रेखाएँ और कोण (Lines and Angles)

ज्यामिति (Geometry) गणित की वह शाखा है जिसमें बिंदु, रेखाएँ, कोण, आकृतियाँ और उनके बीच के संबंधों का अध्ययन किया जाता है।
रेखाएँ और कोण ज्यामिति के बहुत महत्वपूर्ण आधार हैं। इस अध्याय में हम रेखा, रेखाखंड, किरण, प्रतिच्छेदी रेखाएँ और शीर्षाभिमुख कोणों के बारे में समझेंगे।

अध्याय :रेखाएँ और कोण

1. रेखा (Line)

रेखा बिंदुओं का एक सीधा समूह है, जो दोनों दिशाओं में अनंत तक बढ़ती रहती है।

रेखा का कोई निश्चित प्रारंभिक या अंतिम बिंदु नहीं होता।

उदाहरण के लिए, यदि किसी रेखा पर दो बिंदु A और B हैं, तो इसे रेखा AB या रेखा BA लिखा जा सकता है।

रेखा का संकेत:
↔ AB

रेखा की विशेषताएँ

• रेखा दोनों दिशाओं में अनंत तक बढ़ती जाती है।
• इसका कोई अंतिम बिंदु नहीं होता।
• रेखा की लंबाई निश्चित नहीं होती।
• किसी रेखा को दो बिंदुओं से नाम दिया जा सकता है।

उदाहरण: सड़क की सीधी दिशा को हम एक रेखा के रूप में समझ सकते हैं, हालांकि वास्तविक सड़क अनंत नहीं होती।

2. रेखाखंड (Line Segment)

रेखा का वह भाग जिसके दो निश्चित अंतिम बिंदु होते हैं, उसे रेखाखंड (Line Segment) कहते हैं।

यदि रेखाखंड के अंतिम बिंदु A और B हैं, तो इसे रेखाखंड AB लिखा जाता है।

रेखाखंड का संकेत:
A ───────── B

यहाँ A और B रेखाखंड के अंतिम बिंदु हैं।

रेखाखंड की विशेषताएँ

• इसके दो निश्चित अंतिम बिंदु होते हैं।
• इसकी लंबाई निश्चित होती है।
• इसे मापा जा सकता है।
• इसे दो अंतिम बिंदुओं के नाम से दर्शाया जाता है।

उदाहरण: किसी स्केल की लंबाई को एक रेखाखंड के रूप में समझा जा सकता है।

3. किरण (Ray)

किसी रेखा का वह भाग जिसका एक निश्चित प्रारंभिक बिंदु हो और जो दूसरी दिशा में अनंत तक बढ़ता जाए, उसे किरण (Ray) कहते हैं।

यदि किरण का प्रारंभिक बिंदु A है और वह B की दिशा में अनंत तक जाती है, तो इसे किरण AB लिखते हैं।

किरण का संकेत:
A ─────────────→ B

यहाँ A प्रारंभिक बिंदु है।

किरण की विशेषताएँ

• इसका केवल एक निश्चित प्रारंभिक बिंदु होता है।
• यह एक दिशा में अनंत तक जाती है।
• इसकी कोई निश्चित लंबाई नहीं होती।
• किरण का नाम लिखते समय प्रारंभिक बिंदु को पहले लिखा जाता है।

याद रखें

आकृति	निश्चित बिंदु	किस दिशा में बढ़ती है?
रेखा	कोई नहीं	दोनों दिशाओं में
रेखाखंड	दो	नहीं बढ़ता
किरण	एक	एक दिशा में

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Chapter : Lines and Angles के नोटस English Medium

4. प्रतिच्छेदी रेखाएँ (Intersecting Lines)

जब दो रेखाएँ एक-दूसरे को एक बिंदु पर काटती हैं, तो उन्हें प्रतिच्छेदी रेखाएँ (Intersecting Lines) कहते हैं।

जिस बिंदु पर दोनों रेखाएँ एक-दूसरे को काटती हैं, उसे प्रतिच्छेद बिंदु (Point of Intersection) कहते हैं।

उदाहरण:

यहाँ दोनों रेखाएँ O बिंदु पर एक-दूसरे को काटती हैं। इसलिए ये प्रतिच्छेदी रेखाएँ हैं और O प्रतिच्छेद बिंदु है।

प्रतिच्छेदी रेखाओं की विशेषताएँ

• दो रेखाएँ एक बिंदु पर मिलती या काटती हैं।
• उनके मिलने वाले बिंदु को प्रतिच्छेद बिंदु कहते हैं।
• दो प्रतिच्छेदी रेखाएँ चार कोण बनाती हैं।

उदाहरण: सड़क के चौराहे पर एक-दूसरे को काटती हुई दो सीधी सड़कें प्रतिच्छेदी रेखाओं को समझने का आसान उदाहरण हैं।

