Home › CBSE › Class 7 › Maths › PARALLEL AND INTERSECTING LINES
CBSE · Class 7 · 🧮 Maths · Chapter 5

PARALLEL AND INTERSECTING LINES

Intersecting LinesVertically Opposite AnglesLinear PairsPerpendicular LinesParallel LinesTransversal

Chapter 5, 'Parallel and Intersecting Lines', introduces fundamental geometric concepts crucial for higher mathematics. Students learn to identify different types of lines and the angles formed when they intersect or are cut by a transversal. Key topics include understanding vertically opposite angles, linear pairs, corresponding angles, and alternate interior angles, as well as the properties of parallel and perpendicular lines. This chapter builds a strong foundation for geometry.

Intersecting Lines and Angles

Intersecting Lines aur unse banne wale Angles
Intersecting Lines aur unse banne wale Angles

Jab do lines ek dusre ko ek point par cut karti hain, toh unhe intersecting lines kehte hain. Is intersection point par chaar angles bante hain.

  • Point of Intersection: Jis point par lines meet karti hain.
  • Angles Formed: Let's say lines 'l' aur 'm' intersect kar rahi hain, toh 4 angles banenge: \(\angle a, \angle b, \angle c, \angle d\).
  • Ye angles linear pairs aur vertically opposite pairs banate hain.

Linear Pair of Angles

  • Definition: Do adjacent angles jo ek straight line banate hain aur unka sum 180° hota hai.
  • Properties:
  • Common vertex hota hai.
  • Common arm hoti hai.
  • Non-common arms ek straight line banati hain.
  • Sum = 180° (supplementary).
  • Example (from [IMAGE: intersecting_lines_and_angles_fig52]):
  • \(\angle a + \angle b = 180°\)
  • \(\angle b + \angle c = 180°\)
  • \(\angle c + \angle d = 180°\)
  • \(\angle d + \angle a = 180°\)

Vertically Opposite Angles

  • Definition: Jab do lines intersect karti hain, toh opposite side par banne wale angles ko vertically opposite angles kehte hain.
  • Properties:
  • Ye angles hamesha equal hote hain.
  • Inka common arm nahi hota.
  • Example (from [IMAGE: intersecting_lines_and_angles_fig52]):
  • \(\angle a = \angle c\)
  • \(\angle b = \angle d\)
📖Definition

Intersecting Lines: Do lines jo ek common point par meet karti hain.

❗Important

Vertically Opposite Angles are ALWAYS equal. Ye ek fundamental property hai.

Vertically Opposite and Perpendicular Lines

Perpendicular Lines
Perpendicular Lines

Humne dekha ki intersecting lines vertically opposite angles banati hain jo equal hote hain. Ab dekhte hain special cases.

Proof of Vertically Opposite Angles Equality

  • Statement: Jab do lines intersect karti hain, toh vertically opposite angles equal hote hain.
  • Proof:
  1. Let two lines AB aur CD intersect at point O. (Imagine lines AB and CD intersecting at O).
  2. Angles formed hain \(\angle AOC, \angle COB, \angle BOD, \angle DOA\).
  3. Line AB ek straight line hai. So, \(\angle AOC + \angle COB = 180°\) (Linear Pair).
  4. Line CD ek straight line hai. So, \(\angle COB + \angle BOD = 180°\) (Linear Pair).
  5. From (3) and (4): \(\angle AOC + \angle COB = \angle COB + \angle BOD\).
  6. Subtract \(\angle COB\) from both sides: \(\angle AOC = \angle BOD\).
  7. Similarly, hum prove kar sakte hain ki \(\angle COB = \angle DOA\).
  • Hence Proved: Vertically opposite angles equal hote hain.

