exercises & Summary | Parallel and Intersecting Lines | 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 4 — Exercises & Summary | Parallel and Intersecting Lines | Class 7 Maths Ch 5 | MyAiSchool?
Common mistakes in Part 4 — Exercises & Summary | Parallel and Intersecting Lines | 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 4 — Exercises & Summary | Parallel and Intersecting Lines | Class 7 Maths Ch 5 | MyAiSchool?
End-of-chapter NCERT exercises for Part 4 — Exercises & Summary | Parallel and Intersecting Lines | 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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