ML Aggarwal Class 6 Maths Solutions Chapter 11 Understanding Symmetrical Shapes

Question 1. Why is it better to use a divider and a ruler than a ruler only, while measuring the length of a line segment?
Answer: Using just a ruler can lead to mistakes when you measure a line. The thickness of the ruler itself can cause errors. Also, if you look at the ruler from an angle instead of straight on, you get a wrong reading (this is called a parallax error). A divider has two sharp pointy ends. When you use a divider with a ruler, those sharp points show you exactly where the line ends are on the ruler scale. This gives you an exact reading and avoids mistakes from the ruler's thickness or from looking at it at an angle.
In simple words: A divider with a ruler gives you the right answer because the sharp points mark the exact spots, with no mistakes.

Exam Tip: Mention both sources of error (ruler thickness and viewing angle) and explain how a divider fixes them — this shows full understanding of the measuring method.

Question 2. In the adjoining figure, compare the line segments with the help of a divider and fill in the blanks by using the symbol >, = or <:
(i) AB .... CD
(ii) BC .... AB
(iii) AC .... BD
(iv) CD .... BD
Answer:
(i) AB = CD
(ii) BC < AB
(iii) AC > BD
(iv) CD < BD
In simple words: Use a divider to check each pair of line segments. If they match in length, write =. If the first one is longer, write >. If the second one is longer, write <.

Exam Tip: Always measure each segment carefully with the divider and compare them one by one — rushing this step often leads to wrong symbols.

Question 3. If A, B and C are collinear points such that AB = 6 cm, BC = 4 cm and AC = 10 cm, which one of them lies between the other two?
Answer: Given: A, B and C are collinear points. AB = 6 cm, BC = 4 cm, AC = 10 cm.

Let's check: AB + BC = 6 cm + 4 cm = 10 cm = AC

Since AB + BC = AC, point B lies between A and C.

\( \therefore \) Point B lies between A and C.
In simple words: When you add the two shorter distances and get the longest distance, the middle point sits between the other two points.

Exam Tip: Always add the two shorter segments to check if they equal the longest one — this confirms which point lies between the others.

Question 4. In the adjoining figure, verify the following by measurement:
(i) AB + BC = AC
(ii) AC - BC = AB
Answer: By measuring the line segments with a ruler, we get: AB = 3 cm, BC = 2 cm, AC = 5 cm

(i) AB + BC = 3 cm + 2 cm = 5 cm = AC
\( \therefore \) AB + BC = AC is verified.

(ii) AC - BC = 5 cm - 2 cm = 3 cm = AB
\( \therefore \) AC - BC = AB is verified.
In simple words: When points line up in order, adding two parts gives the whole, and taking away one part from the whole gives the other part.

Exam Tip: Show all arithmetic steps clearly — examiners want to see both the addition and subtraction worked out fully, not just the final answer.

Question 5. In the adjoining figure, verify by measurement that:
(i) AC + BD = AD + BC
(ii) AB + CD = AD - BC
Answer: By measuring the line segments with a ruler, we get: AB = 2 cm, BC = 1 cm, CD = 1.5 cm, AC = 3 cm, BD = 2.5 cm, AD = 4.5 cm

(i) AC + BD = 3 cm + 2.5 cm = 5.5 cm
AD + BC = 4.5 cm + 1 cm = 5.5 cm
\( \therefore \) AC + BD = AD + BC is verified.

(ii) AB + CD = 2 cm + 1.5 cm = 3.5 cm
AD - BC = 4.5 cm - 1 cm = 3.5 cm
\( \therefore \) AB + CD = AD - BC is verified.
In simple words: These equations show how segments can be regrouped and still balance out to the same value.

Exam Tip: Write out all the measured values first, then work through each equation step by step to show both sides match exactly.

Question 6. In the adjoining figure, measure the lengths of the sides of the triangle ABC and verify:
(i) AB + BC > AC
(ii) BC + AC > AB
(iii) AC + AB > BC
Answer: By measuring the sides of triangle ABC with a ruler, we get: AB = 4.5 cm, BC = 5 cm, AC = 3.5 cm

(i) AB + BC = 4.5 cm + 5 cm = 9.5 cm
Since 9.5 cm > 3.5 cm,
\( \therefore \) AB + BC > AC is verified.

(ii) BC + AC = 5 cm + 3.5 cm = 8.5 cm
Since 8.5 cm > 4.5 cm,
\( \therefore \) BC + AC > AB is verified.

(iii) AC + AB = 3.5 cm + 4.5 cm = 8 cm
Since 8 cm > 5 cm,
\( \therefore \) AC + AB > BC is verified.
In simple words: In any triangle, when you add any two sides together, they are always longer than the third side. This is the triangle rule.

Exam Tip: The triangle inequality is a key property — make sure to verify all three conditions, not just one, to show full understanding of this rule.

Question 1. What fraction of a clockwise revolution does the hour hand of a clock turn through when it goes from:
(i) 4 to 10
(ii) 2 to 5
(iii) 7 to 10
(iv) 8 to 5
(v) 11 to 5
(vi) 6 to 3

Also find the number of right angles turned in each case.
Answer: In one complete revolution, the hour hand covers 12 hours on the clock face.

So, 1 hour movement = \( \frac{1}{12} \) of a revolution.

Also, one complete revolution = 4 right angles, so 1 hour = \( \frac{1}{3} \) of a right angle and 3 hours = 1 right angle.

(i) From 4 to 10 (clockwise), the hour hand moves through 6 hours.

Fraction of revolution = \( \frac{6}{12} = \frac{1}{2} \)

Number of right angles = \( \frac{6}{3} = 2 \)

\( \therefore \) Fraction of revolution = \( \frac{1}{2} \) and number of right angles = 2.

(ii) From 2 to 5 (clockwise), the hour hand moves through 3 hours.

Fraction of revolution = \( \frac{3}{12} = \frac{1}{4} \)

Number of right angles = \( \frac{3}{3} = 1 \)

\( \therefore \) Fraction of revolution = \( \frac{1}{4} \) and number of right angles = 1.

(iii) From 7 to 10 (clockwise), the hour hand moves through 3 hours.

Fraction of revolution = \( \frac{3}{12} = \frac{1}{4} \)

Number of right angles = \( \frac{3}{3} = 1 \)

\( \therefore \) Fraction of revolution = \( \frac{1}{4} \) and number of right angles = 1.

(iv) From 8 to 5 (clockwise), the hour hand moves through 9 hours.

Fraction of revolution = \( \frac{9}{12} = \frac{3}{4} \)

Number of right angles = \( \frac{9}{3} = 3 \)

\( \therefore \) Fraction of revolution = \( \frac{3}{4} \) and number of right angles = 3.

(v) From 11 to 5 (clockwise), the hour hand moves through 6 hours.

Fraction of revolution = \( \frac{6}{12} = \frac{1}{2} \)

Number of right angles = \( \frac{6}{3} = 2 \)

\( \therefore \) Fraction of revolution = \( \frac{1}{2} \) and number of right angles = 2.

(vi) From 6 to 3 (clockwise), the hour hand moves through 9 hours.

Fraction of revolution = \( \frac{9}{12} = \frac{3}{4} \)

Number of right angles = \( \frac{9}{3} = 3 \)

\( \therefore \) Fraction of revolution = \( \frac{3}{4} \) and number of right angles = 3.
In simple words: Count how many numbers the hour hand passes. Divide that by 12 to get the fraction. Divide the number of hours by 3 to get the right angles.

Exam Tip: Always count the hours the hand moves, not the starting position — use the hour count to find both the fraction and the right angles.

Question 2. Where will the hour hand of a clock stop if it
(i) starts at 10 and makes \( \frac{1}{2} \) of a revolution, clockwise?
(ii) starts at 4 and makes \( \frac{1}{4} \) of a revolution, clockwise?
(iii) starts at 4 and makes \( \frac{3}{4} \) of a revolution, clockwise?
Answer: One complete revolution of the hour hand = 12 hours.

