Understanding Elementary Shapes

By mere observation, we can only judge roughly by looking at two segments. However, our eyes can be deceived, and the comparison may not be accurate. We cannot be certain which is longer when the difference is small. Therefore, comparison by observation is unreliable and inaccurate.

When using a ruler, there may be positioning errors — we might not be able to place the ruler exactly at the endpoint or our eyes may not be directly above the marking (parallax error). A divider allows us to pick up the exact length and then measure it on the ruler, giving a more accurate reading.

Example: Let AB = 7 cm, AC = 3 cm, CB = 4 cm.

AC + CB = 3 + 4 = 7 cm = AB. Yes, AC + CB = AB.

This holds true for any point C between A and B. It confirms that if C is between A and B, the sum of the two smaller segments equals the whole.

Check: AB + BC = 5 + 3 = 8 cm = AC.

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

AD = 3.5 cm and DG = 3.5 cm.

AD + DG = 3.5 + 3.5 = 7 cm = AG.

Since AD = DG, point D divides AG into two equal halves. Therefore, D is the midpoint of AG. ✓

One full revolution = 12 hours on a clock. Each hour = 1/12 of a revolution = 30°.

(a) 3 to 9 = 6 hours = 6/12 = 1/2 revolution (180°)

(b) 4 to 7 = 3 hours = 3/12 = 1/4 revolution (90°)

(c) 7 to 10 = 3 hours = 3/12 = 1/4 revolution (90°)

(d) 12 to 9 = 9 hours = 9/12 = 3/4 revolution (270°)

(e) 1 to 10 = 9 hours = 9/12 = 3/4 revolution (270°)

(f) 6 to 3 = 9 hours = 9/12 = 3/4 revolution (270°)

(a) 12 + 6 hours = 6

(b) 2 + 6 hours = 8

(c) 5 + 3 hours = 8

(d) 5 + 9 hours = 14 = 14 - 12 = 2

Order (clockwise): N → E → S → W → N. 1/4 turn = 90°.

(a) E + 2 right turns = West

(b) E + 6 right turns = E + 1.5 rev = West

(c) W + 3 left turns = W turning anti-clockwise → S → E → N = North

(d) S + 4 left turns = 1 full revolution anti-clockwise = South (same position)

(a) E to N clockwise = 3 right turns = 3/4 revolution

(b) S to E clockwise = 3 right turns = 3/4 revolution

(c) W to E clockwise = 2 right turns = 1/2 revolution

Column 1:
(i) Straight angle   (ii) Right angle   (iii) Acute angle   (iv) Obtuse angle   (v) Reflex angle

Column 2:
(a) Less than one fourth of a revolution
(b) More than half a revolution
(c) Half a revolution
(d) One fourth of a revolution
(e) Between one fourth and half a revolution

Answers:
(i) Straight — (c) Half a revolution
(ii) Right — (d) One fourth of a revolution
(iii) Acute — (a) Less than one fourth of a revolution
(iv) Obtuse — (e) Between one fourth and half a revolution
(v) Reflex — (b) More than half a revolution

(a) 75° → Acute (between 0° and 90°)

(b) 215° → Reflex (between 180° and 360°)

(c) 90° → Right angle

(d) 125° → Obtuse (between 90° and 180°)

(e) 180° → Straight angle

(f) 25° → Acute

(a) 30° → Acute angle

(b) 90° → Right angle

(c) 117° → Obtuse angle

(d) 165° → Obtuse angle

(a) Acute

(b) Obtuse

(c) Right

(d) Straight

(a) Two adjacent walls of a room → Yes (they meet at 90°)

(b) Two adjacent edges of a book → Yes

(c) Railway lines (parallel) → No

(d) Legs of a letter T → Yes

(e) Hands of a clock at 3 o'clock → Yes (they form a 90° angle)

Since PQ is perpendicular to XY, they meet at right angles. All four angles formed are 90° each.

Therefore, ∠PAY = 90°.

First set-square (45°-45°-90° triangle): angles are 45°, 45°, 90°.

Second set-square (30°-60°-90° triangle): angles are 30°, 60°, 90°.

Common angle: 90° (both have a right angle).

(a) If E is the midpoint of CG, then Yes, CE = EG.

(b) The perpendicular from P meets CG at E and bisects it. Yes, PE bisects CG.

