Triangles - Fifth Grade
Complete Notes & Formulas
What is a Triangle?
A triangle is a polygon with 3 sides, 3 angles, and 3 vertices (corners).
Important Triangle Properties
1. Three Sides: Every triangle has exactly 3 sides
2. Three Angles: Every triangle has exactly 3 angles
3. Three Vertices: Three corner points where sides meet
4. Closed Shape: All sides connect to form a closed figure
Most Important Formula
Sum of All Angles in a Triangle = 180°
∠A + ∠B + ∠C = 180°
Two Ways to Classify Triangles
| Classification | Based On | Types |
|---|---|---|
| By Angles | Size of angles | Acute, Right, Obtuse |
| By Sides | Length of sides | Scalene, Isosceles, Equilateral |
1. Acute, Obtuse, and Right Triangles
(Classification by Angles)
A. Acute Triangle
Definition: A triangle in which ALL three angles are less than 90°
∠A < 90°
∠B < 90°
∠C < 90°
All angles are acute (less than 90°)
Examples:
• Triangle with angles: 60°, 70°, 50° (all < 90°)
• Triangle with angles: 45°, 65°, 70° (all < 90°)
• Equilateral triangle: 60°, 60°, 60° (all < 90°)
Key Point: No angle reaches or exceeds 90° in an acute triangle!
B. Right Triangle
Definition: A triangle that has exactly ONE angle that measures 90° (a right angle)
One angle = 90° (right angle)
Other two angles are acute (< 90°)
The two acute angles add up to 90°
Special Terms:
Hypotenuse: The longest side (opposite the right angle)
Legs: The two sides that form the right angle
Right Angle Symbol: A small square (□) at the corner
Examples:
• Triangle with angles: 90°, 60°, 30°
• Triangle with angles: 90°, 45°, 45°
• Triangle with angles: 90°, 70°, 20°
Remember: Only ONE angle is 90°. If two or more angles were 90°, the sum would exceed 180°!
C. Obtuse Triangle
Definition: A triangle that has ONE angle that measures more than 90° (an obtuse angle)
One angle > 90° (obtuse angle)
Other two angles are acute (< 90°)
The obtuse angle is less than 180°
Examples:
• Triangle with angles: 120°, 40°, 20° (120° is obtuse)
• Triangle with angles: 100°, 50°, 30° (100° is obtuse)
• Triangle with angles: 95°, 55°, 30° (95° is obtuse)
Key Point: Only ONE angle can be obtuse. The other two must be acute to keep the sum at 180°!
Quick Comparison
| Type | Angle Requirement | Example |
|---|---|---|
| Acute | All 3 angles < 90° | 60°, 70°, 50° |
| Right | 1 angle = 90° | 90°, 60°, 30° |
| Obtuse | 1 angle > 90° | 120°, 40°, 20° |
2. Scalene, Isosceles, and Equilateral Triangles
(Classification by Sides)
A. Scalene Triangle
Definition: A triangle with ALL three sides of different lengths
Side A ≠ Side B ≠ Side C
All sides have different lengths
All angles have different measures
Properties:
• No equal sides
• No equal angles
• No line of symmetry
Examples:
• Sides: 3 cm, 4 cm, 5 cm
• Sides: 6 inches, 8 inches, 10 inches
• Sides: 7 m, 9 m, 11 m
Remember: "Scalene" means "uneven" - no sides are the same!
B. Isosceles Triangle
Definition: A triangle with EXACTLY TWO sides of equal length
Side A = Side B ≠ Side C
Two sides are equal (legs)
Two angles are equal (base angles)
Special Terms:
Legs: The two equal sides
Base: The third side (different length)
Vertex Angle: The angle between the two equal sides
Base Angles: The two equal angles at the base
Properties:
• Two equal sides
• Two equal angles (opposite the equal sides)
• One line of symmetry
Examples:
• Sides: 5 cm, 5 cm, 3 cm
• Sides: 8 inches, 8 inches, 10 inches
• Sides: 6 m, 4 m, 6 m
Remember: "Iso" means "same" - two sides are the same!
C. Equilateral Triangle
Definition: A triangle with ALL three sides of equal length
Side A = Side B = Side C
All sides are equal
All angles are equal (60° each)
Properties:
• All three sides are equal
• All three angles are 60° (equiangular)
• Three lines of symmetry
• Regular polygon (all sides and angles equal)
Each Angle = 60°
180° ÷ 3 = 60°
Examples:
• Sides: 4 cm, 4 cm, 4 cm
• Sides: 7 inches, 7 inches, 7 inches
• Sides: 10 m, 10 m, 10 m
Remember: "Equi" means "equal" - everything is equal! An equilateral triangle is ALWAYS acute!
