Solutions: Section 2.8 (Types of Angles)

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Q. Is it possible to draw $\overrightarrow{OC}$ such that the two angles are equal to each other in size? Answer: Yes. When $\overrightarrow{OA}$ and $\overrightarrow{OB}$ overlap each other (folding the straight angle), the crease $\overrightarrow{OC}$ divides the straight angle $\angle AOB$ into two equal-sized angles ($90^\circ$ each).

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Q. If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle? Answer: $\frac{1}{4}$ of a full turn.

Figure it Out (Page 29)

Q4. Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease.

• a. How many right angles do you have now? Justify.
• b. Describe how you folded.

Solution:

• a. Four right angles. Each angle is $\frac{1}{4}$ of the complete angle (full turn).
• b. (Activity) Fold the paper once to make the slanting line. Then fold the paper onto itself so the line lies exactly on top of itself. The new crease is perpendicular.

Figure it Out (Page 31)

Q2. Make a few acute angles and a few obtuse angles. Solution:

• Acute: Like arrowheads pointing sharply ($\angle < 90^\circ$).
• Obtuse: Wide “arms” like a clock at 5 o’clock ($\angle > 90^\circ$).

Q3. Why do you think words acute and obtuse were chosen? Answer: Acute means “sharp” (small opening). Obtuse means “blunt” (wide opening).

Q4. Find out the number of acute angles in each figure. Pattern? Solution:

• Fig 1 (Triangle): 3 acute angles.
• Fig 2: 12 acute angles.
• Fig 3: 21 acute angles.
• Pattern: $3 \times 0 + ?$… The solution text suggests a pattern based on inner triangles: $3, 12, 21...$
• Next figure: 30 acute angles.