Perpendicular Lines

  • Definition: Agar do lines intersect karti hain aur unke beech ka angle 90° ho, toh unhe perpendicular lines kehte hain.
  • Notation: Perpendicular lines ko symbol ' \(\perp\) ' se denote karte hain. Jaise, line 'l' is perpendicular to line 'm' ko \(l \perp m\) likhte hain.
  • Properties:
  • Jab do lines perpendicular hoti hain, toh saare chaar angles 90° ke hote hain.
  • Har linear pair ka sum 180° hoga (90° + 90° = 180°).
  • Har vertically opposite pair 90° ka hoga (90° = 90°).
  • Example: Agar \(\angle a = 90°\) hai, toh \(\angle b = 180° - 90° = 90°\), \(\angle c = \angle a = 90°\), aur \(\angle d = \angle b = 90°\).
📖Definition

Perpendicular Lines: Do lines jo ek dusre ko 90° par intersect karti hain.

💡Tip

Vertically opposite angles ki equality ka proof aksar exams mein pucha jaata hai. Steps ko yaad rakho!

Understanding Parallel Lines

Types of Lines and Intersections
Types of Lines and Intersections
Identifying Parallel Lines
Identifying Parallel Lines

Lines sirf intersect hi nahi karti, kuch lines aisi bhi hoti hain jo kabhi meet nahi karti.

Parallel Lines

  • Definition: Do lines jo ek hi plane mein hoti hain aur kabhi intersect nahi karti, chahe unhe kitna bhi extend kar do, unhe parallel lines kehte hain.
  • Notation: Parallel lines ko symbol ' \(||\) ' se denote karte hain. Jaise, line 'l' is parallel to line 'm' ko \(l \parallel m\) likhte hain.
  • Real-life Examples:
  • Railway tracks
  • Table ke opposite edges
  • Notebook ki ruled lines
  • Key Characteristic: Inke beech ki perpendicular distance hamesha same rehti hai.

Non-Parallel Lines

  • Jo lines parallel nahi hoti, woh ya toh intersect karti hain ya phir different planes mein hoti hain (skew lines, jo aap higher classes mein padhoge).
  • Agar do lines intersect nahi kar rahi hain, toh unhe parallel kehne se pehle check karna zaroori hai ki woh ek hi plane mein hain ya nahi.
📖Definition

Parallel Lines: Ek hi plane mein do lines jo kabhi intersect nahi karti.

🚧Misconception

Students aksar 'non-intersecting' aur 'parallel' ko same samajhte hain. Yaad rakho, parallel lines ek hi plane mein honi chahiye.

Parallel and Perpendicular Lines in Paper Folding

Folding a Square Paper in Half
Folding a Square Paper in Half
Paper Folding for Parallel Lines
Paper Folding for Parallel Lines
Notations for Parallel and Perpendicular Lines
Notations for Parallel and Perpendicular Lines

Paper folding ek fun way hai lines ke concepts ko samajhne ka.

Paper Folding se Lines Banana

  • Straight Line: Paper ko fold karke crease banana ek straight line banata hai.
  • Parallel Lines:
  1. Ek paper lo aur usko beech se fold karo (ek crease banegi).
  2. Ab us folded paper ko phir se fold karo, pehle wale fold ke parallel. Aapko do parallel creases milengi.
  3. Aise hi aap aur parallel creases bana sakte ho.
  • Observation: Har naya fold pichle wale fold ke parallel hoga, agar folds ko carefully kiya jaye. [IMAGE: paper_folding_for_parallel_lines_fig58] mein aap dekh sakte ho ki kaise multiple parallel lines banti hain.
  • Perpendicular Lines:
  1. Ek paper lo aur usko fold karke ek line (crease) banao.
  2. Ab us line ko khud par hi fold karo, matlab crease ke upar crease lao, taaki fold ek right angle par ho.
  3. Jo naya crease banega woh pehle wale crease ke perpendicular hoga.
  • Observation: Is method se aap 90° ka angle bana sakte ho.

Notations for Lines

  • Parallel Lines: Do chote arrows (\(\rightarrow\)) ya ticks (||) lines par banate hain, jo indicate karte hain ki woh lines parallel hain. [IMAGE: notations_for_parallel_and_perpendicular_lines_fig59]
  • Perpendicular Lines: Intersection point par ek chota square box banate hain, jo 90° angle ko represent karta hai. [IMAGE: notations_for_parallel_and_perpendicular_lines_fig59]
⭐Remember

Paper folding ek practical tareeka hai geometry concepts ko visualize karne ka. Isse aapko lines aur angles ki properties better samajh aayengi.

