This lesson covers two of the most fundamental and frequently used theorems in geometry. Mastering them is essential for solving almost any problem involving triangles. You will use these concepts to perform calculations and to write formal proofs (riders).

1. Core Concepts (Short Notes)

• Interior Angles: The three angles inside a triangle.

• Exterior Angle: An angle formed outside a triangle when one of its sides is extended.

• Interior Opposite Angles: For any exterior angle, the two interior angles that are not adjacent (next) to it are its interior opposite angles.In this diagram, ∠ACD is the exterior angle. ∠ABC and ∠BAC are the interior opposite angles.

2. Key Theorems

You must memorize the name and meaning of these two theorems for exam questions.

Theorem 1: The Sum of Interior Angles of a Triangle

The sum of the three interior angles of any triangle is always 180°.

• Formula: In ΔABC, ∠ABC + ∠BCA + ∠CAB = 180°.

• Use Case: If you know two angles in a triangle, you can always find the third one.

Theorem 2: The Exterior Angle of a Triangle

The exterior angle formed when a side of a triangle is produced is equal to the sum of the two interior opposite angles.

• Formula: In the diagram above, ∠ACD = ∠ABC + ∠BAC.

• Use Case: This is a powerful shortcut to find an exterior angle without needing to calculate the adjacent interior angle first.

3. Tips & Tricks for Exams

• Draw the Diagram: If a problem is described in words, always draw a rough sketch. It makes the relationships between the angles much clearer. Mark all the information given in the question onto your sketch.

• Look for Hidden Clues: These theorems are often used together with other geometry facts:

  • Angles on a Straight Line: Angles on a straight line add up to 180°. (Useful for finding an exterior angle if you know the adjacent interior angle, or vice-versa).

  • Parallel Lines: Look for Z-angles (alternate angles are equal) and F-angles (corresponding angles are equal). These often give you the first step in a proof.

  • Vertically Opposite Angles: When two lines cross, the angles opposite each other are equal.

• Structure Your Proofs (Riders): When asked to "prove" or "show that", you need to write a formal proof.

  • Start with the information given in the diagram.

  • State a geometric fact (e.g., ∠ACD = ∠ABC + ∠BAC).

  • Provide the theorem as a reason in brackets: (Exterior angle of a triangle).

  • Logically connect your steps until you reach the required conclusion.

• Example: Prove that a = 2x in the figure.

  • ∠AEC = ∠EAD + ∠ADE (Exterior angle of ΔADE)

  • But ∠EAD = 3x and ∠ADE = a (Given)

  • So, ∠AEC = 3x + a

  • Also, ∠AEC = ∠EBC + ∠ECB (Exterior angle of ΔEBC)

  • So, 3x + a = x + 4x

  • 3x + a = 5x

  • a = 5x - 3x

  • a = 2x

4. Important Points to Remember

• Don't Confuse the Angles: The exterior angle is equal to the sum of the two opposite interior angles, NOT the one next to it (the adjacent angle).

• Every Triangle Adds to 180°: This is true for all triangles—scalene, isosceles, equilateral, right-angled. It is the most universal rule for triangles.

• These two theorems are the building blocks for more complex proofs in later lessons (like Parallelograms and Circles). A strong foundation here is crucial.