(i) \( \frac{1}{2} \) of a revolution = \( \frac{1}{2} \times 12 = 6 \) hours

Starting from 10 and moving 6 hours clockwise: 10 → 11 → 12 → 1 → 2 → 3 → 4

\( \therefore \) The hour hand will stop at 4.

(ii) \( \frac{1}{4} \) of a revolution = \( \frac{1}{4} \times 12 = 3 \) hours

Starting from 4 and moving 3 hours clockwise: 4 → 5 → 6 → 7

\( \therefore \) The hour hand will stop at 7.

(iii) \( \frac{3}{4} \) of a revolution = \( \frac{3}{4} \times 12 = 9 \) hours

Starting from 4 and moving 9 hours clockwise: 4 → 5 → 6 → 7 → 8 → 9 → 10 → 11 → 12 → 1

\( \therefore \) The hour hand will stop at 1.
In simple words: Multiply the fraction by 12 to get the number of hours. Count that many hour numbers forward from where you start.

Exam Tip: Convert the fraction to hours first, then count forward on the clock face — writing out the sequence helps avoid mistakes.

Question 3. Where will the hour hand of a clock stop if it starts from
(i) 6 and turns through 1 right angle?
(ii) 8 and turns through 2 right angles?
(iii) 10 and turns through 3 right angles?
(iv) 7 and turns through 2 straight angles?
Answer: One complete revolution of the hour hand = 12 hours.

1 right angle = \( \frac{1}{4} \) of a revolution = \( \frac{1}{4} \times 12 = 3 \) hours.

1 straight angle = \( \frac{1}{2} \) of a revolution = \( \frac{1}{2} \times 12 = 6 \) hours.

(i) Starting from 6 and turning through 1 right angle (3 hours, clockwise): 6 → 7 → 8 → 9

\( \therefore \) The hour hand will stop at 9.

(ii) Starting from 8 and turning through 2 right angles (6 hours, clockwise): 8 → 9 → 10 → 11 → 12 → 1 → 2

\( \therefore \) The hour hand will stop at 2.

(iii) Starting from 10 and turning through 3 right angles (9 hours, clockwise): 10 → 11 → 12 → 1 → 2 → 3 → 4 → 5 → 6 → 7

\( \therefore \) The hour hand will stop at 7.

(iv) Starting from 7 and turning through 2 straight angles (12 hours, clockwise): This is one complete revolution.

\( \therefore \) The hour hand will stop at 7 (same position).
In simple words: Change right angles and straight angles to hours, then count forward from your starting number on the clock.

Exam Tip: Remember: 1 right angle = 3 hours, and 1 straight angle = 6 hours. Converting first makes the rest easy.

Question 4. What fraction of a revolution have you turned through if you stand facing
(i) north and turn clockwise to face west?
(ii) south and turn anti-clockwise to face east?
(iii) east and turn clockwise (or anti-clockwise) to face west?

Also find the number of right angles turned in each case.
Answer: One complete revolution = 4 right angles = 360°.

So, \( \frac{1}{4} \) revolution = 1 right angle = 90°.

(i) Facing north and turning clockwise to face west:

The sequence is: North → East → South → West.

This is 3 right angles or \( \frac{3}{4} \) of a revolution.

\( \therefore \) Fraction of revolution = \( \frac{3}{4} \) and number of right angles = 3.

(ii) Facing south and turning anti-clockwise to face east:

The sequence is: South → East.

This is 1 right angle or \( \frac{1}{4} \) of a revolution.

\( \therefore \) Fraction of revolution = \( \frac{1}{4} \) and number of right angles = 1.

(iii) Facing east and turning (clockwise or anti-clockwise) to face west:

This is 2 right angles or \( \frac{1}{2} \) of a revolution.

\( \therefore \) Fraction of revolution = \( \frac{1}{2} \) and number of right angles = 2.
In simple words: Picture the compass with North at top, East on right, South at bottom, West on left. Count how many quarter-turns you need to move from your start direction to your end direction.

Exam Tip: Draw a compass diagram with N, E, S, W marked — it makes counting the turns much faster and reduces errors.

Question 5. Match the following:

Exam Tip: Memorize the key fraction values for each angle type — this is a standard matching question that tests whether you know these definitions by heart.

Question 6. Classify the angles whose magnitudes are given below:
(i) 56°
(ii) 125°
(iii) 90°
(iv) 180°
(v) 215°
(vi) 328°
(vii) 89°
(viii) 178°
Answer:
(i) 56° lies between 0° and 90°.
\( \therefore \) 56° is an acute angle.

(ii) 125° lies between 90° and 180°.
\( \therefore \) 125° is an obtuse angle.

(iii) 90° is exactly 90°.
\( \therefore \) 90° is a right angle.

(iv) 180° is exactly 180°.
\( \therefore \) 180° is a straight angle.

(v) 215° lies between 180° and 360°.
\( \therefore \) 215° is a reflex angle.

(vi) 328° lies between 180° and 360°.
\( \therefore \) 328° is a reflex angle.

(vii) 89° lies between 0° and 90°.
\( \therefore \) 89° is an acute angle.

(viii) 178° lies between 90° and 180°.
\( \therefore \) 178° is an obtuse angle.
In simple words: Check where the degree measure falls: less than 90 is acute, exactly 90 is right, between 90 and 180 is obtuse, exactly 180 is straight, and more than 180 is reflex.

Exam Tip: Use the boundary values (0°, 90°, 180°, 360°) as guides. Do not round or guess — the exact degree value determines the class.

Question 7. State which of the following angles with a small letter in the following diagrams are acute, which are obtuse and which are reflex:
Answer: (i) In the given figure:
Angle a measures less than 90°, so \( \angle a \) is acute.
Angle b measures greater than 90°, so \( \angle b \) is obtuse.
Angle c measures less than 90°, so \( \angle c \) is acute.

(ii) In the given figure:
\( \angle x \) is greater than 90°, so \( \angle x \) is obtuse.
\( \angle y \) is less than 90°, so \( \angle y \) is acute.
\( \angle z \) lies between 90° and 180°, so \( \angle z \) is obtuse.

(iii) In the given figure:
\( \angle p \) measures greater than 90°, so \( \angle p \) is obtuse.
\( \angle q \) measures less than 90°, so \( \angle q \) is acute.
\( \angle r \) lies between 180° and 360°, so \( \angle r \) is reflex.
\( \angle s \) measures less than 90°, so \( \angle s \) is acute.
In simple words: Check each angle by comparing it to 90° (a right angle). If it is less than 90°, it is acute. If it is more than 90° but less than 180°, it is obtuse. If it is more than 180°, it is reflex.

Exam Tip: Always compare the given angle with the benchmark angles - 90°, 180°, and 360° - to classify it correctly.

Question 8. Use your protractor to measure each of the angles marked in the following figures:
Answer: Using a protractor to measure the angles:
(i) The measure of the angle = 62°
(ii) The measure of the angle = 116°
(iii) The measure of the angle = 121°
In simple words: Place the protractor's center on the angle's point. Line up one ray with zero. Read where the other ray crosses the scale to find the angle measure.

Exam Tip: When using a protractor, ensure the center point is exactly at the angle's vertex and one arm aligns with the zero mark for an accurate reading.

Question 9. Use your protractor to measure the reflex angles marked in the following figures:
Answer: To find a reflex angle, first measure the smaller angle (less than 180°) using a protractor, then subtract it from 360°.

(i) Measuring the smaller angle gives 45°.
Reflex angle = 360° - 45° = 315°

(ii) Measuring the smaller angle gives 125°.
Reflex angle = 360° - 125° = 235°
In simple words: A reflex angle is more than 180°. Measure the angle on the other side, then subtract it from 360° to find the reflex angle.

Exam Tip: Always measure the non-reflex angle first with the protractor, as it is easier and more accurate. Then use subtraction to find the reflex angle.