Introduction to Transversals

Transversal and Angles Formed
Transversal and Angles Formed

Jab ek line do ya do se zyada lines ko different points par cut karti hai, toh us line ko transversal kehte hain.

What is a Transversal?

  • Definition: A line that intersects two or more lines at distinct points.
  • Example: [IMAGE: transversal_and_angles_formed_fig514] mein, line 't' lines 'l' aur 'm' ko cut kar rahi hai. Yahan 't' ek transversal hai.
  • Angles Formed: Jab ek transversal do lines ko cut karti hai, toh 8 angles bante hain. In angles ko different categories mein classify kiya jaata hai:
  1. Interior Angles: Jo angles do lines ke 'andar' bante hain.
  • Example: \(\angle 3, \angle 4, \angle 5, \angle 6\) in [IMAGE: transversal_and_angles_formed_fig514].
  1. Exterior Angles: Jo angles do lines ke 'bahar' bante hain.
  • Example: \(\angle 1, \angle 2, \angle 7, \angle 8\) in [IMAGE: transversal_and_angles_formed_fig514].

Pairs of Angles Formed by a Transversal

Transversal se banne wale 8 angles ke special relationships hote hain:

  • Corresponding Angles
  • Alternate Interior Angles
  • Alternate Exterior Angles
  • Interior Angles on the Same Side of the Transversal (Consecutive Interior Angles)

In relationships ko samajhna bahut zaroori hai, especially jab lines parallel hon.

📖Definition

Transversal: Ek line jo do ya do se zyada lines ko alag-alag points par cut karti hai.

❗Important

Ek transversal hamesha 8 angles banati hai jab woh do lines ko cut karti hai. In angles ki positions aur relationships ko yaad rakhna important hai.

Corresponding Angles and Parallel Lines

Corresponding Angles
Corresponding Angles

Corresponding angles transversal se banne wale angles ke pairs hote hain jo same relative position par hote hain.

Corresponding Angles

  • Definition: Angles jo transversal ke same side par hote hain aur lines ke same relative position par hote hain (jaise, both upper-left, both lower-right, etc.).
  • Pairs (from [IMAGE: transversal_and_angles_formed_fig514]):
  • \(\angle 1\) aur \(\angle 5\)
  • \(\angle 2\) aur \(\angle 6\)
  • \(\angle 3\) aur \(\angle 7\)
  • \(\angle 4\) aur \(\angle 8\)

Corresponding Angles Postulate (Axiom)

  • Statement: Agar ek transversal do parallel lines ko cut karti hai, toh corresponding angles equal hote hain.
  • Iska matlab, agar \(l \parallel m\) hai, toh:
  • \(\angle 1 = \angle 5\)
  • \(\angle 2 = \angle 6\)
  • \(\angle 3 = \angle 7\)
  • \(\angle 4 = \angle 8\)
  • Converse of Corresponding Angles Postulate: Agar ek transversal do lines ko cut karti hai aur corresponding angles equal hain, toh woh lines parallel hoti hain.
  • Ye property bahut important hai parallel lines ko prove karne ke liye.

Activity based understanding

  • [IMAGE: corresponding_angles_fig519] mein, lines l aur m parallel hain aur t transversal hai. \(\angle a\) aur \(\angle b\) corresponding angles hain. Agar aap \(\angle a\) ko trace karke \(\angle b\) par rakhenge, toh woh exactly align honge, matlab \(\angle a = \angle b\).
  • Ye property parallel lines ki pehchan hai.
📖Definition

Corresponding Angles: Transversal ke same side par aur lines ke same relative position par banne wale angles.

❗Important

Parallel lines \(\iff\) Corresponding angles are equal. Ye ek two-way statement hai.

Methods for Drawing Parallel Lines

Drawing Parallel Lines using a Ruler and Set Square
Drawing Parallel Lines using a Ruler and Set Square
A Line and a Point Outside It
A Line and a Point Outside It
Drawing Parallel Lines by Paper Folding
Drawing Parallel Lines by Paper Folding

Parallel lines draw karne ke kai tareeke hain. Yahan hum kuch common methods dekhenge.

