Math class 6 icse chapter 15

EX 15D

1. In Fig. 1, name the following angles.

(i) An acute angle at T

Explanation: An acute angle is an angle less than 90°.
In Fig. 1, the angle at T that is less than 90° is ∠PTS.

(ii) An acute angle at Q

Explanation: An acute angle is an angle less than 90°.
In Fig. 1, the angle at Q that is less than 90° is ∠PQR.

(iii) Three obtuse angles

Explanation: An obtuse angle is an angle greater than 90° but less than 180°.
In Fig. 1, the obtuse angles are:

∠QPR
∠PRS
∠QRS

(iv) A right angle

Explanation: A right angle is an angle equal to 90°.
In Fig. 1, the right angle is ∠QPT.

(v) A straight angle

Explanation: A straight angle is an angle equal to 180°.
In Fig. 1, the straight angle is ∠QPS.

2. Classify the angles whose magnitudes are given below.

(i) 132°

Explanation: 132° is greater than 90° and less than 180°.
Answer: Obtuse angle

(ii) 26°

Explanation: 26° is less than 90°.
Answer: Acute angle

(iii) 170°

Explanation: 170° is greater than 90° and less than 180°.
Answer: Obtuse angle

(iv) 30°

Explanation: 30° is less than 90°.
Answer: Acute angle

(v) 79°

Explanation: 79° is less than 90°.
Answer: Acute angle

(vi) 175°

Explanation: 175° is greater than 90° and less than 180°.
Answer: Obtuse angle

(vii) 98.7°

Explanation: 98.7° is greater than 90° and less than 180°.
Answer: Obtuse angle

(viii) 320°

Explanation: 320° is greater than 180° and less than 360°.
Answer: Reflex angle

(ix) 90°

Explanation: 90° is exactly 90°.
Answer: Right angle

(x) 0°

Explanation: 0° is neither acute, obtuse, right, straight, nor reflex.
Answer: Zero angle

(xi) 180°

Explanation: 180° is exactly 180°.
Answer: Straight angle

(xii) 270°

Explanation: 270° is greater than 180° and less than 360°.
Answer: Reflex angle

3. Find the number of degrees in:

(i) $\frac{1}{5}$ of a right angle

Explanation: A right angle is 90°.
$\frac{1}{5} \times 90° = 18°$ ×90°=18°
Answer: 18°

(ii) 40% of a right angle

Explanation: A right angle is 90°.
$40\% \times 90° = 0.4 \times 90° = 36°$ ×90°=0.4×90°=36°
Answer: 36°

(iii) 120% of a right angle

Explanation: A right angle is 90°.
$120\% \times 90° = 1.2 \times 90° = 108°$ ×90°=1.2×90°=108°
Answer: 108°

(iv) $\frac{4}{9}$ of a straight angle

Explanation: A straight angle is 180°.
$\frac{4}{9} \times 180° = 80°$ ×180°=80°
Answer: 80°

(v) 50% of a straight angle

Explanation: A straight angle is 180°.
$50\% \times 180° = 0.5 \times 180° = 90°$ ×180°=0.5×180°=90°
Answer: 90°

(vi) 130% of a straight angle

Explanation: A straight angle is 180°.
$130\% \times 180° = 1.3 \times 180° = 234°$ ×180°=1.3×180°=234°
Answer: 234°

4. State the type of angle in each case, formed between the following directions.

(i) East and North

Explanation: The angle between East and North is 90°.
Answer: Right angle

(ii) East and West

Explanation: The angle between East and West is 180°.
Answer: Straight angle

(iii) North and South-East

Explanation: The angle between North and South-East is 135°.
Answer: Obtuse angle

(iv) North and North-East

Explanation: The angle between North and North-East is 45°.
Answer: Acute angle

5. Using only a ruler, draw an acute angle, a right angle, and an obtuse angle in your notebook and name them.

Explanation: Use a ruler to draw the following angles:

Acute angle: Less than 90° (e.g., 45°)
Right angle: Exactly 90°
Obtuse angle: Greater than 90° but less than 180° (e.g., 120°)

6. In Fig. Q.6, find ∠x. Is it acute, obtuse, or reflex?

Explanation: The angle given is 120° and ∠x is the other part of the straight line.
$∠x = 180 ° - 120 ° = 60 °$ x=180°−120°=60°
Answer: ∠x is 60°, which is an acute angle.

7. What is the size of the reflex angle between the hands of a clock at 4 O'clock, 3 O'clock, and 10 O'clock?

Explanation:

At 4 O'clock: The angle between the hands is 120° (acute). The reflex angle is $360° - 120° = 240°$ −120°=240°.
At 3 O'clock: The angle between the hands is 90° (right). The reflex angle is $360° - 90° = 270°$ −90°=270°.
At 10 O'clock: The angle between the hands is 60° (acute). The reflex angle is $360° - 60° = 300°$ −60°=300°.

