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State Board · Class 6 · Mathematics

Lesson 2: Lines and Angles

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Point, line and plane

From daily life

A dot made with a sharp pencil, a stretched thread and the top of a table remind us of a point, a line and a plane. These three are the basic ideas of geometry.

Simple definition

A point (बिंदू) has no length, width or thickness. A line (रेषा) is infinite on both sides. A plane (प्रतल) is a flat surface that is infinite on all sides.

Rule 1: Point and line — A point is named with a capital letter, such as point P or point Q. A line has arrow marks at both ends. It is named by any two points on it (line DE or line ED) or by one small letter (line m). A line has infinitely many points.

  • Easy : Easy: Mark two points A and B on paper.
  • Medium : Medium: The line through K and L can be called line KL, line LK or line n.
  • Exam-type : Exam: Exactly one line passes through two given points, but infinitely many lines pass through a single point.

Rule 2: Plane and dimensions — A plane is named with a capital letter, such as plane M. A point has no dimension, a line has one dimension (length) and a plane has two dimensions (length and width).

  • Easy : Easy: The floor of a hall is a part of a plane.
  • Medium : Medium: A notebook page is only a part of a plane, because a plane has no edges.
  • Exam-type : Exam: A line cannot be drawn completely on paper because it is infinite; we draw a part with arrows.
Remember

Point: no size. Line: infinite, 1 dimension. Plane: flat, infinite, 2 dimensions.

Common mistake

Thinking a line ends at the arrow marks. The arrows show that it goes on forever.

Shortcut

Memory clue: "बिंदू ०, रेषा १, प्रतल २ (मिती)" (dimensions 0, 1, 2).

Spot the difference

IdeaDimensions
Point0
Line1
Plane2

Practice questions

Easy
  1. 1What has no length, width or thickness?
  2. 2How do we name a point?
  3. 3How many arrow marks does a line have?
Medium
  1. 4Name the line through K and L in two ways.
  2. 5How many lines pass through two given points?
  3. 6How many lines pass through one given point?
Challenging
  1. 7Is a blackboard a plane?
  2. 8Why can't we draw a whole line on paper?
Show answers (try them yourself first)
  1. 1A point.
  2. 2With a capital letter.
  3. 3Two, one at each end.
  4. 4Line KL and line LK.
  5. 5Exactly one.
  6. 6Infinitely many.
  7. 7It is only a part of a plane.
  8. 8A line is infinite on both sides.

Collinear points, line segment and ray

From daily life

Milestones on a straight road are collinear. The road between two milestones is a segment, and a torch beam leaving the torch is like a ray.

Simple definition

Points on one line are collinear (एकरेषीय). A line segment (रेषाखंड) has two endpoints. A ray (किरण) has one starting point and goes on forever in one direction.

Rule 1: Collinear points — If one line passes through three or more points, they are collinear. If no line can pass through all of them, they are non-collinear (अएकरेषीय).

  • Easy : Easy: Three beads on a stretched thread are collinear.
  • Medium : Medium: The three corners of a triangle are non-collinear.
  • Exam-type : Exam: P, Q, R have PQ = 3 cm, QR = 4 cm, PR = 7 cm. Since 3 + 4 = 7, Q lies on segment PR, so the points are collinear.

Rule 2: Line segment and ray — A segment from P to Q is written seg PQ or seg QP; P and Q are its endpoints. Of all paths between two points A and B, the shortest is the segment AB. Ray AB starts at A and passes through B; it is not the same as ray BA.

  • Easy : Easy: Seg PQ has endpoints P and Q.
  • Medium : Medium: Ray AB starts at A; ray BA starts at B and points the other way.
  • Exam-type : Exam: Asha walks from A to B along a curved path and Sima along the straight segment AB. Sima's path is shorter.
Remember

Collinear = on one line. Segment = 2 endpoints. Ray = 1 starting point.

Common mistake

Writing ray AB and ray BA as the same ray. Their starting points differ.

Shortcut

Memory clue: "रेषाखंड = दोन टोके; किरण = एक टोक".

Spot the difference

FigureEndpoints
LineNone
Ray1
Line segment2

Practice questions

Easy
  1. 1What are collinear points?
  2. 2What are the endpoints of seg PQ?
  3. 3How many starting points does a ray have?
Medium
  1. 4Is seg PQ the same as seg QP?
  2. 5Is ray PQ the same as ray QP?
  3. 6Are the three corners of a triangle collinear?
Challenging
  1. 7Which is the shortest path between two points A and B?
  2. 8Points X, Y, Z have XY = 5 cm, YZ = 2 cm, XZ = 7 cm. Are they collinear?
Show answers (try them yourself first)
  1. 1Points that lie on the same line.
  2. 2P and Q.
  3. 3One.
  4. 4Yes.
  5. 5No, they start at different points.
  6. 6No, they are non-collinear.
  7. 7The line segment AB.
  8. 8Yes, because 5 + 2 = 7, so Y lies on segment XZ.