Question 10. Find the measure of the angle between the hands of the clock in each figure:
Answer: In a clock, the angle between any two consecutive numbers is \( \frac{360°}{12} = 30° \).

(i) The hour hand is at 10 and the minute hand is at 12. The angle covers 2 numbers.
Angle between the hands = 2 \( \times \) 30° = 60°

(ii) The hour hand is at 12 and the minute hand is at 1. The angle covers 1 number.
Angle between the hands = 1 \( \times \) 30° = 30°

(iii) The hour hand is at 5 and the minute hand is at 12. The angle covers 5 numbers.
Angle between the hands = 5 \( \times \) 30° = 150°
In simple words: Count how many numbers are between the two hands. Multiply this count by 30° to get the angle.

Exam Tip: Always count the numbers going the shorter way around the clock between the two hands to avoid measuring the reflex angle by mistake.

Question 11. Write the measure of the smaller angle formed by the hour and minute hands of a clock at 7 o'clock. Also write the measure of the other angle and also state what types of angles these are.
Answer: At 7 o'clock, the hour hand is at 7 and the minute hand is at 12.

Angle between any two consecutive numbers = \( \frac{360°}{12} = 30° \)

The smaller angle covers the distance from 7 to 12 (5 numbers in the shorter way).
Smaller angle = 5 \( \times \) 30° = 150°

The other angle = 360° - 150° = 210°

Since 150° lies between 90° and 180°, the smaller angle 150° is an obtuse angle.

Since 210° lies between 180° and 360°, the other angle 210° is a reflex angle.
In simple words: At 7 o'clock, the two hands are 5 numbers apart, making an angle of 150°. This is an obtuse angle. The angle on the other side is 210°, which is a reflex angle.

Exam Tip: When two angles form a complete turn around a point, they always add up to 360°. Use this to find the second angle once you know the first.

Question 12. There are two set-squares in your geometry box. What are measures of the angles formed at their corners? Do they have any angle measure that is common?
Answer: In a geometry box, there are two set-squares:

(i) One set-square has angles: 30°, 60° and 90°.

(ii) The other set-square has angles: 45°, 45° and 90°.

Yes, both set-squares have a common angle measure of 90° (right angle).
In simple words: Both types of set-squares share one angle in common - a 90° right angle. This makes them useful tools for drawing perpendicular lines and right angles.

Exam Tip: Remember the two standard set-squares: the 30-60-90 and the 45-45-90. The 90° angle is their common feature, useful for constructing perpendiculars.

Exercise 11.3

Question 1. Which of the following are models for perpendicular lines?
(i) The adjacent edges of a postcard.
(ii) The line segment forming the letter 'L'.
(iii) The adjacent edges of your Math book.
(iv) The line segments forming the letter 'V'.
Answer: Two lines are called perpendicular if they intersect at right angles (90°).

(i) The adjacent edges of a postcard meet at right angles.
\( \therefore \) Yes, they are models for perpendicular lines.

(ii) The line segments forming the letter 'L' meet at right angles.
\( \therefore \) Yes, they are models for perpendicular lines.

(iii) The adjacent edges of a Math book meet at right angles.
\( \therefore \) Yes, they are models for perpendicular lines.

(iv) The line segments forming the letter 'V' meet at an acute angle, not at a right angle.
\( \therefore \) No, they are not models for perpendicular lines.
In simple words: Perpendicular lines always meet at 90°. Check each example to see if the two lines form a right angle or not.

Exam Tip: When identifying perpendicular lines, look for a perfect right angle or the symbol ⊥ in the diagram. Do not confuse acute angles with right angles.

Question 2. In the figure given below, line l is perpendicular to line m.
(a) Is CE = EG?
(b) Does \( \overleftrightarrow{PE} \) bisect segment \( \overline{BH} \)?
(c) Identify any two line segments for which \( \overleftrightarrow{PE} \) is the perpendicular bisector.
(d) Are these true?
(i) AC > FG
(ii) CD = GH
(iii) BC < EG
Answer: From the figure, the points are at positions:
A(1), B(2), C(3), D(4), E(5), F(6), G(7), H(8)
on line l, and line m is perpendicular to line l at point E.

(a) CE = 5 - 3 = 2 units
EG = 7 - 5 = 2 units
Since CE = EG = 2 units,
\( \therefore \) Yes, CE = EG.

(b) BE = 5 - 2 = 3 units
EH = 8 - 5 = 3 units
Since BE = EH, point E is the mid-point of \( \overline{BH} \). Also, \( \overleftrightarrow{PE} \) is perpendicular to line l.
\( \therefore \) Yes, \( \overleftrightarrow{PE} \) bisects segment \( \overline{BH} \).

(c) \( \overleftrightarrow{PE} \) is perpendicular bisector of any line segment whose midpoint is E and which lies along line l.
For example:
(i) \( \overline{DF} \), since DE = EF = 1 unit and \( \overleftrightarrow{PE} \) \( \perp \) \( \overline{DF} \).
(ii) \( \overline{BH} \), since BE = EH = 3 units and \( \overleftrightarrow{PE} \) \( \perp \) \( \overline{BH} \).
\( \therefore \) \( \overleftrightarrow{PE} \) is the perpendicular bisector of \( \overline{DF} \) and \( \overline{BH} \).

(d) (i) AC = 3 - 1 = 2 units
FG = 7 - 6 = 1 unit
Since 2 > 1,
\( \therefore \) AC > FG is True.

(ii) CD = 4 - 3 = 1 unit
GH = 8 - 7 = 1 unit
Since CD = GH,
\( \therefore \) CD = GH is True.

(iii) BC = 3 - 2 = 1 unit
EG = 7 - 5 = 2 units
Since 1 < 2,
\( \therefore \) BC < EG is True.
In simple words: A perpendicular bisector divides a line segment into two equal parts and meets it at a right angle. Check distances from point E to find equal parts, and look at the symbol ⊥ to confirm perpendicularity.

Exam Tip: To prove a line is a perpendicular bisector, show two things: (1) it passes through the midpoint of the segment, and (2) it meets the segment at 90°.

Exercise 11.4

Question 1. Name the following triangles with regards to sides:
Answer: (i) In triangle ABC, AB = 5 cm, BC = 4 cm and AC = 5 cm.
Since two sides (AB and AC) are equal,
\( \therefore \) Triangle ABC is an isosceles triangle.

(ii) In triangle PQR, PQ = 4 cm, QR = 3 cm and RP = 5 cm.
Since all three sides are different in length,
\( \therefore \) Triangle PQR is a scalene triangle.

(iii) In triangle XYZ, XY = 4.5 cm, YZ = 4.5 cm and XZ = 4.5 cm.
Since all three sides are equal in length,
\( \therefore \) Triangle XYZ is an equilateral triangle.
In simple words: Count how many sides of the triangle are equal. If two sides match, it is isosceles. If all three match, it is equilateral. If all sides are different, it is scalene.

Exam Tip: Compare the side lengths carefully. Write out the measurements next to each side name before classifying the triangle.

Question 2. Name the following triangles with regards to angles:
Answer: (i) In triangle ABC, \( \angle A = 40° \), \( \angle B = 90° \) and \( \angle C = 50° \).
Since one angle (\( \angle B \)) is a right angle,
\( \therefore \) Triangle ABC is a right-angled triangle.

(ii) In triangle PQR, \( \angle P = 25° \), \( \angle Q = 35° \) and \( \angle R = 120° \).
Since one angle (\( \angle R \)) is greater than 90° (obtuse),
\( \therefore \) Triangle PQR is an obtuse-angled triangle.

(iii) In triangle XYZ, \( \angle X = 59° \), \( \angle Y = 65° \) and \( \angle Z = 56° \).
Since all three angles are less than 90° (acute),
\( \therefore \) Triangle XYZ is an acute-angled triangle.
In simple words: Check the angles. If one is 90°, it is right-angled. If one is more than 90°, it is obtuse-angled. If all are less than 90°, it is acute-angled.

