Properties of Triangles: Angles, Sides, Theorems & Trigonometry

TAKEAWAYS:

• Angles in a triangle always sum to 180° and follow specific relationships (e.g., exterior angle = sum of two opposite interior angles).
• Similar triangles have equal angles and proportional sides, solved using AA (Angle-Angle), SAS (Side-Angle-Side), or SSS (Side-Side-Side) criteria.
• Congruent triangles are identical in shape and size, verified by SSS, SAS, ASA, AAS, or RHS rules.
• Trigonometric ratios (sine, cosine, tangent) relate angles to side lengths in right triangles and are used to solve real-world problems.
• Pythagoras’ theorem applies only to right triangles: $a^2+b^2=c^2$.
• Area and height formulas for triangles include $\frac{1}{2}\times \text{base}\times \text{height}$ and Heron’s formula for scalene triangles.

1. Basic Properties of Triangles

Triangles are three-sided polygons with three vertices and three angles. Their properties depend on their sides and angles.

Types of Triangles by Sides

Types of Triangles by Angles

Key Angle Properties

1. Sum of interior angles = $180^{\circ }$. figure {"type":"right-triangle","sides":{"hypotenuse":"c","opposite":"a","adjacent":"b","angle":"θ"},"labels":{"sin":"a/c","cos":"b/c","tan":"a/b"},"caption":"Trigonometric ratios in a right triangle"}

- **sin θ** = \(\frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{c}\)
- **cos θ** = \(\frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{b}{c}\)
- **tan θ** = \(\frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{b}\)

#### **Worked Example: Finding Side Lengths**
**Problem**: In △ABC (right-angled at B), ∠A = 30°, and AC = 10 cm. Find BC and AB.
**Solution**:
1. Draw the triangle:
   ```figure
   {"type":"right-triangle","sides":{"hypotenuse":"AC = 10 cm","opposite":"BC","adjacent":"AB","angle":"30°"},"caption":"Right triangle with ∠A = 30°"}

2. Use sin 30°: [ \sin 30° = \frac{BC}{AC} \implies \frac{1}{2} = \frac{BC}{10} \implies BC = 5 \text{ cm} ]
3. Use cos 30°: [ \cos 30° = \frac{AB}{AC} \implies \frac{\sqrt{3}}{2} = \frac{AB}{10} \implies AB = 5\sqrt{3} \text{ cm} ]
4. Answer: BC = 5 cm, AB = (5\sqrt{3}) cm.

5. Pythagoras’ Theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: [ a^2 + b^2 = c^2 ] Worked Example: Problem: A ladder leans against a wall at 6 m height. If the base is 8 m from the wall, find the ladder’s length. Solution:

1. Draw the triangle:

2. Apply Pythagoras’ theorem: [ 6^2 + 8^2 = c^2 \implies 36 + 64 = c^2 \implies c^2 = 100 \implies c = 10 \text{ m} ]
3. Answer: The ladder is 10 m long.

6. Area of Triangles

Formula 1: Base × Height / 2

[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ] Example: Base = 8 cm, height = 5 cm → Area = (\frac{1}{2} \times 8 \times 5 = 20 \text{ cm}^2).

Formula 2: Heron’s Formula (for scalene triangles)

1. Find the semi-perimeter (s = \frac{a + b + c}{2}).
2. Area = (\sqrt{s(s - a)(s - b)(s - c)}). Example: Sides = 5 cm, 6 cm, 7 cm.
3. (s = \frac{5 + 6 + 7}{2} = 9).
4. Area = (\sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} = 6\sqrt{6} \text{ cm}^2).

Exam Tip

1. Memorize angle sum properties, similarity/congruence criteria, and trigonometric ratios.
2. Draw diagrams for every problem—visualizing helps!
3. Label sides/angles clearly in proofs (e.g., "Given: AB = DE, ∠B = ∠E").
4. Practice NEB-style questions:
   • Short answer: "State the similarity criterion used if two triangles have angles 50°, 60°, 70°." Answer: AA (Angle-Angle) criterion.
   • Proof: "Prove △ABC ≅ △DEF given AB = DE, BC = EF, and ∠B = ∠E." Answer: Use SSS (all sides equal).
   • Calculation: "Find the height of a triangle with base 12 cm and area 36 cm²." Answer: (36 = \frac{1}{2} \times 12 \times h \implies h = 6 \text{ cm}).

flowchart TD
    A["Memorize Key Properties"] --> B["Draw Diagrams"]
    B --> C["Label Clearly"]
    C --> D["Practice NEB Questions"]
    D --> E["Check Answers"]

NEB-Style Practice Questions

1. In △PQR, ∠P = 40°, ∠Q = 70°. Find ∠R and an exterior angle at Q.
2. Two triangles have sides in ratio 3:5 and included angles 45°. Are they similar? Why?
3. A 10 m pole casts a 6 m shadow. Find the angle of elevation of the sun.
4. Prove △ABC ≅ △DEF if AB = DE, ∠B = ∠E, and BC = EF.
5. Find the area of a triangle with sides 9 cm, 12 cm, and 15 cm using Heron’s formula.

Key Formulas Summary