Angle and its parts

From daily life

The two blades of a pair of scissors meet at a point and open to make an angle. Every angle has two arms and one meeting point.

Simple definition

An angle (कोन) is formed by two rays with a common starting point. The rays are the arms (बाहू) and the common point is the vertex (शिरोबिंदू).

Rule 1: Naming an angle — In ∠PQR the vertex is Q and the arms are ray QP and ray QR. The angle is written ∠PQR or ∠RQP, with the vertex letter in the middle. If only one angle is there at Q, it can also be written ∠Q.

Angles at a pointRays MK and ML start at point M.KLM

In the figure : In the figure the vertex is M and the arms are ray ML and ray MK. The angle is written ∠KML or ∠LMK, with the vertex letter in the middle.

  • Easy : Easy: In ∠XYZ the vertex is Y.
  • Medium : Medium: ∠XYZ and ∠ZYX are two names of the same angle.
  • Exam-type : Exam: Rays BA, BC, BD start from B, so ∠B would be unclear. Name them ∠ABC, ∠ABD, ∠CBD.

Rule 2: Interior and exterior — A point can lie in the interior (अंतर्भाग) of an angle, in its exterior (बहिर्भाग), or on the angle (on an arm).

  • Easy : Easy: A point between the two arms is in the interior.
  • Medium : Medium: A point on ray QP is on the angle.
  • Exam-type : Exam: A point outside both arms' opening is in the exterior of the angle.
Remember

Two rays + common vertex = angle. Vertex letter in the middle.

Common mistake

Writing ∠QPR for the angle with vertex Q. The vertex Q must be in the middle: ∠PQR.

Shortcut

Memory clue: "शिरोबिंदू नेहमी मधला" (the vertex is always the middle letter).

Spot the difference

PartIn ∠PQR
VertexQ
ArmsRay QP, ray QR

Practice questions

Easy
  1. 1What is the vertex of ∠PQR?
  2. 2Name the arms of ∠PQR.
  3. 3Give another name for ∠PQR.
Medium
  1. 4Why is the vertex letter written in the middle?
  2. 5Rays BA, BC, BD start at B. How many angles do pairs of them make?
  3. 6Where does a point between the two arms lie?
Challenging
  1. 7Name the angle whose vertex is M and arms are ray ML and ray MK.
  2. 8Can we write just ∠B when three rays start from B? Why?
Show answers (try them yourself first)
  1. 1Q.
  2. 2Ray QP and ray QR.
  3. 3∠RQP.
  4. 4To show clearly which angle we mean.
  5. 5Three: ∠ABC, ∠ABD, ∠CBD.
  6. 6In the interior of the angle.
  7. 7∠KML (or ∠LMK).
  8. 8No, it would not show which angle is meant.

Types of angles

From daily life

A fan folded shut, a half-open book and a fully opened door show angles of different sizes. We name angles by how wide they are, in degrees (°).

Simple definition

Angles are named by their measure: zero, acute, right, obtuse, straight, reflex, complete.

Rule 1: Zero, acute, right, obtuse — Zero angle (शून्य कोन) = 0°. Acute angle (लघुकोन) is greater than 0° and less than 90°. Right angle (काटकोन) = 90°. Obtuse angle (विशालकोन) is greater than 90° and less than 180°.

Angles at a pointRays OA, OB, OC and OD start at point O. Angle AOB is 30°. Angle BOC is a right angle. Angle COD is 115°.ABCD30°115°O

In the figure : In the figure ∠AOB = 30° is an acute angle, ∠BOC = 90° is a right angle and ∠COD = 115° is an obtuse angle.

Live figure

Types of angles

In the app you can move this figure with your finger.

Open in the app
  • Easy : Easy: 35° is acute.
  • Medium : Medium: 100° is obtuse.
  • Exam-type : Exam: 90° is a right angle, but 91° is already obtuse.

Rule 2: Straight, reflex, complete — Straight angle (सरळ कोन) = 180°, a straight line formed by two edges. Reflex angle (प्रवृद्ध कोन) is greater than 180° and less than 360°. Complete angle (पूर्ण कोन) = 360°, when both arms meet again.

  • Easy : Easy: 180° is a straight angle.
  • Medium : Medium: 250° is a reflex angle.
  • Exam-type : Exam: A clock hand that makes one full turn moves through 360°, a complete angle.
Remember

0° zero, <90° acute, 90° right, 90°–180° obtuse, 180° straight, 180°–360° reflex, 360° complete.

Common mistake

Calling 180° an obtuse angle. It is exactly a straight angle.

