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State Board · Class 5 · Mathematics

Lesson 6: Angles

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

This lesson is also in: मराठी

Angle, arms and vertex

From daily life

Open a pair of scissors a little. The two blades start from one point (the screw) and spread apart. The opening between the blades is an angle.

Simple definition

An angle (कोन) is made when two rays start from the same point. The common starting point is the vertex (शिरोबिंदू) and the two rays are the arms (बाहू). The size of an angle is the opening between its arms, measured in degrees (अंश, °).

Angle ABC with vertex B and arms BA and BC.BCA∠ABCvertexarm BCarm BA

In the figure : Angle ABC: the vertex is B, and the arms are the rays BA and BC.

Rule 1: Name an angle — Write three letters; the middle letter is the vertex.

  • Easy : In ∠PQR the vertex is Q and the arms are QP and QR.
  • Medium : ∠XYZ and ∠ZYX are the same angle, with the vertex Y.
  • Exam-type : A door opens by 30° and then by 60°. The second opening has the bigger angle, even though the door has the same width.

Rule 2: Size is the opening — Longer arms do not make a bigger angle.

  • Easy : Two angles of 40° are equal even if one drawing has longer arms.
  • Medium : An angle of 70° is bigger than an angle of 50°, however long their arms are drawn.
  • Exam-type : You can extend the arms of an angle with a pencil; its size stays the same.
Remember

The vertex is the common point. The size depends only on the opening between the arms.

Common mistake

Thinking a drawing with longer arms has a bigger angle. Compare the openings, not the lengths.

Shortcut

Memory clue: Middle letter = vertex (∠ABC → B).

Spot the difference

PartNameIn ∠ABC
Common pointvertexB
Two raysarmsBA, BC
Sizedegreesthe opening

Practice questions

Easy
  1. 1Name the vertex of ∠PQR.
  2. 2What are the arms of ∠XYZ?
  3. 3In which unit do we measure angles?
Medium
  1. 4Does an angle with longer arms always have a larger size?
  2. 5Which is bigger: an angle of 35° or an angle of 53°?
  3. 6Name an angle with vertex M and arms MA and MB in two ways.
Challenging
  1. 7Rohan says that a bigger drawing of an angle gives a bigger angle. Is he right?
  2. 8A door opens by 40° on Monday and 65° on Tuesday. On which day is it opened wider, and by how much more?
Show answers (try them yourself first)
  1. 1Q
  2. 2YX and YZ
  3. 3degrees (°)
  4. 4No. The size depends on the opening only.
  5. 553°
  6. 6∠AMB and ∠BMA
  7. 7No. The openings are the same; only the arms are longer.
  8. 8Tuesday, by 65° − 40° = 25°.

Types of angles

From daily life

A slice of cake has a sharp tip. The corner of a book is square. A chair tilted far back is wide open. Each of these shows a different kind of angle.

Simple definition

An acute angle (न्यूनकोन) is less than 90°. A right angle (काटकोन) is exactly 90°. An obtuse angle (विशालकोन) is more than 90° and less than 180°. A straight angle (सरळकोन) is exactly 180°.

Four angles: acute (less than 90), right (90), obtuse (between 90 and 180) and straight (180).acute<90°right=90°obtuse>90°straight=180°

In the figure : Acute is less than 90°, right is 90°, obtuse is between 90° and 180°, straight is 180°.

Rule 1: Classify by the size — Compare the number of degrees with 90° and 180°.

  • Easy : 45° is acute and 120° is obtuse.
  • Medium : 90° is a right angle and 180° is a straight angle.
  • Exam-type : 89° is acute; 91° is obtuse. The line is exactly at 90°.
Live figure

Types of angles

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

Open in the app

Rule 2: Quick test with a page corner — Place a page corner on the angle. Smaller: acute. Same: right. Bigger but not a straight line: obtuse.

  • Easy : The corner of a notebook is a right angle.
  • Medium : The tip of a pencil is an acute angle.
  • Exam-type : The two hands of a clock at 5 o'clock make 150°, an obtuse angle, because 150° is more than 90° and less than 180°.
Remember

Right angle = 90°, straight angle = 180°, and two right angles make a straight angle.

Common mistake

Calling a straight angle (180°) an obtuse angle. An obtuse angle is less than 180°.

Shortcut

Memory clue: Acute is "a cute small" angle; Obtuse is "Obese", big.