Exam Tip: Always add the three angles to verify they sum to 180°, which helps confirm your angle measurements are correct.

Question 3. Name each of the following triangles in two different ways (you may judge the nature of the angle by observation):
Answer: (i) The triangle has sides 9 cm, 8 cm and 8 cm. Two sides are equal and all angles appear less than 90°.
\( \therefore \) It is an isosceles acute-angled triangle.

(ii) The triangle has sides 13 cm, 12 cm and 5 cm. All sides are different and the angle opposite to the longest side is 90°.
Check: \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \)
\( \therefore \) It is a scalene right-angled triangle.

(iii) The triangle has sides 14 cm, 8 cm, 8 cm and an obtuse angle (the angle appears greater than 90°). Two sides are the same and one angle is obtuse.
\( \therefore \) It is an isosceles obtuse-angled triangle.

(iv) The triangle has sides 8 cm and 8 cm with the included angle 90°. Two sides are equal and one angle is right.
\( \therefore \) It is an isosceles right-angled triangle.

(v) The triangle has all sides equal to 6.4 cm. All three sides are equal so all angles are 60° (acute).
\( \therefore \) It is an equilateral acute-angled triangle.

(vi) The triangle has sides 11 cm, 9 cm and 17 cm. All sides are different.
Check: \( 9^2 + 11^2 = 81 + 121 = 202 < 17^2 = 289 \)
So, the angle opposite to the longest side is obtuse.
\( \therefore \) It is a scalene obtuse-angled triangle.
In simple words: Look at both the sides and angles. Describe the triangle using one word about sides (equilateral, isosceles, or scalene) and one word about angles (acute, right, or obtuse).

Exam Tip: For side-based classification, count equal sides. For angle-based classification, look for the largest angle. Use the Pythagorean theorem check when a right angle is suspected.

Question 4. Match the following:

Exam Tip: Create a checklist of triangle names and their defining features. Check off the features as you read each description to avoid confusion.

Question 5. State which of the following statements are true and which are false:
(i) A triangle can have two right angles.
(ii) A triangle cannot have more than one obtuse angle.
(iii) A triangle has at least two acute angles.
(iv) If all the three sides of a triangle are equal, it is called a scalene triangle.
(v) A triangle has four sides.
(vi) An isosceles triangle is an equilateral triangle also.
(vii) An equilateral triangle is an isosceles triangle also.
(viii) A scalene triangle has all its angles equal.
Answer:
(i) False
Reason: The sum of all angles in any triangle equals 180°. If two angles were right angles, they would already add up to 180°, leaving nothing for the third angle, which is impossible.

(ii) True
Reason: If a triangle had two obtuse angles (each bigger than 90°), their sum alone would be more than 180°. This breaks the angle sum rule for triangles.

(iii) True
Reason: Because angles in a triangle sum to 180°, a triangle can have at most one angle that is right or obtuse. This means the other two angles must be acute.

(iv) False
Reason: When all three sides of a triangle are equal in length, it is called an equilateral triangle, not a scalene triangle.

(v) False
Reason: A triangle has three sides, not four.

(vi) False
Reason: An isosceles triangle has exactly two sides equal. An equilateral triangle has all three sides equal. So an isosceles triangle is not necessarily equilateral.

(vii) True
Reason: An equilateral triangle has all three sides equal, which means it has at least two sides equal. Therefore, it also meets the definition of an isosceles triangle.

(viii) False
Reason: A scalene triangle has all sides of different lengths. When sides differ in length, the angles opposite them also differ, so all angles cannot be equal.
In simple words: Each statement needs checking against triangle rules. Triangles have angles that sum to 180°, and sides/angles match to define each type.

Exam Tip: Always use the angle sum property (180°) to test angle-based statements, and recall that equilateral means all three sides equal while isosceles means only two sides equal.

Exercise 11.5

Question 1. State whether the following statements are true (T) or false (F):
(i) Each angle of a rectangle is a right angle.
(ii) The opposite sides of a rectangle are equal in length.
(iii) The diagonals of a square are perpendicular to one another.
(iv) All sides of a rhombus are equal in length.
(v) All sides of a parallelogram are equal in length.
(vi) The opposite sides of a trapezium are parallel.
(vii) The diagonals of a parallelogram are equal.
Answer:
(i) True
Reason: By definition, all four interior angles inside a rectangle measure 90° each.

(ii) True
Reason: Every rectangle is a special kind of parallelogram, and in a parallelogram, the sides that are opposite to each other are always equal.

(iii) True
Reason: The two diagonals of a square meet at right angles to each other. This is true because a square is also a type of rhombus, and rhombuses have this property.

(iv) True
Reason: By definition, a rhombus is a four-sided shape where all four sides have the same length.

(v) False
Reason: In a parallelogram, only the sides that are across from each other are equal. If all four sides were equal, the shape would become a rhombus instead.

(vi) False
Reason: A trapezium has only one pair of opposite sides that run parallel to each other, not both pairs.

(vii) False
Reason: The diagonals of a parallelogram cut each other in half but are not necessarily the same length. They are equal only when the shape is a rectangle or a square.
In simple words: Rectangles have four right angles. Opposite sides match in length. Rhombuses and squares have special diagonal properties that rectangles and parallelograms do not always have.

Exam Tip: Distinguish between quadrilaterals by their defining properties - rectangles have right angles, rhombuses have equal sides, and parallelograms have opposite sides equal but not necessarily all angles or sides equal.

Question 2. Examine whether the following figures are polygons. Give reasons.
Answer: A polygon is a closed flat shape built entirely from straight line segments.

(i) This figure is not a polygon because it is open (one side is not connected).

(ii) This figure is a polygon because it is a simple closed shape made completely of straight line segments.

(iii) This figure is not a polygon because the line segments cross each other, making it self-intersecting, which violates the definition of a simple polygon.

(iv) This figure is not a polygon because it contains at least one curved line, and polygons must have only straight sides.
In simple words: A polygon must be closed, have only straight sides, and not cross itself. Curves, open gaps, or crossing lines mean it is not a polygon.

Exam Tip: Check three things: Is the shape closed? Are all sides straight (no curves)? Do the sides cross each other? If any answer is "no" to the first two or "yes" to the third, it is not a polygon.

Question 3. Name each of the following polygons:
Answer: Polygons are named by counting the number of sides.

(i) This polygon has 5 sides.
Therefore, it is a pentagon.

(ii) This polygon has 4 sides.
Therefore, it is a quadrilateral.

(iii) This polygon has 6 sides.
Therefore, it is a hexagon.

(iv) This polygon has 8 sides.
Therefore, it is an octagon.
In simple words: Count the sides: 5 sides = pentagon, 4 sides = quadrilateral, 6 sides = hexagon, 8 sides = octagon.

Exam Tip: Memorize the polygon names: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8). Counting sides correctly is the key step.

Question 4. Draw a rough sketch of a pentagon and draw its diagonals.
Answer: A pentagon has 5 sides. Diagonals are line segments that connect two vertices that are not next to each other. A pentagon has 5 diagonals altogether.

In the figure, ABCDE is a pentagon. The five diagonals are: AC, AD, BD, BE, and CE.
In simple words: Draw five vertices in a circle and label them A, B, C, D, E. A diagonal joins any two non-adjacent vertices. You can draw 5 different diagonals in a pentagon.

Exam Tip: Use the formula for the number of diagonals: n(n-3)/2, where n is the number of sides. For a pentagon, this gives 5(5-3)/2 = 5 diagonals. Make sure all diagonals join non-adjacent vertices only.

Question 5. Draw a rough sketch of a regular hexagon. Connecting three of its vertices draw:
(i) an isosceles triangle
(ii) a right-angled triangle.
Answer: A regular hexagon has 6 equal sides and 6 equal interior angles (each measuring 120°).

