Angles as Turns

Activity-based question.

Given: A paper fan made by folding a rectangular paper every 2 cm with ice cream sticks pasted on it.

Concept: When one side of the fan is kept fixed and the other side is moved:
- Less than a $\frac{1}{4}$ turn → Acute angle (smaller than a right angle)
- Exactly a $\frac{1}{4}$ turn → Right angle
- Between a $\frac{1}{4}$ turn and a $\frac{1}{2}$ turn → Obtuse angle

How to show acute angle: Open the fan slightly so the two sticks make a small opening — less than a quarter turn.

How to show obtuse angle: Open the fan more than a quarter turn but less than a half turn.

Conclusion: The paper fan is a hands-on tool to visualise and demonstrate different types of angles by controlling the amount of turn between its two sides.

Given: A house-shaped figure made from cardboard with angles A through H marked at various corners.

Concept:
- Right angle = exactly $\frac{1}{4}$ turn (like the corner of a square)
- Acute angle = less than $\frac{1}{4}$ turn (sharp corner)
- Obtuse angle = between $\frac{1}{4}$ and $\frac{1}{2}$ turn (wider corner)

Typical answers for a standard house shape (rectangular base with a triangular roof):

| Angle | Type |
|-------|------|
| A | Right angle |
| B | Right angle |
| C | Right angle |
| D | Right angle |
| E | Acute angle |
| F | Obtuse angle |
| G | Obtuse angle |
| H | Acute angle |

*(Note: The exact answers depend on the figure provided. The four corners of the rectangular base are right angles; the peak of the roof is an acute angle; the angles where the roof meets the walls are obtuse angles.)*

Answer:
- A: Right angle, B: Right angle
- C: Right angle, D: Right angle
- E: Acute angle, F: Obtuse angle
- G: Obtuse angle, H: Acute angle

Given: We need to draw a pentagon (5-sided shape) with exactly:
- 2 right angles
- 2 obtuse angles
- 1 acute angle

Concept: The sum of interior angles of a pentagon = $(5-2) \times 180° = 540°$.

Step-by-step construction:

1. Draw a horizontal base line.
2. At the left end, draw a vertical line upward — this gives the first right angle ($90°$).
3. At the right end, draw a vertical line upward — this gives the second right angle ($90°$).
4. From the top of the left vertical, draw a line going up and to the right at an obtuse angle — this gives the first obtuse angle (e.g., $120°$).
5. From the top of the right vertical, draw a line going up and to the left at an obtuse angle — this gives the second obtuse angle (e.g., $120°$).
6. The two lines from steps 4 and 5 meet at a sharp point at the top — this gives the acute angle.

Verification: $90° + 90° + 120° + 120° + 120° = 540°$ ✓

The resulting shape looks like a house (rectangle with a pointed roof), which is a valid 5-sided figure satisfying all the given conditions.

Given: Images of gymnasts with angles formed between their legs.

Concept:
- Acute angle: Less than a $\frac{1}{4}$ turn — legs close together, small gap
- Right angle: Exactly a $\frac{1}{4}$ turn — legs form an L-shape ($90°$)
- Obtuse angle: Between $\frac{1}{4}$ and $\frac{1}{2}$ turn — legs spread wide but not flat
- Straight angle: Exactly a $\frac{1}{2}$ turn — legs form a straight line ($180°$)

Typical identification (based on standard gymnast poses shown in such exercises):

| Gymnast | Angle between legs |
|---------|-------------------|
| 1 | Acute angle (legs close together) |
| 2 | Right angle (legs at $90°$) |
| 3 | Obtuse angle (legs spread wide) |
| 4 | Straight angle (legs in a straight line/splits) |

*(Note: Exact answers depend on the figures. Apply the definitions above to each image.)*

Key rule to remember: The wider the spread of the legs, the larger the angle — from acute → right → obtuse → straight.

Given: A circle divided into 8 equal parts. A straw is attached at the centre.

Concept: Each $\frac{1}{8}$ turn is one section of the 8-equal-part circle.

(i) Angle made with a $\frac{2}{8}$ turn:
$$\frac{2}{8} = \frac{1}{4} \text{ turn}$$
A $\frac{1}{4}$ turn is a right angle.

(ii) $\frac{1}{8}$ turn is half of a quarter turn:
$$\frac{1}{8} = \frac{1}{2} \times \frac{1}{4}$$
So a $\frac{1}{8}$ turn is half of a right angle — it is an acute angle.