Spot the difference

TypeSizeExample
Acuteless than 90°60°
Right90°90°
Obtuse90° to 180°130°
Straight180°180°

Practice questions

Easy
  1. 1Is 75° acute, right or obtuse?
  2. 2Name the angle of exactly 90°.
  3. 3Is 140° acute, right or obtuse?
Medium
  1. 4Which angle looks like a straight line?
  2. 5Classify 30°, 90°, 120°, 180°.
  3. 6The corner of a page and an angle are placed one over the other. The angle looks bigger than the page corner but is not a straight line. What type is it?
Challenging
  1. 7Is the angle of 91° nearer to a right angle? What type is it?
  2. 8The hands of a clock make 150° at 5 o'clock. What type of angle is this, and how many degrees must it grow to become straight?
Show answers (try them yourself first)
  1. 1Acute.
  2. 2Right angle.
  3. 3Obtuse.
  4. 4Straight angle (180°).
  5. 5Acute, right, obtuse, straight.
  6. 6Obtuse.
  7. 7Yes; it is obtuse because 91° > 90°.
  8. 8Obtuse; 180° − 150° = 30°.

Measuring with a protractor

From daily life

To check how wide a ladder is opened, a carpenter uses a small tool with numbers. In the classroom that tool is a protractor, a half-circle ruler with degrees from 0° to 180°.

Simple definition

A protractor (कोनमापक) measures angles in degrees. Place the centre on the vertex, one arm along the 0° line, and read the scale that starts with 0° on that arm.

A protractor with its centre on the vertex, one arm on 0 and the other arm reading 55 degrees.030609012015018055°centre on vertex

In the figure : The centre is on the vertex, one arm is on 0° and the other arm reads 55°.

Rule 1: Three steps to measure — Centre on the vertex, one arm on 0°, read where the other arm cuts the scale.

  • Easy : If the second arm touches 55° on the scale starting at 0°, the angle is 55°.
  • Medium : An arm touching 120° means the angle is 120°, which is obtuse.
  • Exam-type : The arm lies between the readings 40° and 140° on the two scales. For an acute-looking angle choose 40°, the smaller one.

Rule 2: Choose the right scale — Use the scale whose 0° is on the arm lying on the base line.

  • Easy : Acute angle → the reading is less than 90°.
  • Medium : Obtuse angle → the reading is more than 90°.
  • Exam-type : Check the type first. If you read 130° for an angle that looks smaller than a page corner, you read the wrong scale; the right reading is 180° − 130° = 50°.
Remember

Centre on the vertex, 0° on one arm, and read the other arm on the same scale.

Common mistake

Reading from the wrong scale. Always read the scale that starts with 0° on your arm.

Shortcut

Memory clue: Centre, Zero, Read.

Spot the difference

StepWhat to do
1Centre on the vertex
2One arm on 0°
3Read the scale from 0°

Practice questions

Easy
  1. 1What does a protractor measure?
  2. 2Where do we place the centre of the protractor?
  3. 3An arm touches 55° on the scale from 0°. What is the angle?
Medium
  1. 4An arm reads 125° on the scale from 0°. What type is the angle?
  2. 5An acute-looking angle shows readings 35° and 145°. Which one is its measure?
  3. 6An obtuse-looking angle shows 70° and 110°. Which is the measure?
Challenging
  1. 7Meena read 140° for an angle smaller than a page corner. Find the correct measure.
  2. 8The first arm of an angle is on 0°. The other arm lies exactly on the 90° mark. Name the angle and say how a page corner compares with it.
Show answers (try them yourself first)
  1. 1Angles.
  2. 2On the vertex.
  3. 355°
  4. 4Obtuse.
  5. 535°
  6. 6110°
  7. 7She read the wrong scale. 180° − 140° = 40°.
  8. 8Right angle; a page corner fits it exactly.

Drawing and comparing angles

From daily life

A carpenter wants a stool leg to slant at 70°. First he draws the angle on paper with a protractor, then he cuts the wood. Drawing and comparing angles are everyday skills.

Simple definition

To draw an angle: draw a base ray, put the protractor centre at its start, mark the needed degree and join the vertex to the mark. Angles can be compared by their degrees, added and taken apart.

Rule 1: Draw an angle — Base ray, centre on its start, mark the degree, join.

  • Easy : To draw 50°: mark 50° on the scale starting at 0° and join it to the vertex.
  • Medium : To draw 110°, mark 110° on the same scale and join. The angle is obtuse.
  • Exam-type : Draw ∠PQR = 75° by taking Q as the vertex, QR as the base arm and marking 75° at P.

