Angles Class 6 ICSE — Maths Chapter 17

Written by Greya LakshmiPublished on 14 October 2026

Angles Class 6 ICSE is Chapter 17 of the Class 6 Mathematics book. It teaches what an angle is, how to name it, and how to measure it in degrees. The chapter covers the five angle types, the rule that angles around a point sum to 360°, and complementary and supplementary angles. Each section below follows the book’s own order and numbering. For more from this book, visit the Class 6 Maths hub.

ICSE Class 6 Mathematics Chapter 17 Angles PDF

What is inside Chapter 17

10 pages · 13 sections · 7 worked examples · 38 exercise questions

Section	Title
17.1	CONCEPT OF AN ANGLE
17.2	INTERIOR OF AN ANGLE
17.3	EXTERIOR OF AN ANGLE
17.4	ANGLE FORMED BY ROTATION
17.5	MEASURING AN ANGLE
17.6	USING A PROTRACTOR FOR MEASURING AN ANGLE
17.7	TYPES OF ANGLES
17.8	AN IMPORTANT RESULT
17.9	ADJACENT ANGLES
17.10	VERTICALLY OPPOSITE ANGLES
17.11	CONGRUENT (EQUAL) ANGLES
17.12	COMPLEMENTARY ANGLES
17.13	SUPPLEMENTARY ANGLES

EXERCISE 17(A): 13 questions
EXERCISE 17(B): 25 questions

17.1 CONCEPT OF AN ANGLE

An angle is formed by two different rays starting from the same fixed point. In the book’s figure, rays OA and OB start from point O to form angle AOB. The common point O is the vertex. The rays OA and OB are the arms or sides of the angle.

The symbol $\angle$ represents an angle. So angle AOB is written as $\angle AOB$ . When an angle is named with three capital letters, the middle letter is always the vertex. This means $\angle ABC = \angle CBA = \angle B$ , since B is the vertex in each case.

Two line segments that share an end point also form an angle there. The measure stays the same whether the arms are rays or line segments. One angle can carry several names when its arms hold many labelled points. The book shows $\angle O = \angle AOB = \angle BOA = \angle COD$ , and more.

Common naming mistakes:

Keep the vertex letter in the middle. $\angle ABC$ and $\angle CBA$ name the same angle.
Do not use the vertex name alone when several angles share it.
Use three letters so the angle is clear.

17.2 INTERIOR OF AN ANGLE

The interior of an angle is the region that lies within the angle. It is the region bounded by the arms. In the book’s figure, the shaded portion shows the interior of angle AOB.

Exercise 17(A) question 2 tests this idea directly. It asks you to name the points that lie inside angle PQR on a given diagram.

A point is in the interior when it sits between the two arms. Points on the arms themselves are not in the interior.

17.3 EXTERIOR OF AN ANGLE

The exterior of an angle is the region that lies outside the angle. Together, the arms split the plane into two regions: the interior and the exterior.

In the book’s figure, the shaded portion shows the exterior of angle AOB. The second part of exercise 17(A) question 2 asks you to name the points in the exterior of angle PQR.

A point is in the exterior when it does not lie between the arms. It sits outside the wedge the arms create.

17.4 ANGLE FORMED BY ROTATION

Consider a ray OA rotating about its end point O. Position OA is the initial position. As the ray turns, it reaches several positions before returning to where it started.

At any moment, the current position OA’ is the final position. The initial position OA and the final position OA’ form angle $\angle AOA'$ . As the ray rotates, the angle grows bigger and bigger.

The angle keeps growing until the ray returns to its starting position. That full turn is a complete rotation. This model prepares the 360° full rotation in section 17.5. It also sets up the reflex angle definition in section 17.7, where an angle can be larger than 180°.

17.5 MEASURING AN ANGLE

Degree is the unit of angle measure. Its symbol is °. An angle of 60 degrees is written as 60°.

One complete rotation measures 360°. Divide that rotation into 360 equal parts, and each part is one degree. Each degree divides into 60 minutes, written with one prime. Each minute divides into 60 seconds, written with two primes.

1 \text{ complete rotation} = 360^\circ

1^\circ = 60'

1' = 60''

Compact notation joins all three. For example, 25 degrees 30 minutes 15 seconds is 25° 30′ 15″. And 5 minutes 30 seconds is 5′ 30″.

The carry rule keeps addition correct:

60 seconds make one minute.
60 minutes make one degree.
Carry over every group of 60.

Worked pattern: add 41° 28′ 46″ and 27° 35′ 38″.

Step 1: Add seconds. $46'' + 38'' = 84''$ . Since $60'' = 1'$ , write $24''$ and carry $1'$ .