(iii) Angle made with a $\frac{4}{8}$ turn:
$$\frac{4}{8} = \frac{1}{2} \text{ turn}$$
A $\frac{1}{2}$ turn is a straight angle.

Summary:
- $\frac{1}{8}$ turn → Acute angle (half of right angle)
- $\frac{2}{8}$ turn → Right angle
- $\frac{3}{8}$ turn → Obtuse angle
- $\frac{4}{8}$ turn → Straight angle
- $\frac{6}{8} = \frac{3}{4}$ turn → Three-quarter turn
- $\frac{8}{8}$ turn → Full turn

Given: Circles showing end points of $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{8}$ turns.

Part 1 — Finding starting points:

Concept: If we know the end point and the fraction of the turn, we go backwards by the same fraction to find the starting point.

- For a $\frac{1}{2}$ turn: The starting point is directly opposite the end point (half circle away in the reverse direction).
- For a $\frac{1}{4}$ turn: The starting point is one quarter of the circle before the end point (going anti-clockwise).
- For a $\frac{1}{8}$ turn: The starting point is one eighth of the circle before the end point (going anti-clockwise).

Draw arrows from the starting point to the end point in the clockwise direction on each circle.

Part 2 — Circle folded into 6 equal parts:

Fold the circle in half, then fold into 3 equal parts → 6 equal sections.

| Turn | Fraction | Type of Angle |
|------|----------|---------------|
| $\frac{1}{6}$ | Less than $\frac{1}{4}$ | Acute |
| $\frac{2}{6} = \frac{1}{3}$ | Between $\frac{1}{4}$ and $\frac{1}{2}$ | Obtuse |
| $\frac{3}{6} = \frac{1}{2}$ | Half turn | Straight angle |
| $\frac{4}{6}$ | More than half | Reflex |
| $\frac{6}{6}$ | Full turn | Full circle |

Part 3 — Half of a $\frac{1}{6}$ turn:
$$\frac{1}{2} \times \frac{1}{6} = \frac{1}{12} \text{ turn}$$

Answer: Half of a $\frac{1}{6}$ turn equals a $\frac{1}{12}$ turn.

This is a very small acute angle — the same as the angle the minute hand of a clock makes in 5 minutes.

Given: Various angles shown in figures (images not visible, so general method is described).

Concept and Method:

Step 1 — Guess: Look at the angle and compare it mentally with a right angle ($\frac{1}{4}$ turn).
- If it looks smaller → guess it is acute (less than $\frac{1}{4}$ turn)
- If it looks like a corner of a square → guess right angle ($\frac{1}{4}$ turn)
- If it looks wider than a right angle but less than a straight line → guess obtuse (between $\frac{1}{4}$ and $\frac{1}{2}$ turn)

Step 2 — Measure: Place your angle measuring tool (the cut-out sectors of $\frac{1}{8}$, $\frac{1}{6}$, $\frac{1}{4}$, etc.) over the angle. Try combinations:
- Does the $\frac{1}{4}$ piece fit exactly? → Right angle
- Is the angle smaller than $\frac{1}{4}$? → Acute; try $\frac{1}{8}$ or $\frac{1}{6}$
- Is the angle between $\frac{1}{4}$ and $\frac{1}{2}$? → Obtuse; try $\frac{1}{4} + \frac{1}{8}$ or $\frac{1}{4} + \frac{1}{6}$

Step 3 — Record:

| Angle | Estimated Turn | Type |
|-------|---------------|------|
| Angle 1 | $\frac{1}{8}$ turn | Acute |
| Angle 2 | $\frac{1}{4}$ turn | Right |
| Angle 3 | $\frac{3}{8}$ turn | Obtuse |
| Angle 4 | $\frac{1}{6}$ turn | Acute |

*(Actual answers depend on the figures provided. Apply the above method to each angle.)*

Given: Four diagrams (a), (b), (c), (d) showing an arrow that has turned from an initial position to a final position.

Concept: The fraction of the turn = (arc covered by arrow) ÷ (full circle).

Method:
1. Note the starting position of the arrow.
2. Note the ending position of the arrow.
3. Count how many equal sections (out of 8 or 6 or 12) the arrow has moved through.
4. Express as a fraction of the full turn.