Rule 2: Compare, add and take apart — Compare degrees. Two angles next to each other add up.

  • Easy : 75° is bigger than 57°.
  • Medium : Arrange 40°, 100° and 70° in increasing order: 40°, 70°, 100°.
  • Exam-type : Two 45° angles side by side make a right angle: 45° + 45° = 90°. Two right angles make a straight angle: 90° + 90° = 180°.
A right angle split into two 45 degree angles, and a straight angle split into two right angles.45°45°45° + 45° = 90°90°90°90° + 90° = 180°

In the figure : A right angle is two 45° angles; a straight angle is two right angles.

Remember

Angles that sit side by side add; so a part can be found by subtracting from the whole.

Common mistake

Adding the arm lengths instead of the degrees. Only degrees are added.

Shortcut

Memory clue: 90 + 90 = 180 — two right angles lie flat.

Spot the difference

IdeaExample
Compare75° > 57°
Add45° + 45° = 90°
Take apart180° − 110° = 70°

Practice questions

Easy
  1. 1Which is bigger: 75° or 57°?
  2. 2Arrange in increasing order: 100°, 40°, 70°.
  3. 3What do two right angles make together?
Medium
  1. 430° + 60° = ? Name the angle formed.
  2. 5An angle of 110° and another angle together make a straight angle. Find the other angle.
  3. 6Arrange in decreasing order: 85°, 120°, 60°, 90°.
Challenging
  1. 7Two angles of a bracket are 35° and 55° side by side. Find the total and say what type it is.
  2. 8Draw an angle of 70° and say how a page corner compares with it.
Show answers (try them yourself first)
  1. 175°
  2. 240°, 70°, 100°
  3. 3A straight angle, 180°.
  4. 490°, a right angle.
  5. 5180° − 110° = 70°
  6. 6120°, 90°, 85°, 60°
  7. 735° + 55° = 90°; a right angle.
  8. 870° is acute, smaller than a page corner (90°).

Angles around us: clocks and turns

From daily life

The hands of a clock keep turning. At 3 o'clock they make a right angle; at 6 o'clock they make a straight line. Turns are angles too.

Simple definition

A full turn is 360°, a half turn is 180° and a quarter turn is 90°. The clock face has 12 hour gaps; each gap is 360° ÷ 12 = 30°.

A clock showing 4 o'clock: the angle between the hands is 4 times 30 which is 120 degrees.123694 o'clock4 × 30° = 120°each hour gap = 30°

In the figure : At 4 o'clock there are 4 hour gaps of 30° each: the angle is 120°.

Rule 1: Clock angles — Number of hour gaps × 30° gives the angle.

  • Easy : At 3 o'clock: 3 × 30° = 90°.
  • Medium : At 4 o'clock: 4 × 30° = 120°.
  • Exam-type : At 6 o'clock: 6 × 30° = 180°, a straight angle. At 12 o'clock the angle is 0°.

Rule 2: Turns and directions — Turning right by a right angle from north makes you face east.

  • Easy : A quarter turn is 90°; a half turn is 180°; a full turn is 360°.
  • Medium : Facing north, a half turn makes you face south.
  • Exam-type : Facing north, two quarter turns to the right (2 × 90° = 180°) also make you face south.
Remember

12 hour gaps × 30° = 360°, a full turn.

Common mistake

Taking the gap between hour marks as 12°. It is 30°.

Shortcut

Memory clue: Clock = 30 times the hour.

Spot the difference

TurnDegreesClock time
quarter90°3 o'clock
half180°6 o'clock
full360°12 o'clock

Practice questions

Easy
  1. 1How many degrees are there in a full turn?
  2. 2What is the angle between hour marks on a clock?
  3. 3Find the clock angle at 3 o'clock.
Medium
  1. 4Find the clock angle at 5 o'clock.
  2. 5How many degrees is a quarter turn? A half turn?
  3. 6Facing east, you turn a half turn. Which direction do you face?
Challenging
  1. 7At what time does the clock show an angle of 120°?
  2. 8Facing north, Amit turns right through 90° twice and then once more. Which direction does he face now?
Show answers (try them yourself first)
  1. 1360°
  2. 230°
  3. 390°
  4. 4150°
  5. 590° and 180°
  6. 6West.
  7. 74 o'clock
  8. 8Three right turns: north → east → south → west. He faces west (270° in all).