(i) Isosceles triangle: Connect vertices A, B, and C of hexagon ABCDEF. Triangle ABC has two sides of equal length. Therefore, ABC is an isosceles triangle.

(ii) Right-angled triangle: Connect vertices A, B, and D of hexagon ABCDEF (where AD passes through the center). Since all interior angles in a regular hexagon equal 120°, and all sides are equal, sides BC and CD are equal. This makes triangle BCD isosceles, so angles CBD and CDB are equal (angles opposite equal sides are equal).

Angle C in the hexagon = 120°

Using the angle sum property of triangles:
∠CBD + ∠CDB + ∠C = 180°
2∠CBD + 120° = 180°
2∠CBD = 60°
∠CBD = 30°

Since ∠ABD + ∠CBD = ∠B = 120°:
∠ABD + 30° = 120°
∠ABD = 90°

Triangle ABD is a right-angled triangle with the right angle at B.
In simple words: A regular hexagon has all sides and angles equal. By picking three vertices cleverly, you can make an isosceles triangle (two equal sides) or a right-angled triangle (one 90° angle).

Exam Tip: Remember that each interior angle of a regular hexagon is 120°. Use this fact along with the angle sum property and properties of isosceles triangles to find the right angle in part (ii).

Question 6. Can you identify the regular quadrilateral?
Answer: A regular polygon is one where all sides are equal in length and all angles are equal in measure.

A quadrilateral has 4 sides. The quadrilateral with all 4 sides equal and all 4 angles equal (90° each) is a square.

Therefore, the regular quadrilateral is a square.
In simple words: A regular shape has all sides the same length and all angles the same size. The only four-sided shape that fits this is a square.

Exam Tip: Understand what "regular" means - all sides and angles must be identical. A rectangle has equal angles but not equal sides; a rhombus has equal sides but not equal angles. Only a square satisfies both conditions.

Exercise 11.6

Question 1. What is the shape of
(i) your geometry box?
(ii) a brick?
(iii) a matchbox?
(iv) a drum?
(v) a playing die?
(vi) a sweet laddu?
Answer:
(i) A geometry box is shaped like a cuboid.

(ii) A brick is shaped like a cuboid.

(iii) A matchbox is shaped like a cuboid.

(iv) A drum is shaped like a cylinder.

(v) A playing die is shaped like a cube.

(vi) A sweet laddu is shaped like a sphere.
In simple words: Cuboid = a box shape with rectangular faces. Cylinder = a tube shape with circular ends. Cube = a box with square faces. Sphere = a ball shape that is round all over.

Exam Tip: Recognize 3D shapes from everyday objects. Cuboids have rectangular faces (most boxes). Cylinders have circular tops and bottoms (drums, cans). Cubes have square faces (dice). Spheres are perfectly round (balls, sweets).

Question 2. Match the following:
Answer:
(i) Cone - (b) Cone shape with a pointed top and circular base

(ii) Sphere - (d) A soccer ball shape - perfectly round

(iii) Cube - (e) A die shape with square faces

(iv) Pyramid - (a) A pyramid shape with a pointed top

(v) Cylinder - (f) A cylinder shape with circular ends

(vi) Cuboid - (c) A cuboid or box shape
In simple words: Match each 3D shape with its matching picture by recognizing its key features - pointed or round ends, number of faces, and the type of base.

Exam Tip: Learn the main features of each shape: cone (one circular base, one point), sphere (no flat faces), cube (six square faces), pyramid (pointed top, polygon base), cylinder (two circular bases, curved side), cuboid (rectangular faces).

Question 3. Fill in the blanks:
(i) A cube has ..... square faces, ..... edges and ..... vertices.
(ii) A triangular prism has ..... triangular faces, ..... rectangular faces, ..... edges and ..... vertices.
(iii) A triangular pyramid has ..... faces, ..... edges and ..... vertices.
Answer:
(i) A cube has 6 square faces, 12 edges and 8 vertices.

(ii) A triangular prism has 2 triangular faces, 3 rectangular faces, 9 edges and 6 vertices.

(iii) A triangular pyramid has 4 faces, 6 edges and 4 vertices.
In simple words: A cube is like a dice - 6 square sides, 12 edges (where sides meet), and 8 corners. A triangular prism has a triangle top and bottom. A triangular pyramid is like a tent with a triangle base.

Exam Tip: Use Euler's formula for checking: Vertices - Edges + Faces = 2. For a cube: 8 - 12 + 6 = 2. For a triangular pyramid: 4 - 6 + 4 = 2. This helps verify your counts.

Objective Type Questions - Mental Maths

Question 1. Fill in the following blanks:
(i) An angle whose measure is less than that of a right angle is .....
(ii) An angle greater than 180° and less than a complete angle is called .... .
(iii) An angle whose measure is the sum of the measures of two right angles is ..... .
(iv) When the sum of measures of two angles is that of a right angle, then each one of them is .....
(v) When the sum of measures of two angles is that of a straight angle and if one of them is acute then the other is ..... .
(vi) A triangle having one of its angles as right angle and with lengths of two sides equal is called ..... triangle.
(vii) A cuboid has ..... faces, ..... edges and ..... vertices.
(viii) A rectangular pyramid has ..... faces, ..... edges and ..... vertices.
Answer:
(i) An angle whose measure is less than that of a right angle is an acute angle.

(ii) An angle greater than 180° and less than a complete angle is called a reflex angle.

(iii) An angle whose measure is the sum of the measures of two right angles is a straight angle (since 90° + 90° = 180°).

(iv) When the sum of measures of two angles is that of a right angle, then each one of them is an acute angle.

(v) When the sum of measures of two angles is that of a straight angle and if one of them is acute, then the other is an obtuse angle.

(vi) A triangle having one of its angles as right angle and with lengths of two sides equal is called an isosceles right-angled triangle.

(vii) A cuboid has 6 faces, 12 edges and 8 vertices.

(viii) A rectangular pyramid has 5 faces, 8 edges and 5 vertices.
In simple words: Acute = smaller than 90°. Reflex = between 180° and 360°. Straight angle = exactly 180°. Two angles that add to 90° are both acute. Two angles that add to 180° can be acute and obtuse. Isosceles right-angled triangle has a 90° angle and two equal sides.

Exam Tip: Learn angle types by degrees: acute (0-90°), right (90°), obtuse (90-180°), straight (180°), reflex (180-360°). For shapes, count carefully - visualize the 3D object to count faces, edges, and vertices correctly.

Question 2. State whether the following statements are true (T) or false (F):
(i) Each angle of an equilateral triangle is a right angle.
(ii) The adjacent sides of a rectangle are equal in length.
(iii) The diagonals of a rectangle are equal in length.
(iv) The diagonals of a rectangle are perpendicular to one another.
(v) The diagonals of a rhombus are equal in length.
(vi) Any three line segments make up a triangle.
(vii) All the faces of a triangular prism are triangles.
(viii) All the faces of a triangular pyramid are triangles.
Answer:
(i) False - In an equilateral triangle, each angle measures 60°, not 90°.
(ii) False - In a rectangle, only opposite sides have equal length. When adjacent sides are equal, the shape becomes a square.
(iii) True - The diagonals of a rectangle are equal in length.
(iv) False - The diagonals of a rectangle bisect each other but do not meet at right angles. Perpendicularity occurs only in a square.
(v) False - The diagonals of a rhombus intersect at right angles and bisect each other, but they are generally not equal in length. Equal diagonals occur only in a square.
(vi) False - Three line segments can form a triangle only when the sum of any two sides exceeds the third side (triangle inequality rule).
(vii) False - A triangular prism has 2 triangular faces and 3 rectangular faces, so not all faces are triangular.
(viii) True - A triangular pyramid (tetrahedron) has 4 triangular faces - one triangular base and 3 triangular side faces.
In simple words: Check each statement by thinking about the shapes' properties. Triangles have 60° angles, not 90°. Rectangles have equal opposite sides but unequal adjacent sides. Some 3D shapes mix different face shapes.

