Deductive geometry uses known facts and logical steps to prove new geometric results. A valid proof must justify each statement with a theorem, definition, or given condition.

Reasoning language

• Given: information provided in the question.
• To prove: target statement.
• Because / therefore: links each conclusion to a reason.

Core angle facts

• Angles on a straight line sum to \(180^\circ\).
• Angles around a point sum to \(360^\circ\).
• Vertically opposite angles are equal.
• In parallel lines:
  • alternate interior angles are equal,
  • corresponding angles are equal,
  • co-interior angles sum to \(180^\circ\).

Triangle facts

• Interior angles sum to \(180^\circ\).
• Exterior angle equals sum of two opposite interior angles.
• Isosceles triangle: equal sides oppose equal angles.

Polygon facts

Sum of interior angles of \(n\)-gon:

\[ (n-2)\times180^\circ. \]

Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\).

Proof structure

1. Write what is given and what must be shown.
2. Mark known equal angles/lengths on diagram.
3. Use relevant theorems in short logical chain.
4. End with clear conclusion sentence.

Example proof pattern

If \(AB\parallel CD\) and line \(AD\) intersects them, prove one pair of alternate interior angles is equal.

Since \(AB\parallel CD\), alternate interior angles are equal (parallel-line theorem). Therefore required pair is equal.

Common proof errors

• Stating result without reason.
• Using theorem conditions not satisfied by diagram.
• Assuming diagram scale implies equality.
• Skipping steps between statements.

Exam strategy

• Use short statements with explicit reasons.
• Reference standard theorem names precisely.
• Keep symbols consistent with question labels.
• Do not rely on appearance; rely on given facts and proven steps.