Complete Triangle Notes
For 5th Grade Math Students
📐 Topic 1: Acute, Obtuse, and Right Triangles
Triangles can be classified based on their angles. There are three types:
🔺 Acute Triangle
Definition: A triangle where ALL three angles are less than 90°.
Key Formula:
∠A < 90°, ∠B < 90°, ∠C < 90°
Properties:
- All three angles are acute (less than 90°)
- The sum of all angles = 180°
- All angles are sharp and pointed
Example: A triangle with angles 60°, 70°, and 50° is an acute triangle.
📏 Right Triangle
Definition: A triangle where ONE angle is exactly 90° (a right angle).
Key Formula:
One angle = 90°, Other two angles < 90°
Properties:
- Has exactly one 90° angle (right angle)
- The other two angles are acute
- Forms an "L" shape at the right angle
- The side opposite to the right angle is called the hypotenuse
Example: A triangle with angles 90°, 60°, and 30° is a right triangle.
🔻 Obtuse Triangle
Definition: A triangle where ONE angle is greater than 90° (an obtuse angle).
Key Formula:
One angle > 90°, Other two angles < 90°
Properties:
- Has exactly one obtuse angle (greater than 90°)
- The other two angles are acute
- Looks "wider" or "stretched out"
- The sum of all angles still = 180°
Example: A triangle with angles 110°, 40°, and 30° is an obtuse triangle.
⭐ Important Formula
Sum of all angles in ANY triangle = 180°
∠A + ∠B + ∠C = 180°
📏 Topic 2: Scalene, Isosceles, and Equilateral Triangles
Triangles can also be classified based on their side lengths. There are three types:
🔺 Scalene Triangle
Definition: A triangle where ALL three sides have DIFFERENT lengths.
Key Formula:
Side A ≠ Side B ≠ Side C
Properties:
- All three sides have different lengths
- All three angles have different measures
- No sides are equal
- No line of symmetry
Example: A triangle with sides 3 cm, 5 cm, and 7 cm is a scalene triangle.
🔺 Isosceles Triangle
Definition: A triangle where TWO sides have EQUAL lengths.
Key Formula:
Side A = Side B ≠ Side C
Properties:
- Two sides have equal length
- The angles opposite to the equal sides are also equal
- Has one line of symmetry
- The third side is called the "base"
Example: A triangle with sides 5 cm, 5 cm, and 8 cm is an isosceles triangle.
🔺 Equilateral Triangle
Definition: A triangle where ALL three sides have EQUAL lengths.
Key Formula:
Side A = Side B = Side C
Properties:
- All three sides have equal length
- All three angles are equal (each angle = 60°)
- Has three lines of symmetry
- Most symmetrical triangle
Each angle in an equilateral triangle = 60°
Example: A triangle with all sides measuring 6 cm is an equilateral triangle.
📊 Quick Comparison
| Type | Equal Sides | Equal Angles |
|---|---|---|
| Scalene | 0 (No equal sides) | 0 (No equal angles) |
| Isosceles | 2 equal sides | 2 equal angles |
| Equilateral | 3 equal sides | 3 equal angles (60° each) |
🎯 Topic 3: How to Classify Triangles
Triangles can be classified using BOTH angles AND sides at the same time!
📝 Steps to Classify Any Triangle
Step 1: Look at the ANGLES
- Are all angles less than 90°? → Acute
- Is one angle exactly 90°? → Right
- Is one angle greater than 90°? → Obtuse
Step 2: Look at the SIDES
- Are all sides different? → Scalene
- Are two sides equal? → Isosceles
- Are all sides equal? → Equilateral
Step 3: Combine Both Classifications!
You can describe a triangle using both names. For example: "Right Isosceles Triangle" or "Acute Scalene Triangle"
💡 Examples of Combined Classification
Example 1: Right Scalene Triangle
- Has one 90° angle (Right)
- All three sides have different lengths (Scalene)
- Example: Sides = 3 cm, 4 cm, 5 cm; Angles = 90°, 53°, 37°
Example 2: Acute Isosceles Triangle
- All angles are less than 90° (Acute)
- Two sides have equal length (Isosceles)
- Example: Sides = 5 cm, 5 cm, 6 cm; Angles = 70°, 70°, 40°
Example 3: Acute Equilateral Triangle
- All angles are exactly 60° (Acute)
- All three sides are equal (Equilateral)
- Example: Sides = 7 cm, 7 cm, 7 cm; Angles = 60°, 60°, 60°
- Note: All equilateral triangles are always acute!
Example 4: Obtuse Scalene Triangle
- One angle is greater than 90° (Obtuse)
- All three sides have different lengths (Scalene)
- Example: Sides = 4 cm, 6 cm, 8 cm; Angles = 110°, 40°, 30°
⚠️ Important Notes
- An equilateral triangle is ALWAYS acute (all angles = 60°)
- A triangle CANNOT be both right and obtuse
- A triangle CANNOT be both right and acute
- A triangle can have at most ONE right angle or ONE obtuse angle
- Every triangle fits into exactly ONE category by angles AND ONE category by sides
📚 Quick Reference Summary
By Angles
- ✓ Acute: All < 90°
- ✓ Right: One = 90°
- ✓ Obtuse: One > 90°
By Sides
- ✓ Scalene: All different
- ✓ Isosceles: 2 equal
- ✓ Equilateral: All equal
🔑 Key Formulas to Remember
Sum of Interior Angles = 180°
∠A + ∠B + ∠C = 180°
Each Angle in Equilateral Triangle = 60°
Perimeter = Side A + Side B + Side C
📐 Master these triangle types and you'll be a geometry expert! 📐