Exam Tip: For true/false shape questions, use specific examples and measurements - naming the exact angle measure or face count shows clear understanding to the examiner.

Question 3. State whether the following statement is true or false. Justify your answer. 'An angle whose measure is greater than that of a right angle is obtuse'.
Answer: False. While an obtuse angle is larger than a right angle (90°), being larger than 90° does not automatically make an angle obtuse. An obtuse angle lies specifically between 90° and 180°. Angles greater than 90° can also be straight angles (exactly 180°), reflex angles (between 180° and 360°), or complete angles (exactly 360°). For example, a 200° angle is greater than 90°, but it is a reflex angle, not an obtuse angle. Therefore, the given statement is false.
In simple words: Just because an angle is bigger than 90° doesn't mean it's obtuse. Obtuse angles must be between 90° and 180°. Angles of 200° or 270° are bigger than 90° but they are different types.

Exam Tip: When justifying a false statement, provide at least one counterexample with a specific angle measure - examiners award full marks when a single clear example proves the statement wrong.

Question 4. Comparison of lengths is possible in case of 1. two lines 2. two line segments 3. two rays 4. a ray and a line segment
Answer: (2) Two line segments
In simple words: A line goes on forever in both directions, so it has no measurable length. A ray also continues endlessly in one direction. Only line segments have a fixed starting and ending point, so you can measure and compare them.

Exam Tip: Remember the key difference - lines and rays extend infinitely (no end point = no length), while segments have two definite endpoints that you can measure between.

Question 5. A reflex angle measures 1. more than 90° but less than 180° 2. more than 180° but less than 270° 3. more than 180° but less than 360° 4. none of these
Answer: (3) More than 180° but less than 360°
In simple words: By definition, a reflex angle is any angle that falls between 180° and 360°. It is bigger than a straight line (180°) but smaller than a complete turn (360°).

Exam Tip: Memorize the angle ranges - acute (0-90°), right (90°), obtuse (90-180°), straight (180°), reflex (180-360°). Knowing these ranges lets you instantly identify any angle type.

Question 6. A scalene triangle cannot be 1. an acute-angled triangle 2. an obtuse-angled triangle 3. a right-angled triangle 4. an equilateral triangle
Answer: (4) An equilateral triangle
In simple words: A scalene triangle has all three sides of different lengths. An equilateral triangle has all three sides of equal length. These two are opposite, so a scalene triangle can never be equilateral.

Exam Tip: Remember - scalene triangles can still be acute, obtuse, or right-angled depending on their angles. The distinction between scalene and equilateral is about side lengths, not angle types.

Question 7. An obtuse-angled triangle can be 1. right-angled 2. isosceles 3. equilateral 4. none of these
Answer: (2) Isosceles
In simple words: An obtuse-angled triangle has one angle larger than 90°. It cannot be right-angled (that would be two 90° angles, leaving no room for another). It cannot be equilateral (all angles would be 60°). But it can be isosceles - for example, a triangle with angles 100°, 40°, and 40° has one obtuse angle and two equal sides.

Exam Tip: Test each option with a concrete angle example - this prevents confusion and shows the examiner your clear reasoning.

Question 8. If you are facing north and turn through \( \frac{3}{4} \) of a turn in anti-clockwise direction, in which direction will you face? 1. east 2. south 3. west 4. north
Answer: (1) East
\( \frac{3}{4} \) of a turn = 3 right angles. Starting from north and turning anti-clockwise: North - West (1 right angle) - South (2 right angles) - East (3 right angles).
In simple words: When you turn anti-clockwise from north, you go to west, then south, then east. Three right angles means three 90° turns, landing you at east.

Exam Tip: Draw a compass directions diagram for these questions - showing north at top, east at right, south at bottom, west at left helps you count the turns accurately.

Question 9. Open any two adjacent fingers of your hand. What kind of angle you get? 1. acute 2. right 3. obtuse 4. straight
Answer: (1) Acute
In simple words: When you open two fingers next to each other, the angle formed between them is less than 90°. This makes it an acute angle.

Exam Tip: Physical exploration questions like this reward you for actually testing the scenario - try it yourself to confirm the answer matches reality, which also helps you remember the concept.

Question 10. If the sum of two angles is an obtuse angle, then which of the following is not possible? 1. one right angle and one acute angle 2. one obtuse angle and one acute angle 3. two acute angles 4. two right angles
Answer: (4) Two right angles
An obtuse angle lies between 90° and 180°. Testing each option: (1) 90° + 20° = 110° (obtuse) - possible. (2) 100° + 20° = 120° (obtuse) - possible. (3) 60° + 60° = 120° (obtuse) - possible. (4) 90° + 90° = 180° (straight angle, not obtuse) - not possible.
In simple words: Two right angles add to exactly 180°, which is a straight line, not an obtuse angle. The other combinations can add up to something between 90° and 180°.

Exam Tip: For "not possible" questions, test each option by computing its sum - the option that produces a sum outside the given range is your answer.

Question 11. If the sum of two angles is greater than 180°, then which of the following is not possible? 1. two obtuse angles 2. two right angles 3. one obtuse and one acute angle 4. one reflex and one acute angle
Answer: (2) Two right angles
For the sum to exceed 180°: (1) Two obtuse angles (each > 90°): sum > 180° - possible. (2) Two right angles: 90° + 90° = 180° (not greater) - not possible. (3) Obtuse + acute, e.g., 120° + 70° = 190° > 180° - possible. (4) Reflex (> 180°) + acute: sum always > 180° - possible.
In simple words: Two right angles make exactly 180°, which does not exceed it. You need both angles large enough (like two obtuse angles) or at least one very large angle (like reflex) to go over 180°.

Exam Tip: When the question asks "is not possible," systematically check the sum for each option - the one that equals or stays below the limit is your answer.

Question 12. Which of the following statements is false? 1. Every equilateral triangle is an isosceles triangle. 2. Every isosceles triangle is an equilateral triangle. 3. Every parallelogram is a trapezium. 4. Every trapezium is a quadrilateral.
Answer: (2) Every isosceles triangle is an equilateral triangle
Testing each: (1) Equilateral has all sides equal, so it certainly has at least two equal sides - true. (2) Isosceles has only 2 equal sides, not necessarily all 3 - false. (3) A parallelogram has two pairs of parallel sides, so at least one pair - making it a trapezium - true. (4) A trapezium has 4 sides by definition - true.
In simple words: An equilateral triangle is always isosceles, but an isosceles triangle is not always equilateral. Isosceles means just 2 equal sides; equilateral means all 3 equal.

Exam Tip: For "which is false" questions, evaluate each statement independently - use a counterexample to prove a statement false (like an isosceles triangle with sides 5, 5, 8).

Question 13. Which of the following statements is correct? 1. Every rhombus is a square. 2. Every parallelogram is a rectangle. 3. Every square is a rhombus. 4. Every rectangle is a square.
Answer: (3) Every square is a rhombus
Testing each: (1) A rhombus has equal sides but not necessarily right angles - false. (2) A parallelogram lacks the requirement of right angles - false. (3) A square has all four sides equal, which is the defining property of a rhombus - true. (4) A rectangle has opposite sides equal but not all sides equal - false.
In simple words: A square has all sides equal, which makes it a special rhombus. But a rhombus doesn't need right angles, so not every rhombus is a square.

Exam Tip: Remember the hierarchy - square is the most restrictive (all sides equal + all angles 90°), rhombus requires equal sides only, rectangle requires right angles only, parallelogram is the broadest.

Question 14. A quadrilateral whose each angle is a right angle is a 1. trapezium 2. parallelogram 3. rhombus 4. rectangle
Answer: (4) Rectangle
In simple words: By definition, a rectangle is a quadrilateral where all four angles equal 90°. That is exactly what the question describes.