Step 2: Add minutes. $28' + 35' + 1' = 64'$ . Since $60' = 1^\circ$ , write $4'$ and carry $1^\circ$ .

Step 3: Add degrees. $41^\circ + 27^\circ + 1^\circ = 69^\circ$ .

$41^\circ 28' 46'' + 27^\circ 35' 38'' = 69^\circ 4' 24''$

Final answer: $69^\circ 4' 24''$ .

17.6 USING A PROTRACTOR FOR MEASURING AN ANGLE

A protractor is a semi-circular disc marked in degrees from 0° to 180°. The centre O is the midpoint of its base-line.

Steps to measure angle PQR:

Keep the base-line on one arm, QR.
Put the centre O on the vertex, Q.
Read where the other arm, PQ, crosses the scale.

In the book’s figure, arm PQ crosses at 40°, so $\angle PQR = 40^\circ$ .

A protractor has two scales, one reading from each end. Read the scale that the arm actually touches. Reading the opposite side of the base-line gives the wrong measure. This is the most common protractor error.

17.7 TYPES OF ANGLES

The chapter names five types of angles by their degree range.

Angle type	Measure
Acute	less than $90^\circ$
Right	exactly $90^\circ$
Obtuse	more than $90^\circ$ , less than $180^\circ$
Straight	exactly $180^\circ$
Reflex	more than $180^\circ$ , less than $360^\circ$

A straight angle equals two right angles: $180^\circ = 2 \times 90^\circ$ . Two lines, rays, or segments that form $90^\circ$ are perpendicular to each other.

Memory aid tied to a book corner: an acute angle is smaller than a book corner. An obtuse angle is wider than a book corner but flatter than a straight line. A reflex angle is larger than a straight line.

17.8 AN IMPORTANT RESULT

The sum of the angles around a point is always $360^\circ$ .

In the book’s figure, four angles meet at point O.

\angle AOB + \angle BOC + \angle COD + \angle DOA = 360^\circ

This result lets you find an unknown angle when the others are known.

Worked pattern: around a point, angles of $2x^\circ$ , $3x^\circ$ and $60^\circ$ are given.

Step 1: Write the sum of the angles.

$2x + 3x + 60^\circ = 360^\circ$

Step 2: Combine the x terms.

$5x = 300^\circ$

Step 3: Divide by 5.

$x = 60^\circ$

Final answer: $x = 60^\circ$ . The angles are $120^\circ$ , $180^\circ$ and $60^\circ$ .

17.9 ADJACENT ANGLES

Two angles are adjacent if they meet three conditions:

They have a common vertex.
They have one common arm.
The other arms lie on opposite sides of the common arm.

In the book’s pair, $\angle AOB$ and $\angle BOC$ share vertex O and arm OB. Arms OA and OC lie on opposite sides of OB.

Each condition matters. Without a shared arm, two angles cannot be adjacent even if they sit next to each other. Exercise 17(A) question 5 asks you to name three pairs of adjacent angles in one figure.

17.10 VERTICALLY OPPOSITE ANGLES

When two straight lines intersect, the angles on opposite sides of the point of intersection are vertically opposite angles.

In the figure, lines AB and CD intersect at O. $\angle AOC$ and $\angle BOD$ are vertically opposite. So are $\angle BOC$ and $\angle AOD$ .

Vertically opposite angles are always equal:

\angle AOC = \angle BOD

\angle BOC = \angle AOD

The same four angles also sum to $360^\circ$ around O, linking this to section 17.8. Exercise 17(A) question 12 asks you to name the pairs and find reflex measures. A reflex angle here is the larger turn around the intersection.

17.11 CONGRUENT (EQUAL) ANGLES

Congruent angles have the same measure. The book shows three angles, each equal to $45^\circ$ , so they are congruent.

Two equation patterns appear here. The first uses a straight line. On straight line AOB, angles of $(2x + 20)^\circ$ and $(x - 5)^\circ$ lie together.

Step 1: Parts on a straight line sum to $180^\circ$ .

$(2x + 20)^\circ + (x - 5)^\circ = 180^\circ$

Step 2: Simplify.

$3x + 15^\circ = 180^\circ \Rightarrow x = 55^\circ$

Final answer: The angles are $130^\circ$ and $50^\circ$ .

The second pattern works around a point. Angles $x^\circ$ , $3x^\circ$ , $40^\circ$ and $20^\circ$ meet at one point.

Step 1: Angles around a point sum to $360^\circ$ .

$x + 3x + 40^\circ + 20^\circ = 360^\circ$

Step 2: Simplify.

$4x = 300^\circ \Rightarrow x = 75^\circ$

Final answer: $x = 75^\circ$ .