(a) Two sides equal (BC = CA) → Isosceles triangle

(b) All sides equal → Equilateral triangle

(c) One angle = 90° → Right-angled triangle

(d) One angle = 150° > 90° → Obtuse-angled triangle

(e) Right angle + two equal sides → Right-angled isosceles triangle

(f) All sides unequal → Scalene triangle

Matching:
(i) Equilateral triangle — 3 equal sides
(ii) Isosceles triangle — 2 equal sides
(iii) Scalene triangle — All sides different
(iv) Acute triangle — Each angle less than 90°
(v) Obtuse triangle — One angle greater than 90°
(vi) Right triangle — One angle exactly 90°

(a) True — all four angles = 90°.

(b) True — opposite sides are equal and parallel.

(c) True — diagonals of a square bisect each other at 90°.

(d) True — all four sides of a rhombus are equal.

(e) False — only opposite sides are equal in a parallelogram.

(f) False — only ONE pair of opposite sides is parallel in a trapezium.

(a) Not a polygon — it's not a closed curve.

(b) Not a polygon — sides must be straight line segments.

(c) Yes, a polygon — a triangle is a 3-sided polygon.

(d) Yes, a polygon — a hexagon is a 6-sided polygon.

Match:
Cone — Ice cream cone
Sphere — Football
Cylinder — Battery / Tin can
Cube — Dice
Cuboid — Duster / Brick
Triangular pyramid — Pyramid shape

(a) Instrument box → Cuboid

(b) Brick → Cuboid

(c) Match box → Cuboid

(d) Road roller → Cylinder

(e) Sweet laddu → Sphere

The minute hand completes one full revolution (360°) in 60 minutes. So, it turns 6° per minute.

(a) 15 min: 15 × 6 = 90°

(b) 30 min: 30 × 6 = 180°

(c) 45 min: 45 × 6 = 270°

(d) 60 min: 60 × 6 = 360°

Each hour mark = 360 ÷ 12 = 30° apart.

At 4 o'clock: 4 divisions × 30° = 120°.

At 6 o'clock: 6 divisions × 30° = 180° (straight angle).

All three angles are equal (60° each), so it is an Acute-angled triangle.
Also, all three angles are equal, making it an Equilateral triangle.
Special property: all three sides are also equal, and all three angles = 60°.

A cube has:
Faces = 6 (all square)
Edges = 12
Vertices = 8

Verify using Euler's formula: F + V − E = 6 + 8 − 12 = 2 ✓

Clockwise order: N → E → S → W → N.
From South, a 90° clockwise turn brings her to West.

If C is the midpoint, then AC = CB = AB ÷ 2 = 6.5 ÷ 2 = 3.25 cm.

Let angles be x, 2x, 3x.
x + 2x + 3x = 180° → 6x = 180 → x = 30°.
Angles = 30°, 60°, 90°.
Since one angle = 90°, it is a right-angled triangle.

No. The sum of angles in a triangle is always 180°. If there were two obtuse angles, each would be greater than 90°, so their sum would exceed 180° — which is impossible. Therefore, at most one obtuse angle can be present.

Reflex angle = 360° − given angle = 360 − 75 = 285°.

Complementary angles add to 90°.
Other angle = 90 − 35 = 55°.

Supplement = 180 − 112 = 68°.

Perimeter = 2(l + w) = 2(8 + 5) = 2 × 13 = 26 cm.
Diagonal = square root of (8² + 5²) = square root of (64 + 25) = square root of 89 ≈ 9.43 cm.

A straight angle = 180°. A right angle = 90°.
180 ÷ 90 = 2 right angles.

Sum of all angles in a quadrilateral = 360°.
Fourth angle = 360 − (70 + 80 + 120) = 360 − 270 = 90°.

Check if 5² + 12² = 13²:
25 + 144 = 169 = 13². Yes!
This is a right-angled scalene triangle (all sides unequal, one angle = 90°).

A Rhombus. All four sides are equal, but angles are not necessarily 90°. (If all angles were 90°, it would be a square).

A sphere has 0 flat faces. It has 1 curved surface. A sphere has 0 edges and 0 vertices.

(a) Two pairs of equal sides: Parallelogram, Rectangle, Square, Rhombus

(b) All sides equal: Square, Rhombus

(c) Only one pair of parallel sides: Trapezium

At 3:00, the hour hand points to 3 and the minute hand points to 12. The angle between them is 3 divisions = 3 × 30° = 90°. It is a right angle.