5. शीर्षाभिमुख कोण (Vertically Opposite Angles)

जब दो रेखाएँ एक-दूसरे को काटती हैं, तो उनके प्रतिच्छेद बिंदु पर चार कोण बनते हैं।

इन चार कोणों में जो कोण एक-दूसरे के ठीक सामने होते हैं, उन्हें शीर्षाभिमुख कोण (Vertically Opposite Angles) कहते हैं।

इसमें एक-दूसरे के सामने वाले कोण शीर्षाभिमुख कोण कहलाते हैं।

शीर्षाभिमुख कोणों का महत्वपूर्ण नियम

शीर्षाभिमुख कोण हमेशा बराबर होते हैं।

अर्थात,

शीर्षाभिमुख कोण = शीर्षाभिमुख कोण

याद रखने की आसान ट्रिक

जब दो रेखाएँ एक-दूसरे को काटें:

सामने-सामने वाले कोण = बराबर

लेकिन बगल-बगल वाले कोण शीर्षाभिमुख कोण नहीं होते।

संक्षिप्त पुनरावृत्ति

Topic	आसान परिभाषा
रेखा (Line)	दोनों दिशाओं में अनंत तक बढ़ने वाली सीधी रेखा
रेखाखंड (Line Segment)	दो निश्चित अंतिम बिंदुओं वाली रेखा
किरण (Ray)	एक निश्चित प्रारंभिक बिंदु से एक दिशा में अनंत तक जाने वाली रेखा
प्रतिच्छेदी रेखाएँ	जो दो रेखाएँ एक बिंदु पर एक-दूसरे को काटती हैं
शीर्षाभिमुख कोण	प्रतिच्छेदी रेखाओं से बने आमने-सामने के कोण

महत्वपूर्ण बातें याद रखें

1. रेखा की कोई निश्चित लंबाई नहीं होती।
2. रेखाखंड के दो अंतिम बिंदु होते हैं।
3. किरण का केवल एक प्रारंभिक बिंदु होता है।
4. दो रेखाएँ यदि एक बिंदु पर काटती हैं, तो वे प्रतिच्छेदी रेखाएँ कहलाती हैं।
5. प्रतिच्छेदी रेखाओं से बनने वाले आमने-सामने के कोण शीर्षाभिमुख कोण कहलाते हैं।
6. शीर्षाभिमुख कोण हमेशा बराबर होते हैं।

Lines and Angles

Geometry is the mathematical study that deal with shapes, directions, distances, and turns etc. Lines and angles are Basic Parts of Geometry.

1. Line

A line is a perfectly straight path that proceeds/exceed without limit in both directions. It has neither a beginning nor an ending point.

Arrowheads placed at both end indicate that the line continues indefinitely.

Essential Features

• A line has no endpoints.
• It extends indefinitely in both directions.
• Any two points lying on it may be used to identify the line.

2. Line Segment

A line segment is a bounded portion of a line enclosed by two fixed endpoints.

Unlike a line, which stretches without limit, a line segment possesses a measurable and definite length.

Diagram

A ●────────────────────────● B
          Line Segment AB

Here, A and B mark the endpoints.

The line segment may be denoted by AB or BA.

Everyday Example

The straight edge of a ruler can be regarded as a familiar example of a line segment because it has two definite ends and a fixed length.

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3. Ray

A ray is a portion of a line that begins at one fixed point and proceeds indefinitely in a single direction.

Diagram

A ●───────────────────────────►                                                                          B
   Endpoint

The ray originates at A and continues endlessly through B.

It is written as Ray AB.

Naming Rule

When a ray is named with two letters, the first letter must identify its endpoint.

Thus, Ray AB begins at A, whereas Ray BA begins at B.

4.Intersecting Lines

Two lines are known as intersecting lines when they meet at a common point.

The location at which they meet is called the point of intersection.

5. Vertically Opposite Angles

When two lines cross, four angles are produced around their point of intersection.

The pairs positioned directly opposite one another are called vertically opposite angles.

Fundamental Rule

Vertically opposite angles are always equal.

If one of these angles measures 75°, the angle directly across from it also measures 75°.

Chapter : Lines and Angles

6. Angle and its types

An angle is created when two rays emerge from the same endpoint.

The shared endpoint is known as the vertex, while the two rays forming the angle are its arms.

Diagram

                 B
                ●
               /
              /
             /
            /
           ● O────────────────● A
         Vertex             Arm OA
             \
              \
               \
                Arm OB

In this figure:

• O is the vertex.
• OA is one arm.
• OB is the other arm.
• The angle may be named ∠AOB or ∠BOA.

Naming Rule

When an angle is expressed using three letters, the vertex must occupy the middle position.