8. If the sum of two angles is equal to an obtuse angle, then which of the following is not possible?

· (a) One obtuse angle and one acute angle

· (b) One right angle and one acute angle

· (c) Two acute angles

· (d) Two right angles

· Explanation:

(a) Possible: e.g., 100° + 30° = 130° (obtuse)
(b) Possible: e.g., 90° + 30° = 120° (obtuse)
(c) Possible: e.g., 80° + 40° = 120° (obtuse)
(d) Not possible: 90° + 90° = 180° (straight, not obtuse)

· Answer: (d) Two right angles

9. 250% of a right angle is a/an _______ angle.

Explanation: A right angle is 90°.
$250\% \times 90° = 2.5 \times 90° = 225°$ ×90°=2.5×90°=225°
225° is greater than 180° and less than 360°.
Answer: Reflex angle

EX-15B

Concept Notes (Basics to Remember):

· Full Circle: A complete rotation is 360 degrees.

· Clock Basics:

o A clock circle is divided into 12 hours.

o Angle of 1 hour gap: 360 / 12 = 30 degrees.

o Minute Hand Speed: It moves 360 degrees in 60 minutes. Speed = 6 degrees per minute.

o Hour Hand Speed: It moves 30 degrees in 60 minutes. Speed = 0.5 degrees per minute.

· Directions: A compass has 8 main points (N, NE, E, SE, S, SW, W, NW). The angle between any two adjacent points is 45 degrees.

1. Fill in the blanks

· (i) The unit of measurement of an angle is degree.

o Explanation: Just like length is measured in meters, angles are measured in degrees.

· (ii) 1/2 rotation of a line about a point is equal to 180 degrees.

o Explanation: A full rotation is 360 degrees. Half of that is 360 / 2 = 180 degrees. This forms a straight line.

· (iii) The angle between North and West directions is 90 degrees.

o Explanation: North is at the top (12 o'clock) and West is at the left (9 o'clock). The angle between them is a right angle, or 90 degrees.

· (iv) The measure of the angle between the hands of the clock at 7 O’clock is 150 degrees.

o Explanation: At 7:00, the minute hand is at 12 and the hour hand is at 7. Counting from 12 backwards to 7 is complicated, so counting the "gaps" inside the smaller angle (7 to 12) is usually standard, but strictly 12 to 7 clockwise is 7 gaps.

o Let's count the gaps between 7 and 12 on the left side: 7-8, 8-9, 9-10, 10-11, 11-12. That is 5 gaps.

o Calculation: 5 gaps x 30 degrees per gap = 150 degrees.

· (v) The measure of the angle when the minute hand moves by 10 minutes is equal to 60 degrees.

o Explanation: The minute hand moves 6 degrees every minute.

o Calculation: 10 minutes x 6 degrees = 60 degrees.

2. Domestic Appliance Dial

The dial is a circle cut into 8 slices.

Value of one slice: 360 degrees / 8 = 45 degrees.

· (i) Clockwise from 'off' to 'cold': 90 degrees

o Steps: Start at Off -> Move 1 step to Very Cold -> Move 1 step to Cold.

o Calculation: 2 steps x 45 degrees = 90 degrees.

· (ii) Anticlockwise from 'hot' to 'warm': 45 degrees

o Steps: Start at Hot -> Move back 1 step to Warm.

o Calculation: 1 step x 45 degrees = 45 degrees.

· (iii) Anticlockwise from 'very cold' to 'very hot': 90 degrees

o Steps: Start at Very Cold -> Move back to Off -> Move back to Very Hot.

o Calculation: 2 steps x 45 degrees = 90 degrees.

· (iv) Clockwise from 'warm' to 'cool': 270 degrees

o Steps: Warm -> Hot -> Very Hot -> Off -> Very Cold -> Cold -> Cool.

o Calculation: That is 6 steps. 6 x 45 degrees = 270 degrees.

3. Clock Questions

· (i) When the second hand has moved from 12 to 6, how many degrees has it turned through? 180 degrees

o Explanation: 12 to 6 is exactly half of the clock circle. Half of 360 is 180.

· (ii) When the second hand has moved from 5 to 8, how many degrees has it turned through? 90 degrees

o Explanation: Count the gaps: 5 to 6, 6 to 7, 7 to 8. That is 3 gaps.

o Calculation: 3 gaps x 30 degrees = 90 degrees.

· (iii) What is the time on the clock when the hour hand moves clockwise...

o (a) 60 degrees from 6 O’clock: 8 O'clock

§ Since 30 degrees = 1 hour, then 60 degrees = 2 hours.

§ 6 O'clock + 2 hours = 8 O'clock.

o (b) 180 degrees from 10 O’clock: 4 O'clock

§ 180 degrees is a straight line (6 hours difference).

§ Directly opposite 10 on a clock is 4.

o (c) 270 degrees from 12 O’clock: 9 O'clock

§ 270 degrees is 3/4 of the circle, or 9 hours (270 / 30 = 9).

§ 12 + 9 hours = 9 O'clock.

