Acute, Right, and Obtuse Triangles
Classifying Triangles by Their Angles — Grade 7 Geometry
📐 Introduction: Classifying Triangles by Angles
Triangles can be classified in two main ways: by their sides (equilateral, isosceles, scalene) and by their angles. In this lesson, we focus on classifying triangles based on the types of angles they contain.
🔄 Quick Angle Review
Acute Angle
Less than $90°$
Example: $30°$, $60°$, $85°$
Right Angle
Exactly $90°$
Marked with a small square
Obtuse Angle
Greater than $90°$ but less than $180°$
Example: $100°$, $120°$, $150°$
🔑 Connection to the Angle Sum Theorem
Angle Sum Theorem
The sum of the interior angles of any triangle is exactly $180°$:
$$\angle A + \angle B + \angle C = 180°$$
Key Observation
From this theorem, we can draw a powerful conclusion about what kinds of angle combinations are possible in a triangle:
If one angle is $90°$ or greater...
Then the sum of the other two angles is $90°$ or less. This means each of the other two angles must be acute (less than $90°$).
Example: If $\angle A = 110°$, then $\angle B + \angle C = 70°$, so both $\angle B$ and $\angle C$ are less than $90°$.
Fundamental Conclusion
In ANY triangle, either:
- All three angles are acute, OR
- Two angles are acute and the third is either right or obtuse
Every triangle has at least two acute angles!
Why Only ONE Non-Acute Angle Is Possible
| Impossible Combination | Why It Fails |
|---|---|
| Two right angles ($90° + 90°$) | Sum = $180°$, leaving $0°$ for the third angle — impossible! |
| Two obtuse angles (e.g., $100° + 95°$) | Sum already exceeds $180°$ — impossible! |
| One right + one obtuse (e.g., $90° + 100°$) | Sum = $190°$, which exceeds $180°$ — impossible! |