Types of Angle:

7. Acute Angle

An acute angle whose measures is more than 0° but less than 90°.

Rule

0° < Acute Angle < 90°

8. Right Angle

A right angle has an exact measure of 90°.

Familiar Examples

Right angles occur at the corners of:

• Books
• Squares
• Rectangular tables
• Rooms

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9. Obtuse Angle

An obtuse angle is whose measure is greater than 90° and less than 180°.

Rule

90° < Obtuse Angle < 180°

---

10. Straight Angle

A straight angle measures exactly 180°.

Its two arms point in precisely opposite directions, producing the appearance of a straight line.

---

11. Reflex Angle

A reflex angle measures more than 180° but less than 360°.

When two rays form a reflex angle, the larger rotational region around the vertex is considered, rather than the smaller opening between the rays.

Rule

180° < Reflex Angle < 360°

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12. Complete Angle

A complete angle measures exactly 360°.

It represents one entire revolution around a point.

Imagine a ray beginning at O, rotating through a full turn, and eventually returning to its original position. The total rotation is 360°.

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13. Comparison of Angle Types

Type	Measurement
Acute angle	Greater than 0° and less than 90°
Right angle	Exactly 90°
Obtuse angle	Greater than 90° and less than 180°
Straight angle	Exactly 180°
Reflex angle	Greater than 180° and less than 360°
Complete angle	Exactly 360°

14. Adjacent Angles

Two angles are described as adjacent angles when they sit beside one another and satisfy several conditions:

• They possess the same vertex.
• They share one arm.
• Their remaining arms lie on opposite sides of the common arm.
• Their interiors do not overlap.

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15. Complementary Angles

Two angles are complementary angles when their measures together total 90°.

Example

Suppose the two angles measure 30° and 60°.

30° + 60° = 90°

Hence, they form a pair of complementary angles.

Important Point

Complementary angles do not necessarily need to be adjacent. Their defining feature is simply that their measures have a combined value of 90°.

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16. Supplementary Angles

Two angles are supplementary angles when their combined measure is exactly 180°.

For example:

120° + 60° = 180°

Therefore, 120° and 60° are supplementary.

In the particular diagram below, the two angles together constitute a straight angle.

Important Point

Supplementary angles need not always be adjacent. Their essential requirement is that their measures total 180°.

Chapter : Lines and Angles

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17. Parallel Lines

Two lines are parallel when they remain in the same plane and do not meet, even if both are extended indefinitely.

Their separation remains constant.

Diagram

Line l:

A ●─────────────────────● B


Line m:

C ●────────────────● D

The relationship is written as:

l ∥ m

The symbol ∥ signifies “is parallel to.”

Familiar Examples

• Railway tracks
• Opposite sides of a rectangle
• Lines printed on ruled paper

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18. Transversal

A transversal is a line that intersects two or more other lines at distinct points.

19. A Transversal Cutting Two Parallel Lines

Consider two parallel lines, l and m, crossed by a transversal t.

l ∥ m

The space enclosed between the two parallel lines is the interior region.

The portions beyond the two parallel lines are the exterior regions.

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20. Corresponding Angles

Corresponding angles occupy the same relative position at the two points where a transversal intersects two lines.

The essential idea is simple: if one angle occupies a particular corner at the first intersection, its corresponding partner occupies the equivalent corner at the second intersection.

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21. Alternate Interior Angles

Alternate interior angles satisfy two conditions:

• They lie between the two lines.
• They are positioned on opposite sides of the transversal.

---

22. Alternate Exterior Angles

Alternate exterior angles lie beyond the two lines and occur on opposite sides of the transversal.

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Chapter : Lines And Angles

23. Co-Interior Angles

Co-interior angles, also called same-side interior angles, occupy the region between the two lines while remaining on the same side of the transversal.

They:

• Lie inside the two lines.
• Remain on the same side of the transversal.

Rule

When two lines are parallel, co-interior angles have a combined measure of 180°.

Chapter :Lines and Angles

24. The Four Important Transversal Relationships

Suppose:

l ∥ m

                         t
                        /
                       /
              1       /       2
─────────────────────●───────────── l
              3     /       4
                   /
                  /
              5  /       6
────────────────●───────────────── m
             7 /       8
              /

Corresponding Angles

Same relative position

• ∠1 = ∠5
• ∠2 = ∠6
• ∠3 = ∠7
• ∠4 = ∠8

Alternate Interior Angles

Inside the lines + opposite sides

• ∠3 = ∠6
• ∠4 = ∠5

Alternate Exterior Angles

Outside the lines + opposite sides

• ∠1 = ∠8
• ∠2 = ∠7

Co-Interior Angles

Inside the lines + same side

• ∠3 + ∠5 = 180°
• ∠4 + ∠6 = 180°

These relationships form the central pattern students need to recognize when a transversal intersects parallel lines.