Parts on a straight line sum to $180^\circ$ . Angles around a point sum to $360^\circ$ . These two rules drive most of this chapter’s questions.

17.12 COMPLEMENTARY ANGLES

Two angles are complementary if their sum is $90^\circ$ . Each angle is the complement of the other.

x + y = 90^\circ

If $x + y = 90^\circ$ , then x is the complement of y, and y is the complement of x.

Worked values: the complement of $27^\circ$ is $90^\circ - 27^\circ = 63^\circ$ . To find the complement of $\frac{3}{5}$ of a right angle, work in two steps. First find the angle: $\frac{3}{5} \times 90^\circ = 54^\circ$ . Then subtract: $90^\circ - 54^\circ = 36^\circ$ .

\frac{3}{5} \times 90^\circ = 54^\circ

90^\circ - 54^\circ = 36^\circ

Common confusion: mixing complements with supplements. Complements sum to $90^\circ$ . Supplements sum to $180^\circ$ .

17.13 SUPPLEMENTARY ANGLES

Two angles are supplementary if their sum is $180^\circ$ . Each angle is the supplement of the other.

x + y = 180^\circ

Worked value: the supplement of $118^\circ$ is $180^\circ - 118^\circ = 62^\circ$ .

Ratio method with fresh numbers: supplementary angles in the ratio 7 : 5. Add the parts: $7 + 5 = 12$ . The larger angle is $\frac{7}{12}$ of $180^\circ = 105^\circ$ . The smaller is $\frac{5}{12}$ of $180^\circ = 75^\circ$ .

\frac{7}{12} \times 180^\circ = 105^\circ

\frac{5}{12} \times 180^\circ = 75^\circ

Algebraic method with the same ratio: let the angles be $7x$ and $5x$ .

7x + 5x = 180^\circ

12x = 180^\circ

x = 15^\circ

So the angles are $105^\circ$ and $75^\circ$ .

Mnemonic: c comes before s in the alphabet, just as 90 comes before 180. So complements sum to $90^\circ$ and supplements sum to $180^\circ$ .

What to practise from this chapter

This chapter has two exercise blocks.

EXERCISE 17(A) — 13 questions.
EXERCISE 17(B) — 25 questions.

Practice mapping:

17(A) questions 1 to 3 test sections 17.1 to 17.3: naming the vertex and arms, and interior and exterior points.
17(A) question 4 tests section 17.5: adding degree-minute-second measures.
17(A) questions 5 to 13 test sections 17.7 to 17.11: adjacent, reflex, and vertically opposite angles, plus unknown angles on a straight line and around a point.
17(B) questions 1 to 9 and 14 to 15 test sections 17.12 and 17.13: complements and supplements with fractions, percentages, and ratios.
17(B) question 10 applies sections 17.5 and 17.7 to a clock face.

The two rules that turn up most: parts on a straight line sum to $180^\circ$ , and angles around a point sum to $360^\circ$ . Both exercise blocks follow the ICSE Class 6 Mathematics syllabus. For official syllabus and examination information, see the CISCE website. More chapters from this book are listed on the Class 6 books hub.

Frequently asked questions

Why must the middle letter be the vertex when an angle is named with three letters?

The middle letter names the vertex because the convention fixes one reading order. In $\angle ABC$ , B is the vertex. Reading $\angle CBA$ gives the same angle because B stays in the middle. If the vertex moves to the end, you name a different angle.

How do I remember the difference between complementary and supplementary angles?

Use the alphabet. C comes before S, just as 90 comes before 180. So complements sum to $90^\circ$ and supplements sum to $180^\circ$ . Another check: the complement of $27^\circ$ is $63^\circ$ , and the supplement of $118^\circ$ is $62^\circ$ .

How do I add angles written in degrees, minutes and seconds?

Add the seconds, minutes, and degrees separately. Carry 60 seconds as one minute, and 60 minutes as one degree. For example, $41^\circ 28' 46''$ plus $27^\circ 35' 38''$ gives $69^\circ 4' 24''$ .

What angle do the clock hands make at 3, 6, 9 and 12 o’clock?

At 3 o’clock the hands form a right angle, $90^\circ$ . At 6 o’clock they form a straight angle, $180^\circ$ . At 9 o’clock they form $90^\circ$ again. At 12 o’clock both hands point up, so the angle is $0^\circ$ . Each hour mark on the clock face is $30^\circ$ apart.

Are vertically opposite angles always equal?

Yes. When two straight lines intersect, the opposite pairs are always equal. If $\angle AOC = \angle BOD$ and $\angle BOC = \angle AOD$ , the two pairs match. This holds for any pair of intersecting lines.

Reference: ICSE Class 6 Mathematics textbook material, chapter 17: Angles — CISCE-aligned.

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