Question

Obtuse Angles (greater than 90° but less than 180°)

Answer 1

Definition
- An obtuse angle measures greater than 90° and less than 180° (90° < θ < 180°). Informally, it’s a “wide” or “blunt” angle.

Quick examples
- 91°, 120°, 135°, 179° are obtuse.
- 90° is right, <90° is acute, >180° is reflex.

Key properties
- An obtuse angle cannot have a complement (complement = 90°).
- The supplement of an obtuse angle is acute (supplement = 180° − θ, which is <90°).
- A triangle can have at most one obtuse interior angle (if one angle >90°, the other two must be acute).
- On the unit circle, an angle is obtuse when its cosine is negative (cos θ < 0).
- For two vectors u and v, the angle between them is obtuse exactly when their dot product u·v < 0.

How to tell (practical checks)
- With a protractor: align the baseline, read the measure; if >90° and <180° it’s obtuse.
- From vectors: compute cos θ = (u·v)/(|u||v|). If cos θ < 0, then 90° < θ < 180°.
Example: u = (1,0), v = (−1,1). u·v = −1 → angle is obtuse.
- From coordinates: find direction vectors of the two rays and use the dot-product test above.

Simple constructions / common obtuse angles
- 120° is common: it is the supplement of 60° (i.e., a straight line with a 60° angle on one side gives 120° on the other).
- You can construct many obtuse angles by constructing an acute angle and taking its supplement.

Practice problems (answers below)
1. Is 145° obtuse?
2. For u = (2, 3) and v = (−1, 1), is the angle between u and v obtuse?
3. Can a triangle have two obtuse angles?
4. If an angle measures θ and cos θ = −0.5, is θ obtuse?

Answers
1. Yes (145° > 90° and < 180°).
2. u·v = 2(−1)+3(1) = 1 > 0 → not obtuse (acute).
3. No (sum of angles would exceed 180°).
4. Yes (cos θ < 0 ⇒ θ is between 90° and 180°).

If you want, I can draw a diagram, show step-by-step protractor use, or give more practice problems with solutions.