Exercise 4.2

import os from weasyprint import HTML html_content = """

Chapter 4: Geometry

Exercise 4.2 Solutions

1. Use any number of the given dots to make different angles.

Solution:

Below are illustrations showing how to connect the dots to form each specific angle type:

1) An Acute Angle (< 90°)

2) An Obtuse Angle (> 90° and < 180°)

3) A Right Angle (= 90°)

4) A Straight Angle (= 180°)

2. Name the vertex and arms that form each angle.

Solution:

First Figure (Left):
• Vertex: D
• Arms: DE and DF (or rays D→E, D→F)

Second Figure (Right):
• Vertex: D
• Arms: DC and DE (or rays D→C, D→E)

3. Pick out the Right angles from the given figures.

Solution:

A right angle is exactly equal to 90°.

• Figure (i) is exactly 90°.
• Figure (iii) has a square marking at the vertex indicating a right angle (90°).
• Figure (v) also features a right-angle perpendicular sign (90°).

Correct Answer: Figures (i), (iii), and (v).

4. Pick out the Acute angles from the given figures.

Solution:

An acute angle is strictly less than 90°.

• Figure (i) = 33° (< 90°)
• Figure (ii) = 56° (< 90°)
• Figure (iii) = 89° (< 90°)

Correct Answer: Figures (i), (ii), and (iii).

5. Pick out the Obtuse angles from the given figures.

Solution:

An obtuse angle is greater than 90° and less than 180°.

• Figure (i) = 101° (> 90°)
• Figure (iii) = 129° (> 90°)

Correct Answer: Figures (i) and (iii).

6. Name the angle in each figure given below in all the possible ways.

Solution:

Angles can be named using three letters (with the vertex in the middle) or by the vertex letter alone.

Figure	Vertex	Possible Names
(i)	N	∠MNL, ∠LNM, or ∠N
(ii)	Q	∠PQR, ∠RQP, or ∠Q
(iii)	N	∠MNO, ∠ONM, or ∠N
(iv)	A	∠SAT, ∠TAS, or ∠A

7. Say True or False.

Solution:

Statement	Answer	Reason / Explanation
(i) 20° and 70° are complementary.	True	20° + 70° = 90° (Sum is exactly 90°)
(ii) 88° and 12° are complementary.	False	88° + 12° = 100° (Not equal to 90°)
(iii) 80° and 180° are supplementary.	False	80° + 180° = 260° (Not equal to 180°)
(iv) 0° and 180° are supplementary.	True	0° + 180° = 180° (Sum is exactly 180°)

8. Draw and label each of the angles.

Solution:

(i) ∠ABC = 90° (Right)

(ii) ∠BIG = 35° (Acute)

(iii) ∠SMC = 145° (Obtuse)

(iv) ∠NAS = 180° (Straight)

*Note: Labels adjusted to correct standard cyclic orientation matching mathematical notation rules.

9. Identify the types of angles shown by the hands of the given clock.

Solution:

• (i) Right Angle: The hands point exactly to 9 and 3 (forming a straight line horizontally but here showing a 90-degree corner variant or 3:00/9:00 positioning setup). Obtuse / Right Angle (Visually looking at the exact separation: it forms an Obtuse Angle spanning roughly 4-hour markings).
• (ii) Straight Angle: Hands are perfectly opposite in a vertical straight line pointing to 12 and 6. Straight Angle (180°)
• (iii) Zero Angle / Complete Angle: Both hands overlap directly on 12. Zero Angle (0°)
• (iv) Acute Angle: The gap between hands is small (less than 90°). Acute Angle
• (v) Obtuse Angle: The gap is wide, explicitly greater than a corner 90° frame. Obtuse Angle

10. Find the complementary angle of:

Solution:

Formula: $\text{Complementary Angle} = 90^\circ - \text{Given Angle}$

• (i) 30° → $90^\circ - 30^\circ =$ 60°
• (ii) 26° → $90^\circ - 26^\circ =$ 64°
• (iii) 85° → $90^\circ - 85^\circ =$ 5°
• (iv) 0° → $90^\circ - 0^\circ =$ 90°
• (v) 90° → $90^\circ - 90^\circ =$ 0°

11. Find the supplementary angle of:

Solution:

Formula: $\text{Supplementary Angle} = 180^\circ - \text{Given Angle}$

• (i) 70° → $180^\circ - 70^\circ =$ 110°
• (ii) 35° → $180^\circ - 35^\circ =$ 145°
• (iii) 165° → $180^\circ - 165^\circ =$ 15°
• (iv) 90° → $180^\circ - 90^\circ =$ 90°
• (v) 0° → $180^\circ - 0^\circ =$ 180°

12. Find the supplementary / complementary angles in each case.

Solution:

• Case (i): Angles lie on a straight line $ABC$. They are supplementary.
  $\text{Missing angle } (?) = 180^\circ - 25^\circ =$ 155°


• Case (ii): Together they form a right angle ($90^\circ$) as indicated by the square corner. They are complementary.
  $\text{Missing angle } (?) = 90^\circ - 30^\circ =$ 60°


• Case (iii): Vertically opposite / Straight line configuration. Angles on straight line $ABC$ are supplementary.
  $\text{Missing angle } (?) = 180^\circ - 67^\circ =$ 113°


• Case (iv): The main outer angle forms a right angle ($90^\circ$). The two inner segments are complementary.
  $\text{Missing angle } (?) = 90^\circ - 46^\circ =$ 44°

""" with open("exercise_4.2_solutions_part2.html", "w") as f: f.write(html_content) HTML("exercise_4.2_solutions_part2.html").write_pdf("exercise_4.2_solutions_part2.pdf") print("PDF generated successfully")