Typical answers for standard versions of this exercise:

(a) Arrow turns $\frac{1}{4}$ of the circle → Right angle ($\frac{1}{4}$ turn)

(b) Arrow turns $\frac{1}{8}$ of the circle → Acute angle ($\frac{1}{8}$ turn)

(c) Arrow turns $\frac{3}{8}$ of the circle → Obtuse angle ($\frac{3}{8}$ turn)

(d) Arrow turns $\frac{1}{2}$ of the circle → Straight angle ($\frac{1}{2}$ turn)

Verification: Place the cut-out sector tools over the diagram. Combine $\frac{1}{8}$ and $\frac{1}{4}$ pieces as needed to match the turn shown.

*(Note: Apply the same method to the actual figures in your book.)*

Given: Three shapes (a), (b), (c) with angles to be measured.

Concept: Use the angle measuring tools (cut-out sectors) to measure each interior angle of the shapes.

Method:
1. Place the centre of the sector tool at the vertex of the angle.
2. Align one edge of the tool with one arm of the angle.
3. Check which fraction of the circle matches the angle.
4. Classify as acute, right, or obtuse.

General answers (based on typical shapes used in Class 5 textbooks):

(a) Triangle-like shape:
- Angle 1: $\frac{1}{8}$ turn → Acute
- Angle 2: $\frac{1}{4}$ turn → Right
- Angle 3: $\frac{1}{8}$ turn → Acute

(b) Quadrilateral:
- Angle 1: $\frac{1}{4}$ turn → Right
- Angle 2: $\frac{3}{8}$ turn → Obtuse
- Angle 3: $\frac{1}{4}$ turn → Right
- Angle 4: $\frac{1}{8}$ turn → Acute

(c) Another polygon:
- Angles vary; measure each using the tool and classify accordingly.

*(Apply the method above to the actual figures in your textbook for precise answers.)*

Given: Lines are provided and we need to draw angles of specified turn measures.

Concept: Each fraction of a turn corresponds to a specific angle opening.

Key reference values:
$$\frac{1}{8} \text{ turn} = \text{half of right angle (acute)}$$
$$\frac{1}{6} \text{ turn} = \text{one-sixth of full circle (acute)}$$
$$\frac{1}{4} \text{ turn} = \text{right angle}$$
$$\frac{1}{3} \text{ turn} = \frac{2}{6} \text{ turn} = \text{obtuse angle}$$
$$\frac{3}{8} \text{ turn} = \text{obtuse angle}$$
$$\frac{1}{2} \text{ turn} = \text{straight angle}$$

Steps to draw each angle:
1. Keep one given line fixed as the base arm.
2. Place your angle measuring tool (cut-out sector) with its straight edge along the base line.
3. Mark the point where the curved edge of the sector meets the paper.
4. Draw the second arm of the angle from the vertex through this marked point.
5. Label the angle with its turn measure.

Example: To draw a $\frac{3}{8}$ turn angle:
- Place the $\frac{1}{4}$ sector and then the $\frac{1}{8}$ sector together along the base line.
- The combined arc shows the $\frac{3}{8}$ turn.
- Draw the second arm along the outer edge of the combined sectors.

Given: We need to draw angles for each of the listed turns.

Concept: Use angle measuring tools (cut-out sectors) to draw each angle.

Step-by-step for each turn:

| Turn | Simplified | Type of Angle | Description |
|------|-----------|---------------|-------------|
| $\frac{1}{2}$ turn | $\frac{1}{2}$ | Straight angle | A straight line |
| $\frac{1}{4}$ turn | $\frac{1}{4}$ | Right angle | Like a corner of a square |
| $\frac{2}{4}$ turn | $\frac{1}{2}$ | Straight angle | Same as $\frac{1}{2}$ turn |
| $\frac{1}{6}$ turn | $\frac{1}{6}$ | Acute angle | Small opening |
| $\frac{4}{6}$ turn | $\frac{2}{3}$ | Obtuse/Reflex | More than straight |
| $\frac{3}{12}$ turn | $\frac{1}{4}$ | Right angle | Same as $\frac{1}{4}$ turn |
| $\frac{1}{2} + \frac{1}{4}$ turn | $\frac{3}{4}$ | Three-quarter turn | Reflex angle |
| $\frac{1}{8} + \frac{1}{6}$ turn | $\frac{7}{24}$ | Obtuse angle | Between right and straight |

How to draw:
1. Draw a base line (one arm of the angle).
2. Place the appropriate sector tool(s) along the base line.
3. For combined turns (e.g., $\frac{1}{2} + \frac{1}{4}$), place the $\frac{1}{2}$ sector first, then add the $\frac{1}{4}$ sector next to it.
4. Mark the endpoint of the arc and draw the second arm.
5. Label each angle with its turn fraction.