Exam Tip: This is a direct definitional question - a quadrilateral with all right angles is always a rectangle, regardless of side lengths.

Question 15. If a solid shape is completely bounded by plane faces, then the least number of faces it may have is 1. 3 2. 4 3. 5 4. 6
Answer: (2) 4
A solid bounded by plane faces is called a polyhedron. The minimum number of plane faces needed to enclose a 3-dimensional region is 4, as shown by a tetrahedron (triangular pyramid). A tetrahedron has exactly 4 triangular faces.
In simple words: You need at least 4 flat faces to fully enclose a 3D space. A triangular pyramid (tetrahedron) with 4 triangle faces is the simplest solid.

Exam Tip: Visualize or sketch a tetrahedron - it is the simplest polyhedron and serves as the answer to "minimum faces" questions.

Question 16. Statement I: From 8:00 PM to 8:50 PM, the minute hand of a clock covers an angle of 300°. Statement II: \( \frac{5}{6} \times 360° = 300° \)
Answer: (3) Both Statement I and statement II are true
Statement I: From 8:00 PM to 8:50 PM, the minute hand moves through 50 minutes. In 60 minutes, the minute hand covers 360°. In 50 minutes: \( \frac{50}{60} \times 360° = \frac{5}{6} \times 360° = 300° \). Statement I is true.
Statement II: \( \frac{5}{6} \times 360° = \frac{5 \times 360°}{6} = \frac{1800°}{6} = 300° \). Statement II is true.
In simple words: The minute hand moves 360° in 60 minutes. In 50 minutes, it moves \( \frac{50}{60} \) of the full turn, which is 300°. Both statements say the same thing correctly.

Exam Tip: For statement-pair questions, evaluate each independently - even if they say the same thing, both must be correctly reasoned to score full marks.

Question 17. Statement I: The sum of two right angles and one acute angle is a reflex angle. Statement II: 135° is an obtuse angle.
Answer: (3) Both Statement I and statement II are true
Statement I: Sum of two right angles + one acute angle = 90° + 90° + acute angle = 180° + acute angle. Since an acute angle is between 0° and 90°, the sum falls between 180° and 270°, which is a reflex angle. Statement I is true.
Statement II: 135° lies between 90° and 180°, making it an obtuse angle. Statement II is true.
In simple words: Two right angles give 180°. Adding any acute angle pushes it past 180° but below 270°, which defines a reflex angle. 135° is bigger than 90° and smaller than 180°, so it's obtuse.

Exam Tip: When combining angles, always calculate the range (minimum to maximum) rather than guessing - this ensures you identify the correct angle type.

Question 18. Statement I: An equilateral triangle is an isosceles triangle as well. Statement II: 135° is an obtuse angle.
Answer: (3) Both Statement I and statement II are true
Statement I: An equilateral triangle has all three sides equal. Since it has at least two equal sides, it also satisfies the definition of an isosceles triangle. Statement I is true.
Statement II: 135° lies between 90° and 180°, so it is an obtuse angle. Statement II is true.
In simple words: Every equilateral triangle is isosceles because it has at least two equal sides. 135° is bigger than a right angle but not as big as a straight angle, so it's obtuse.

Exam Tip: Remember the subset relationship - equilateral triangles form a smaller group inside the larger group of isosceles triangles, so every equilateral is also isosceles.

Question 18. Statement I: Every equilateral triangle is also an isosceles triangle. Statement II: In an equilateral triangle, all the sides are equal, but in an isosceles triangle at least two sides are equal.
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (3) Both Statement I and statement II are true.
In simple words: An equilateral triangle has all three sides equal, which also means it has at least two sides equal, so it fits the definition of an isosceles triangle. Both statements describe true facts about these two types of triangles.

Exam Tip: Recognize that equilateral triangles are a special case of isosceles triangles - understanding this relationship between shapes helps you classify them correctly.

Question 19. Statement I: Every rhombus is a parallelogram. Statement II: The opposite sides of a parallelogram, as well as a rhombus, are parallel to each other.
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (3) Both Statement I and statement II are true.
In simple words: A rhombus is a special kind of parallelogram where all sides are the same length. In both shapes, the sides that sit across from each other always run in the same direction.

Exam Tip: Remember that a rhombus inherits all the properties of a parallelogram because it is one - it just has the added feature of equal sides.

Question 20. Statement I: A cube is a special cuboid in which the length, width and height are equal. Statement II: A cuboid has 6 faces, 12 edges and 6 vertices.
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (1) Statement I is true but statement II is false.
In simple words: A cube is indeed a special cuboid where length, width, and height all match. However, a cuboid has 8 corners (vertices), not 6.

Exam Tip: When counting vertices of 3D shapes, visualize the shape or sketch it - a cuboid has a corner at each point where edges meet, giving 8 vertices total (4 on top, 4 on bottom).

Check Your Progress

Question 1. In the adjoining figure, identify the longest and shortest line segments by measuring their lengths.
Answer: By measuring the line segments in the figure (triangle PQR with point S on QR), we get:
PQ = 5 cm
PR = 6 cm
PS = 4 cm
QS = 3 cm
SR = 4 cm
QR = 7 cm

The longest line segment is QR (7 cm). The shortest line segment is QS (3 cm).
In simple words: After measuring all the line segments, QR is the longest at 7 cm, and QS is the shortest at 3 cm.

Exam Tip: Always use a ruler and measure carefully; even small errors in measurement can lead to wrong identification of the longest and shortest segments.

Question 2. Where will the hour hand of a clock stop if it starts from 10 and turns through 3 right angles?
Answer: One right angle equals \( \frac{1}{4} \) of a full rotation, which equals 3 hours (since a full rotation = 12 hours). So 3 right angles = 3 × 3 = 9 hours. Starting from 10 o'clock and moving 9 hours forward in a clockwise direction: 10 → 11 → 12 → 1 → 2 → 3 → 4 → 5 → 6 → 7. The hour hand will stop at 7.
In simple words: Each right angle moves the hour hand by 3 hours. Three right angles move it 9 hours forward. Starting at 10 and adding 9 hours brings you to 7.

Exam Tip: Convert right angles to hours first (1 right angle = 3 hours) before counting forward on the clock face.

Question 3. Classify the angles whose measures are given below:
(i) 56°
(ii) 125°
(iii) 90°
(iv) 180°
(v) 215°
(vi) 328°
Answer:
(i) 56° falls between 0° and 90°. It is an acute angle.
(ii) 125° falls between 90° and 180°. It is an obtuse angle.
(iii) 90° is exactly a right angle. It is a right angle.
(iv) 180° is exactly a straight angle. It is a straight angle.
(v) 215° falls between 180° and 360°. It is a reflex angle.
(vi) 328° falls between 180° and 360°. It is a reflex angle.
In simple words: Angles between 0° and 90° are acute. Those between 90° and 180° are obtuse. 90° is a right angle, and 180° is a straight angle. Angles larger than 180° but smaller than 360° are reflex angles.

Exam Tip: Learn the angle ranges: acute (0-90°), right (90°), obtuse (90-180°), straight (180°), reflex (180-360°) - knowing these boundaries makes classification quick.

Question 4. Name the types of the following triangles:
(i) ΔAB C with AB = 8 cm, AC = 7 cm and BC = 5.5 cm.
(ii) ΔPQR with PQ = RP = 5 cm and QR = 7.3 cm.
(iii) ΔDEF with ∠D = 90°.
(iv) ΔXYZ with ∠Y = 90° and XY = YZ.
(v) ΔLMN with ∠L = 30°, ∠M = 70° and ∠N = 80°.
Answer:
(i) In ΔABC, all three sides have different lengths (8 cm, 7 cm, and 5.5 cm). It is a scalene triangle.
(ii) In ΔPQR, two sides are equal (PQ = RP = 5 cm). It is an isosceles triangle.
(iii) In ΔDEF, one angle is a right angle (∠D = 90°). It is a right-angled triangle.
(iv) In ΔXYZ, one angle is a right angle (∠Y = 90°) and two sides are equal (XY = YZ). It is an isosceles right-angled triangle.
(v) In ΔLMN, all three angles are less than 90° (all acute: 30°, 70°, and 80°), and all are different. It is an acute-angled triangle.
In simple words: A scalene triangle has all different sides. An isosceles triangle has two matching sides. A right-angled triangle has one 90° angle. An acute-angled triangle has all angles less than 90°.