Method 1: Using Ruler and Set Square

  • Steps:
  1. Ek line 'l' draw karo.
  2. Ek set square lo aur uski ek side ko line 'l' par rakho.
  3. Ek ruler ko set square ki dusri side ke along rakho (jo line 'l' par nahi hai).
  4. Ruler ko fix rakhte hue, set square ko ruler ke along slide karo.
  5. Set square ko slide karte hue, ek aur line 'm' draw karo uski side ke along jo pehle line 'l' par thi.
  • Result: Line 'm' line 'l' ke parallel hogi. [IMAGE: drawing_parallel_lines_using_a_ruler_and_set_square_fig521]

Method 2: Using Corresponding Angles Property

  • Steps:
  1. Ek line 'l' draw karo aur uske bahar ek point 'A' mark karo. [IMAGE: a_line_and_a_point_outside_it_fig523]
  2. Point 'A' se line 'l' par ek transversal 't' draw karo, jo line 'l' ko point 'B' par cut kare.
  3. Point 'B' par ek angle banao (e.g., \(\angle ABX\)).
  4. Ab point 'A' par, transversal 't' ke same side par, ek aur angle banao (e.g., \(\angle YAB\)) jo \(\angle ABX\) ke equal ho. Iske liye compass ka use kar sakte hain.
  5. Line 'AY' ko extend karo. Ye line 'l' ke parallel hogi.
  • Reason: Humne corresponding angles equal banaye hain, isliye lines parallel hongi.

Method 3: Paper Folding (Revisit)

  • Steps:
  1. Ek line 'l' draw karo aur uske bahar ek point 'A' mark karo.
  2. Paper ko fold karo taaki line 'l' khud par hi fold ho aur fold point 'A' se pass ho. Ye ek crease (Fold 1) banayega jo 'l' ke perpendicular hogi.
  3. Ab paper ko phir se fold karo taaki Fold 1 khud par hi fold ho aur ye naya fold bhi point 'A' se pass ho. Ye ek naya crease (Fold 2) banayega.
  • Result: Fold 2 line 'l' ke parallel hogi. [IMAGE: drawing_parallel_lines_by_paper_folding_fig524]
  • Reason: Do lines jo ek hi line ke perpendicular hoti hain, woh aapas mein parallel hoti hain.
💡Tip

Ruler aur set square method ya compass method se parallel lines draw karna construction questions mein aa sakta hai. Practice zaroor karna.

Alternate Angles and Interior Angles

Alternate Angles
Alternate Angles
Alternate Interior Angles
Alternate Interior Angles

Corresponding angles ke alawa, transversal aur parallel lines ke beech kuch aur special angle relationships hote hain.

Alternate Interior Angles

  • Definition: Angles jo transversal ke opposite sides par hote hain aur do lines ke interior mein hote hain.
  • Pairs (from [IMAGE: transversal_and_angles_formed_fig514]):
  • \(\angle 3\) aur \(\angle 6\)
  • \(\angle 4\) aur \(\angle 5\)
  • Property: Agar ek transversal do parallel lines ko cut karti hai, toh alternate interior angles equal hote hain.
  • Agar \(l \parallel m\) hai, toh \(\angle 3 = \angle 6\) aur \(\angle 4 = \angle 5\).
  • Converse: Agar alternate interior angles equal hain, toh lines parallel hoti hain.

Alternate Exterior Angles

  • Definition: Angles jo transversal ke opposite sides par hote hain aur do lines ke exterior mein hote hain.
  • Pairs (from [IMAGE: transversal_and_angles_formed_fig514]):
  • \(\angle 1\) aur \(\angle 8\)
  • \(\angle 2\) aur \(\angle 7\)
  • Property: Agar ek transversal do parallel lines ko cut karti hai, toh alternate exterior angles equal hote hain.
  • Agar \(l \parallel m\) hai, toh \(\angle 1 = \angle 8\) aur \(\angle 2 = \angle 7\).
  • Converse: Agar alternate exterior angles equal hain, toh lines parallel hoti hain.