Quick Comparison
| Type | Equal Sides | Example |
|---|---|---|
| Scalene | 0 (all different) | 3, 4, 5 |
| Isosceles | 2 (two equal) | 5, 5, 3 |
| Equilateral | 3 (all equal) | 6, 6, 6 |
3. Classify Triangles
(Combining Both Classifications)
Two-Way Classification
A triangle can be classified BOTH by its angles AND by its sides at the same time!
Every triangle has TWO names:
1. One name for ANGLES (acute, right, or obtuse)
2. One name for SIDES (scalene, isosceles, or equilateral)
Common Combinations
| Combined Name | Description | Example |
|---|---|---|
| Acute Scalene | All angles < 90°, all sides different | Sides: 3,4,5 Angles: 50°,60°,70° |
| Acute Isosceles | All angles < 90°, two equal sides | Sides: 5,5,4 Angles: 70°,70°,40° |
| Acute Equilateral | All angles 60°, all sides equal | Sides: 6,6,6 Angles: 60°,60°,60° |
| Right Scalene | One 90° angle, all sides different | Sides: 3,4,5 Angles: 90°,60°,30° |
| Right Isosceles | One 90° angle, two equal sides | Sides: 5,5,7 Angles: 90°,45°,45° |
| Obtuse Scalene | One angle > 90°, all sides different | Sides: 3,5,7 Angles: 120°,40°,20° |
| Obtuse Isosceles | One angle > 90°, two equal sides | Sides: 4,4,7 Angles: 110°,35°,35° |
Important Notes
1. No Right Equilateral: An equilateral triangle cannot be right because all angles are 60°
2. No Obtuse Equilateral: An equilateral triangle is always acute (all 60°)
3. Equilateral is Always Acute: Since all angles are 60° (< 90°)
How to Classify a Triangle
Step 1: Look at the ANGLES
• All < 90°? → Acute
• One = 90°? → Right
• One > 90°? → Obtuse
Step 2: Look at the SIDES
• All different? → Scalene
• Two equal? → Isosceles
• All equal? → Equilateral
Step 3: Combine both names
Example: "Acute Isosceles Triangle"
Practice Examples
Example 1: Triangle with sides 5, 5, 8 and angles 70°, 70°, 40°
Angles: All < 90° → Acute
Sides: Two equal → Isosceles
Classification: Acute Isosceles Triangle
Example 2: Triangle with sides 3, 4, 5 and angles 90°, 60°, 30°
Angles: One = 90° → Right
Sides: All different → Scalene
Classification: Right Scalene Triangle
Example 3: Triangle with sides 6, 6, 6 and angles 60°, 60°, 60°
Angles: All < 90° (all 60°) → Acute
Sides: All equal → Equilateral
Classification: Acute Equilateral Triangle
Quick Reference Chart
Triangle Classification Summary
By Angles:
• Acute: All angles < 90°
• Right: One angle = 90°
• Obtuse: One angle > 90°
By Sides:
• Scalene: 0 equal sides (all different)
• Isosceles: 2 equal sides
• Equilateral: 3 equal sides (all 60°)
Key Formula:
∠A + ∠B + ∠C = 180°
💡 Important Tips to Remember
✓ All triangle angles always add up to 180°
✓ A triangle can have at most ONE right or obtuse angle
✓ An equilateral triangle is always acute (60° angles)
✓ An equilateral triangle is also isosceles (has 2 equal sides)
✓ In an isosceles triangle, the angles opposite equal sides are equal
✓ Every triangle has TWO names (one for angles, one for sides)
✓ The longest side is always opposite the largest angle
✓ A right triangle's longest side is called the hypotenuse
✓ Look for the square symbol (□) to identify right angles
✓ Use tick marks (') on sides to show equal lengths
🧠 Memory Tricks
For Angle Types:
Acute: "A-cute little angles" (all small, less than 90°)
Right: Has a "right" angle (90° - makes a corner)
Obtuse: "Oh, be obtuse!" (one big angle > 90°)
For Side Types:
Scalene: "Scale is uneven" - all sides different
Isosceles: "Two legs are same" (iso = same)
Equilateral: "Equal all around" (equi = equal)
Count Equal Sides:
• 0 equal sides = Scalene
• 2 equal sides = Isosceles
• 3 equal sides = Equilateral
The 180° Rule:
"Triangle Total = Turn around = 180°"
Master Triangles! 📐🔺
Triangles are everywhere - learn to identify and classify them!