Exam Tip: Triangles can be classified by sides (scalene, isosceles, equilateral) or by angles (acute, right, obtuse) - always check what the question asks for and provide both classifications if required.

Question 5. Name each of the following triangles in two different ways (you may use ruler and protractor):
Answer: After measuring the sides and angles of each triangle using a ruler and protractor:

(i) This triangle has 2 equal sides and one different side, with one angle greater than 90°. It is an isosceles obtuse-angled triangle.

(ii) This triangle has all three sides of different lengths and one angle measuring 90°. It is a scalene right-angled triangle.

(iii) This triangle has two sides equal and all three angles less than 90°. It is an isosceles acute-angled triangle.
In simple words: Each triangle can be named in two ways - one based on its sides (scalene, isosceles, equilateral) and one based on its angles (acute, right, obtuse).

Exam Tip: Use the ruler to compare side lengths and the protractor to measure angles carefully - this dual information lets you name each triangle by both its side and angle properties.

Question 6. State whether the following statements are true or false:
(i) A rectangle is a regular quadrilateral.
(ii) A rhombus is a regular quadrilateral.
(iii) Every parallelogram is a rhombus.
(iv) The diagonals of a rhombus intersect at right angles.
(v) A polygon having 6 sides is called an octagon.
(vi) A road roller has two plane circular faces and one curved face.
(vii) A rectangular pyramid has 5 rectangular faces.
Answer:
(i) False - A regular quadrilateral must have all sides equal and all angles equal. A rectangle has all angles equal (90° each) but not all sides equal.

(ii) False - A rhombus has all sides equal, but its angles are not necessarily equal, so it fails the definition of a regular quadrilateral.

(iii) False - A parallelogram has opposite sides equal but not all four sides equal in general. A rhombus requires all four sides to be equal.

(iv) True - The diagonals of a rhombus always cross each other at 90° angles.

(v) False - A polygon with 6 sides is called a hexagon. An octagon has 8 sides.

(vi) True - A road roller has the shape of a cylinder, which includes two flat circular faces at the ends and one curved face around the side.

(vii) False - A rectangular pyramid has 1 rectangular base and 4 triangular side faces, totaling 5 faces. Only the base is rectangular.
In simple words: A regular shape must have all sides and all angles matching. Know the names of polygons by side count. Understand the key properties of shapes like rhombuses, rectangles, and pyramids.

Exam Tip: For "true or false" questions, give the reason in every case - state what property is missing or what rule is broken when a statement is false.

Question 7. If the lengths of two sides of an isosceles triangle are 3 cm and 7 cm, then what is the length of the third side?
Answer: In an isosceles triangle, two sides must be equal. The third side is either 3 cm or 7 cm.

Case 1: If the sides are 3 cm, 3 cm, and 7 cm - The sum of the two smaller sides is 3 + 3 = 6 cm, which is less than 7 cm. This violates the triangle inequality rule (the sum of any two sides must be greater than the third side). So this triangle cannot form.

Case 2: If the sides are 7 cm, 7 cm, and 3 cm - Check: 7 + 7 = 14 > 3 ✓, and 7 + 3 = 10 > 7 ✓. The triangle inequality is satisfied. This triangle can form.

Therefore, the length of the third side is 7 cm.
In simple words: In an isosceles triangle with sides 3 cm and 7 cm, the third side must match one of them. If it is 3 cm, the triangle cannot exist. If it is 7 cm, the triangle works.

Exam Tip: Always check the triangle inequality - the sum of any two sides must be strictly greater than the third side - to determine if a triangle can actually be formed.

Question 8. If the lengths of three consecutive sides of an isosceles trapezium are 5 cm, 6 cm and 8 cm, then what is the length of the fourth side?
Answer: In an isosceles trapezium, the two non-parallel sides (legs) are equal in length, while the two parallel sides (bases) have different lengths.

Label the four consecutive sides as AB, BC, CD, and DA, where AB and CD are the parallel sides, and BC and DA are the non-parallel sides (legs).

Given: AB = 5 cm, BC = 6 cm, and CD = 8 cm.

Since BC and DA are the non-parallel sides of an isosceles trapezium, they must be equal: DA = BC = 6 cm.

Therefore, the length of the fourth side is 6 cm.
In simple words: In an isosceles trapezium, the two slanted sides (the ones that are not parallel) must always be the same length. So if one is 6 cm, the other must also be 6 cm.

Exam Tip: Identify which sides of a trapezium are parallel and which are not - the non-parallel sides in an isosceles trapezium are always equal, which is the key property to solve these problems.

Question 9. Find out the number of acute angles in each of the figures below:
Answer:
Figure 1 (Equilateral triangle): An equilateral triangle has 3 angles that are all equal, each measuring 60°. Since 60° is less than 90°, all three angles are acute. Number of acute angles = 3.

Figure 2 (Equilateral triangle with an inverted smaller triangle inside, forming 4 small triangles): When an equilateral triangle is divided by connecting the midpoints of its three sides, four smaller equilateral triangles are created. Each of these smaller triangles has 3 angles of 60° each (all acute). Total acute angles = 4 × 3 = 12 acute angles.
In simple words: An equilateral triangle always has three 60° angles, all of which are acute. If you divide it into four smaller equilateral triangles, you get four times as many acute angles.

Exam Tip: Remember that in an equilateral triangle, all angles are 60°, which is always acute - this makes counting acute angles straightforward when a figure contains equilateral triangles.

Question 10. The circle has been divided into 2, 3, 4, 6, 8 and 12 parts below. What are the degree measures of the resulting angles? Do not use protractor to answer this question.
Answer: The total angle around the center of a circle is 360°. When divided into n equal parts, each angle = \( \frac{360°}{n} \).

(i) Circle divided into 2 equal parts: Each angle = \( \frac{360°}{2} \) = 180°

(ii) Circle divided into 3 equal parts: Each angle = \( \frac{360°}{3} \) = 120°

(iii) Circle divided into 4 equal parts: Each angle = \( \frac{360°}{4} \) = 90°

(iv) Circle divided into 6 equal parts: Each angle = \( \frac{360°}{6} \) = 60°

(v) Circle divided into 8 equal parts: Each angle = \( \frac{360°}{8} \) = 45°

(vi) Circle divided into 12 equal parts: Each angle = \( \frac{360°}{12} \) = 30°
In simple words: Divide 360° by the number of equal parts you want. Each part gets an angle measure equal to 360 divided by that number.

Exam Tip: Memorize the formula \( \frac{360°}{n} \) for dividing a circle into n equal parts - this avoids needing a protractor and ensures accuracy.

Question 11. The Ashoka Chakra in Indian Flag is navy blue in colour, and signifies Dharma Chakra or Wheel of law made by 3rd century Mauryan Emperor Ashoka. It has 24 spokes (or lines from the centre). What is the degree measure of the angle between two spokes next to each other?
Answer: At the centre of the Ashoka Chakra, the total angle is 360°. Since the Ashoka Chakra has 24 spokes, they split the centre into 24 equal angles. To find the angle between two adjacent spokes, divide 360° by 24. This gives us \( \frac{360°}{24} = 15° \). Therefore, the degree measure of the angle between two spokes next to each other is 15°.
In simple words: A full circle is 360°. If you split it into 24 equal parts, each part is 15°.

Exam Tip: Always remember that a complete circle at the centre measures 360°. Divide this by the number of equal divisions to find the angle between adjacent rays or spokes.