Note on $\frac{1}{8} + \frac{1}{6}$ turn:
$$\frac{1}{8} + \frac{1}{6} = \frac{3}{24} + \frac{4}{24} = \frac{7}{24} \text{ turn}$$
This is between $\frac{1}{4}$ ($= \frac{6}{24}$) and $\frac{1}{3}$ ($= \frac{8}{24}$) — an obtuse angle.

Given: Minute hand moves 15 minutes forward.

Concept: A clock face is a full circle. The minute hand completes one full turn in 60 minutes.

Calculation:
$$\text{Fraction of turn} = \frac{\text{minutes moved}}{\text{total minutes in full turn}} = \frac{15}{60} = \frac{1}{4}$$

Answer: When the minute hand moves by 15 minutes, it has made a $\dfrac{1}{4}$ turn of the circle.

This is a right angle.

Given: Minute hand moves 30 minutes forward.

Calculation:
$$\text{Fraction of turn} = \frac{30}{60} = \frac{1}{2}$$

Answer: When the minute hand moves by 30 minutes, it has made a $\dfrac{1}{2}$ turn of the circle.

This is a straight angle.

Given: Minute hand moves 45 minutes forward.

Calculation:
$$\text{Fraction of turn} = \frac{45}{60} = \frac{3}{4}$$

Answer: When the minute hand moves by 45 minutes, it has made a $\dfrac{3}{4}$ turn of the circle.

This is a three-quarter turn (reflex angle).

Given: Minute hand turns $\dfrac{1}{12}$ of a full circle.

Concept: Full turn = 60 minutes.

Calculation:
$$\text{Minutes moved} = \frac{1}{12} \times 60 = 5 \text{ minutes}$$

Answer: When the minute hand has turned by $\dfrac{1}{12}$ of a full turn, it has moved by 5 minutes.

Given: Minute hand completes a full turn.

Concept: One full turn of the minute hand = one complete revolution = 60 minutes.

Answer: When the minute hand has turned a full circle, it has moved by 60 minutes.

Given: Minute hand turns $\dfrac{1}{6}$ of a full circle.

Calculation:
$$\text{Minutes moved} = \frac{1}{6} \times 60 = 10 \text{ minutes}$$

Answer: When the minute hand has turned by $\dfrac{1}{6}$ of a full turn, it has moved by 10 minutes.

Given: Minute hand turns $\dfrac{4}{12}$ of a full circle.

Simplification:
$$\frac{4}{12} = \frac{1}{3}$$

Calculation:
$$\text{Minutes moved} = \frac{4}{12} \times 60 = \frac{1}{3} \times 60 = 20 \text{ minutes}$$

Answer: When the minute hand has turned by $\dfrac{4}{12}$ of a full turn, it has moved by 20 minutes.

Given: Children start facing a particular direction and make turns as instructed.

Concept:
- A full turn (clockwise or anti-clockwise) brings you back to the same direction.
- A $\frac{1}{2}$ turn (clockwise or anti-clockwise) makes you face the opposite direction.
- A $\frac{1}{4}$ turn clockwise → you turn to your right.
- A $\frac{1}{4}$ turn anti-clockwise → you turn to your left.
- A $\frac{3}{4}$ turn clockwise = same as $\frac{1}{4}$ turn anti-clockwise → you face left.

Reference Table (if starting facing North):

| Starting Direction | Turn | Final Direction |
|-------------------|------|-----------------|
| North | $\frac{1}{4}$ clockwise | East |
| North | $\frac{1}{2}$ clockwise | South |
| North | $\frac{3}{4}$ clockwise | West |
| North | Full turn | North |
| North | $\frac{1}{4}$ anti-clockwise | West |
| East | $\frac{1}{4}$ clockwise | South |
| South | $\frac{1}{2}$ turn | North |

Key rule: Each $\frac{1}{4}$ clockwise turn moves you: North → East → South → West → North.

*(Fill in the table in your textbook using the above rules based on the starting directions and turns given in each row.)*