· (iv) Through how many degrees does the minute hand turn in... (Speed: 6 degrees/min)

o (a) 1 minute = 6 degrees

o (b) 8 minutes = 8 x 6 = 48 degrees

o (c) 3/4 hour (45 minutes) = 45 x 6 = 270 degrees

o (d) 1 1/2 hours (90 minutes) = 90 x 6 = 540 degrees

· (v) Through how many degrees does the hour hand turn in... (Speed: 0.5 degrees/min)

o (a) 1 minute = 0.5 degrees

o (b) 10 minutes = 10 x 0.5 = 5 degrees

o (c) 30 minutes = 30 x 0.5 = 15 degrees

o (d) 2 hours (120 minutes) = 120 x 0.5 = 60 degrees

4. Compass Directions

There are 8 directions. One step between neighbors = 45 degrees.

· (i) Clockwise from NW to SW: 270 degrees

o Path: NW -> N -> NE -> E -> SE -> S -> SW.

o Count: 6 steps. 6 x 45 = 270.

· (ii) Clockwise from N to S: 180 degrees

o Path: N -> S is a straight line across the compass.

· (iii) Clockwise from N to E: 90 degrees

o Path: N -> NE -> E (2 steps). 2 x 45 = 90.

· (iv) Clockwise from NE to W: 225 degrees

o Path: NE -> E -> SE -> S -> SW -> W.

o Count: 5 steps. 5 x 45 = 225.

5. Circle divided into 16 equal parts

Value of one small part (gap): 360 degrees / 16 = 22.5 degrees.

· (i) Angle A1-O-A2: 22.5 degrees

o Explanation: This is exactly 1 gap.

· (ii) Angle A1-O-A3: 45 degrees

o Explanation: From A1 to A3 is 2 gaps. 2 x 22.5 = 45.

· (iii) Angle A1-O-A5: 90 degrees

o Explanation: From A1 to A5 is 4 gaps. 4 x 22.5 = 90.

· (iv) How many times is angle A1-O-A12 of angle A1-O-A2? 11 times

o Explanation: Angle A1-A2 is 1 gap. Angle A1-A12 is 11 gaps. 11 is 11 times bigger than 1.

· (v) How many times is angle A1-O-A4 of angle A1-O-A2? 3 times

o Explanation: Angle A1-A4 is 3 gaps. Angle A1-A2 is 1 gap. 3 is 3 times bigger than 1.

· (vi) What fraction of the whole revolution is the angle A5-O-A7? 1/8

o Explanation: A5 to A7 is 2 gaps. The whole circle is 16 gaps.

o Fraction = 2/16. Simplifying this fraction (divide top and bottom by 2) gives 1/8.

6. Multiple Choice Question (Boat)

· Problem: A boat sails North-East. Later it is found sailing South.

· Step 1: Draw a compass. Locate North-East (NE).

· Step 2: Locate South (S).

· Step 3: Count the degrees to turn from NE to S in a clockwise direction.

o NE to E = 45 degrees.

o E to SE = 45 degrees.

o SE to S = 45 degrees.

· Total: 45 + 45 + 45 = 135 degrees.

· Answer: (c) 135 degrees

EX-15A

1. Three Examples of Angles from Your Environment

The corner of a table or a book.

The angle formed by the hands of a clock at 3:00.

The angle formed by the blades of an open pair of scissors.

2. Naming the Angles in Three Different Ways

(i)

∠BAC (or ∠CAB)

∠A

∠a

(ii)

∠XZY (or ∠YZX)

∠Z

∠1

(iii)

∠PQR (or ∠RQP)

∠Q

∠2

(iv)

∠3 (Note: Since the vertex is not labeled with a letter, we can only refer to it as ∠3.)

3. AC is a Line. Are BA, BC, and BD Rays?

Yes, BA, BC, and BD are rays because they start at point B and extend infinitely in one direction.

Other Angles in the Figure:

∠ABD

∠CBD (or ∠DBC)

4. Fig. Q.4 Shows Two Intersecting Lines PQ and RS

(i) Can the Figure be Thought of as Being Formed by Four Rays with a Common Endpoint?

Yes, the figure can be thought of as being formed by four rays (OP, OQ, OR, and OS) with a common endpoint O.

(ii) Name at Least Six Angles Formed by the Rays

∠POR

∠ROQ

∠QOS

∠SOP

∠POQ (Straight angle)

∠ROS (Straight angle)

5. Name the Six Angles in Fig. Q.5 That Have O as a Vertex

∠AOB

∠BOC

∠COD

∠AOC (combination of ∠AOB and ∠BOC)

∠BOD (combination of ∠BOC and ∠COD)

∠AOD (combination of ∠AOB, ∠BOC, and ∠COD)

6. In Fig. Q.6, List the Points Which Are:

(i) In the Interior of ∠O

Points M and N

(ii) In the Exterior of ∠O

Points X and Y

(iii) Lying on ∠O

Points A and B

Sushovan Pal