Puzzle: I am an acute angle. If you double my measure, you get an acute angle. If you triple my measure, you will get an acute angle again. If you quadruple (four times) my measure, you will get an acute angle yet again! But if you multiply my measure by 5, you will get an obtuse angle measure. What are the possibilities for my measure?
Step 1: Let the angle be x°.
Step 2: x is acute: 0° < x < 90°.
Step 3: 2x is acute: 2x < 90°, so x < 45°.
Step 4: 3x is acute: 3x < 90°, so x < 30°.
Step 5: 4x is acute: 4x < 90°, so x < 22.5°.
Step 6: 5x is obtuse: 90° < 5x < 180°, so 18° < x < 36°.
Step 7: Combine: 18° < x < 22.5°.
Answer: Possible values: 19°, 20°, 21°, 22° (angles between 18° and 22.5°).
Explanation
The chapter defines acute angles as 'less than 90°' and obtuse angles as 'greater than 90° and less than 180°.' We need to find an angle x such that: x, 2x, 3x, and 4x are all acute (less than 90°), but 5x is obtuse (between 90° and 180°). This gives us the inequalities: x < 90°, 2x < 90°, 3x < 90°, 4x < 90°, and 90° < 5x < 180°. The most restrictive condition for acute angles is 4x < 90°, giving x < 22.5°. For 5x to be obtuse, we need 18° < x < 36°. Combining: 18° < x < 22.5°.
Solution Steps
Step 1: Let the angle be x°.
Step 2: x is acute: 0° < x < 90°.
Step 3: 2x is acute: 2x < 90°, so x < 45°.
Step 4: 3x is acute: 3x < 90°, so x < 30°.
Step 5: 4x is acute: 4x < 90°, so x < 22.5°.
Step 6: 5x is obtuse: 90° < 5x < 180°, so 18° < x < 36°.
Step 7: Combine: 18° < x < 22.5°.
Step 8: Possible values: 19°, 20°, 21°, 22°.