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25. A Simple Way to Remember Angle Relationships

Corresponding Angles

Think:

Same position.

A useful mental image is:

Top-left ↔ Top-left

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Alternate Angles

Think:

Opposite sides of the transversal.

The word alternate suggests a change from one side to the other.

---

Co-Interior Angles

Think:

Same side + inside.

The word interior means that the angles occupy the region between the two lines.

---

Vertically Opposite Angles

Think:

Directly opposite at an intersection.

Such angles are always equal.

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Chapter :Lines and Angles

26. Important Rules at a Glance

Angle Relationship	Rule
Vertically opposite angles	Equal
Corresponding angles	Equal when lines are parallel
Alternate interior angles	Equal when lines are parallel
Alternate exterior angles	Equal when lines are parallel
Co-interior angles	Add up to 180° when lines are parallel
Angles on a straight line	Add up to 180°
Angles around a point	Add up to 360°
Complementary angles	Add up to 90°
Supplementary angles	Add up to 180°

---

Chapter : Lines and Angles

27. Solved Example 1: Complementary Angles

Suppose one angle measures 35°. Determine its complementary angle.

Complementary angles have a combined measure of 90°.

Therefore:

Missing angle = 90° − 35°

Missing angle = 55°

Hence, the complementary pair is:

35° and 55°

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28. Solved Example 2: Supplementary Angles

Suppose one angle measures 125°. Determine its supplementary angle.

Supplementary angles total 180°.

Therefore:

Missing angle = 180° − 125°

Missing angle = 55°

Verification:

125° + 55° = 180°

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29. Solved Example 3: Vertically Opposite Angles

Two lines intersect, and one of the resulting angles measures 75°.

The angle directly opposite it is vertically opposite.

Since vertically opposite angles are equal:

Opposite angle = 75°

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30. Solved Example 4: Corresponding Angles

Two parallel lines are intersected by a transversal. One corresponding angle measures 65°.

Corresponding angles are equal when the lines are parallel.

Therefore:

Corresponding angle = 65°

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31. Solved Example 5: Co-Interior Angles

Two parallel lines are intersected by a transversal. One co-interior angle measures 115°.

Co-interior angles together measure 180°.

Therefore:

Missing angle = 180° − 115°

Missing angle = 65°

Verification:

115° + 65° = 180°

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32. Lines and Angles in Everyday Life

Geometry is not confined to diagrams and classroom exercises. Straight paths, turns, corners, and parallel arrangements appear throughout ordinary surroundings.

Lines

• Roads
• Electric wires
• Long straight edges
• Railway tracks

Line Segments

• Rulers
• Book edges
• Table edges
• Window frames

Rays

• Light rays
• Sunlight spreading outward
• A flashlight beam

Angles

• Clock hands
• Scissors
• Open doors
• Roof structures

Parallel Lines

• Railway tracks
• Opposite sides of a rectangle
• Lines in a notebook

Right Angles

• Corners of rooms
• Books
• Screens
• Tiles

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33. Final Revision of Chapter : Lines and Angles

Line

A straight path extending indefinitely in both directions.

Line Segment

A portion of a line bounded by two endpoints.

Ray

A straight path with one endpoint that continues indefinitely in one direction.

Angle

A figure formed by two rays sharing a common endpoint.

Acute Angle

Greater than 0° but less than 90°.

Right Angle

Exactly 90°.

Obtuse Angle

Greater than 90° but less than 180°.

Straight Angle

Exactly 180°.

Reflex Angle

Greater than 180° but less than 360°.

Complete Angle

Exactly 360°.

Adjacent Angles

Angles sharing a vertex and one arm without overlapping.

Complementary Angles

Angles whose measures total 90°.

Supplementary Angles

Angles whose measures total 180°.

Intersecting Lines

Lines that meet at a point.

Parallel Lines

Lines in the same plane that do not meet, even when extended indefinitely.

Transversal

A line that intersects two or more lines at different points.

Vertically Opposite Angles

Opposite angles produced when two lines intersect; they are equal.

Corresponding Angles

Angles occupying matching relative positions; they are equal when the lines are parallel.

Alternate Interior Angles

Angles situated between the two lines and on opposite sides of the transversal; they are equal when the lines are parallel.

Alternate Exterior Angles

Angles situated outside the two lines and on opposite sides of the transversal; they are equal when the lines are parallel.

Co-Interior Angles

Angles situated between the two lines on the same side of the transversal; when the lines are parallel, their measures total 180°.

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Quick Memory Trick

The principal transversal relationships can be recalled through four compact ideas:

Corresponding → Same position

Alternate → Opposite sides

Interior → Inside

Co-interior → Same side + inside