Shortcut

Memory clue: "लघु < काटकोन < विशाल < सरळ < प्रवृद्ध < पूर्ण" (small to big).

Spot the difference

Angle typeMeasure
Acute0° to 90°
Right90°
Obtuse90° to 180°
Straight180°

Practice questions

Easy
  1. 1Classify 25°.
  2. 2Classify 90°.
  3. 3Classify 120°.
Medium
  1. 4Classify 180°.
  2. 5Classify 300°.
  3. 6Classify 360°.
Challenging
  1. 7An angle is greater than 90° and less than 180°. What type is it?
  2. 8Is 91° acute or obtuse? Why?
Show answers (try them yourself first)
  1. 1Acute.
  2. 2Right angle.
  3. 3Obtuse.
  4. 4Straight angle.
  5. 5Reflex.
  6. 6Complete angle.
  7. 7Obtuse.
  8. 8Obtuse, because it is greater than 90°.

Using angle measures

From daily life

The hands of a clock make a different angle every hour. Using angle measures, we can name those angles and find missing ones.

Simple definition

A straight angle is 180° (2 right angles) and a complete angle is 360° (4 right angles). Angles on a straight line add up to 180°; angles around a point add up to 360°.

Rule 1: Classifying and the clock — 62° is acute, 130° is obtuse and 200° is reflex. Each hour on the clock is 30°. At 12 the hands make a zero angle, at 2 an acute angle (60°), at 3 a right angle, at 4 an obtuse angle (120°) and at 6 a straight angle.

  • Easy : Easy: Three right angles make 3 × 90° = 270°, a reflex angle.
  • Medium : Medium: A straight angle divided into 4 equal parts gives 180° ÷ 4 = 45° each.
  • Exam-type : Exam: At 5 o'clock the angle is 5 × 30° = 150°, an obtuse angle.

Rule 2: Angles on a straight line — Angles on one side of a straight line add up to 180°. If 65° + x = 180°, then x = 180° − 65° = 115°.

Angles at a pointRays OA, OB and OC start at point O. Angle AOB is 48°. Angle BOC is marked x.ABC48°xO

In the figure : In the figure 48° + x = 180°, so x = 132°.

  • Easy : Easy: 70° + x = 180° gives x = 110°.
  • Medium : Medium: 110° + y = 180° gives y = 70°.
  • Exam-type : Exam: Two angles on a straight line are x and 2x. Then 3x = 180°, so x = 60° and the other is 120°.

Rule 3: Angles around a point — Angles around a point add up to 360°. If 120° + 100° + z = 360°, then z = 360° − 220° = 140°.

Angles at a pointRays OA, OB and OC start at point O. Angle AOB is 100°. Angle BOC is a right angle. Angle COA is marked y.ABC100°yO

In the figure : In the figure around a point 100° + 90° + y = 360°, so y = 170°.

  • Easy : Easy: Three right angles and w around a point: 270° + w = 360°, so w = 90°.
  • Medium : Medium: 150° + 90° + y = 360° gives y = 120°.
  • Exam-type : Exam: Angles 80°, 95° and u around a point: u = 360° − 175° = 185°, a reflex angle.
Remember

Straight line: 180°. Around a point: 360°. Each clock hour: 30°.

Common mistake

Using 360° for angles on a straight line. It is 180° there.

Shortcut

Memory clue: "रेषेवर १८०, बिंदूभोवती ३६०" (line 180, point 360).

Spot the difference

SituationSum of angles
On a straight line180°
Around a point360°
Each clock hour30°

Practice questions

Easy
  1. 1Classify 62°.
  2. 2Classify 200°.
  3. 3How many right angles are in a straight angle?
Medium
  1. 4Find x if x + 70° = 180°.
  2. 5Around a point 150°, 90° and y fit. Find y.
  3. 6What angle do the clock hands make at 4 o'clock? What type?
Challenging
  1. 7A straight angle is divided into three equal parts. Find each part and its type.
  2. 8Find an acute angle whose 2, 3 and 4 times are acute and whose 5 times is obtuse.
Show answers (try them yourself first)
  1. 1Acute.
  2. 2Reflex.
  3. 3Two.
  4. 4110°.
  5. 5360° − 240° = 120°.
  6. 64 × 30° = 120°, obtuse.
  7. 7180° ÷ 3 = 60°, acute.
  8. 819°: 38°, 57°, 76° are acute and 95° is obtuse.

Quick revision

  • Point (no size), line (infinite both sides), plane (flat, infinite on all sides); segment has two endpoints; ray has one starting point.
  • Angle = two rays with a common vertex; the vertex is the middle letter.
  • 0° zero, below 90° acute, 90° right, 90°–180° obtuse, 180° straight, 180°–360° reflex, 360° complete.