Interior Angles on the Same Side of the Transversal (Consecutive Interior Angles)

  • Definition: Angles jo transversal ke same side par hote hain aur do lines ke interior mein hote hain.
  • Pairs (from [IMAGE: transversal_and_angles_formed_fig514]):
  • \(\angle 3\) aur \(\angle 5\)
  • \(\angle 4\) aur \(\angle 6\)
  • Property: Agar ek transversal do parallel lines ko cut karti hai, toh interior angles on the same side of the transversal supplementary hote hain (unka sum 180° hota hai).
  • Agar \(l \parallel m\) hai, toh \(\angle 3 + \angle 5 = 180°\) aur \(\angle 4 + \angle 6 = 180°\).
  • Converse: Agar interior angles on the same side of the transversal supplementary hain, toh lines parallel hoti hain.

Summary of Angle Relationships with Parallel Lines

| Angle Pair Type | Relationship (if lines parallel) | Converse (if angles have this relationship, lines are parallel) | |---|---|---| | Corresponding Angles | Equal | True | | Alternate Interior Angles | Equal | True | | Alternate Exterior Angles | Equal | True | | Consecutive Interior Angles | Supplementary (sum = 180°) | True |

❗Important

Ye saare angle relationships sirf tabhi hold karte hain jab lines parallel hon. Agar lines parallel nahi hain, toh ye angles equal ya supplementary nahi honge.

🚧Misconception

Students aksar parallel lines ki condition bhool jaate hain aur non-parallel lines par bhi ye angle properties apply kar dete hain. Hamesha check karo ki lines parallel hain ya nahi.

Solving Problems with Parallel Lines and Transversals

Ab tak jo properties padhi hain, unhe use karke hum unknown angles find kar sakte hain aur prove kar sakte hain ki lines parallel hain ya nahi.

Problem Solving Strategy

  1. Identify Given Information: Kya lines parallel hain? Konsa angle diya hua hai?
  2. Identify Unknown: Konsa angle find karna hai?
  3. Identify Relationship: Diye gaye angle aur unknown angle ke beech konsa relationship hai (corresponding, alternate, linear pair, vertically opposite, etc.)?
  4. Apply Property: Us relationship ki property ko apply karo (equal, supplementary).
  5. Calculate: Unknown angle ki value calculate karo.

Example Scenarios

  • Finding Unknown Angles:
  • Agar \(l \parallel m\) aur \(\angle 1 = 70°\) diya hai, aur \(\angle 5\) find karna hai. \(\angle 1\) aur \(\angle 5\) corresponding angles hain, so \(\angle 5 = 70°\).
  • Agar \(l \parallel m\) aur \(\angle 3 = 110°\) diya hai, aur \(\angle 6\) find karna hai. \(\angle 3\) aur \(\angle 6\) alternate interior angles hain, so \(\angle 6 = 110°\).
  • Agar \(l \parallel m\) aur \(\angle 4 = 60°\) diya hai, aur \(\angle 6\) find karna hai. \(\angle 4\) aur \(\angle 6\) consecutive interior angles hain, so \(\angle 4 + \angle 6 = 180° \implies 60° + \angle 6 = 180° \implies \angle 6 = 120°\).
  • Proving Lines Parallel:
  • Agar ek transversal do lines ko cut karti hai aur corresponding angles equal hain (e.g., \(\angle 1 = \angle 5\)), toh lines parallel hongi.
  • Agar alternate interior angles equal hain (e.g., \(\angle 3 = \angle 6\)), toh lines parallel hongi.
  • Agar consecutive interior angles supplementary hain (e.g., \(\angle 4 + \angle 6 = 180°\)), toh lines parallel hongi.

Ye concepts geometry ke base hain aur aage bhi bahut kaam aayenge.

💡Tip

Problems solve karte waqt, har step ka reason likhna bahut zaroori hai (e.g., 'Corresponding Angles', 'Linear Pair', 'Vertically Opposite Angles'). Isse aapke marks nahi katenge.

⭐Remember

Ek angle find karne ke kai tareeke ho sakte hain. Jo sabse seedha aur easy lage, woh method use karo.

Ask SAAVI — Free