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How to Write a Geometry Proof
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Theorem 1: Angle Sum Property of a Triangle
Statement: The sum of the three angles of a triangle is 180°.
To Prove: ∠A + ∠B + ∠C = 180°
Construction: Draw line XY through A parallel to BC.
Proof:
| Statement | Reason |
|---|---|
| ∠XAB = ∠ABC | Alternate interior angles (XY ∥ BC, AB is transversal) |
| ∠YAC = ∠ACB | Alternate interior angles (XY ∥ BC, AC is transversal) |
| ∠XAB + ∠BAC + ∠YAC = 180° | Angles on a straight line (X-A-Y is a straight line) |
| ∠ABC + ∠BAC + ∠ACB = 180° | Substituting from above |
| ∴ ∠A + ∠B + ∠C = 180° | Proved |
Theorem 2: Exterior Angle Theorem
Statement: The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
To Prove: ∠ACD = ∠A + ∠B
Proof:
| Statement | Reason |
|---|---|
| ∠A + ∠B + ∠ACB = 180° | Angle sum property of triangle |
| ∠ACB + ∠ACD = 180° | Linear pair (angles on a straight line) |
| ∴ ∠ACD = ∠A + ∠B | Subtracting ∠ACB from both equations |
Theorem 3: Angles in an Isosceles Triangle
Statement: In an isosceles triangle, the base angles are equal.
To Prove: ∠B = ∠C
Construction: Draw AD as bisector of ∠A (D on BC).
Proof:
| Statement | Reason |
|---|---|
| AB = AC | Given |
| ∠BAD = ∠CAD | AD is angle bisector |
| AD = AD | Common side |
| △ABD ≅ △ACD | SAS congruence |
| ∴ ∠B = ∠C | Corresponding parts of congruent triangles (CPCT) |
Theorem 4: Pythagoras' Theorem
Statement: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
To Prove: c² = a² + b² (i.e., AB² = BC² + AC²)
Proof Approach: 1. Construct altitude CD from C to AB (D on AB). 2. In △ACD and △ABC: ∠A = ∠A (common); ∠ACD = ∠ACB = 90° → △ACD ~ △ABC (AA similarity) → AC/AB = AD/AC → AC² = AB × AD ... (i) 3. In △BCD and △ABC: ∠B = ∠B (common); ∠BCD = ∠BCA = 90° → △BCD ~ △ABC (AA similarity) → BC/AB = BD/BC → BC² = AB × BD ... (ii) 4. Adding (i) and (ii): AC² + BC² = AB(AD + BD) = AB × AB = AB² 5. ∴ c² = a² + b². Proved.
Converse: If c² = a² + b², then the triangle is right-angled.
Theorem 5: Trigonometric Identity Proofs
Identity 1: sin²θ + cos²θ = 1
Proof:
- In right △ABC with ∠B = 90°, let ∠A = θ
- sin θ = BC/AC; cos θ = AB/AC
- sin²θ + cos²θ = BC²/AC² + AB²/AC² = (BC² + AB²)/AC²
- By Pythagoras: BC² + AB² = AC²
- ∴ sin²θ + cos²θ = AC²/AC² = 1 ✓
Identity 2: 1 + tan²θ = sec²θ
Proof:
- Divide sin²θ + cos²θ = 1 throughout by cos²θ:
- sin²θ/cos²θ + 1 = 1/cos²θ
- tan²θ + 1 = sec²θ ✓
Identity 3: 1 + cot²θ = cosec²θ
Proof:
- Divide sin²θ + cos²θ = 1 throughout by sin²θ:
- 1 + cos²θ/sin²θ = 1/sin²θ
- 1 + cot²θ = cosec²θ ✓
Congruence Criteria — Summary
| Criterion | Full Form | What Must Match |
|---|---|---|
| SSS | Side-Side-Side | All 3 sides equal |
| SAS | Side-Angle-Side | 2 sides + included angle |
| ASA | Angle-Side-Angle | 2 angles + included side |
| AAS | Angle-Angle-Side | 2 angles + non-included side |
| RHS | Right-Hypotenuse-Side | Right angle + hypotenuse + one side |
Similarity Criteria — Summary
| Criterion | What Must Match |
|---|---|
| AA | Two pairs of corresponding angles equal |
| SAS | One pair of angles + ratio of including sides = 1 |
| SSS | All three sides in proportion |
Factor Theorem and Remainder Theorem
Remainder Theorem: When a polynomial f(x) is divided by (x−a), the remainder is f(a).
Factor Theorem: (x−a) is a factor of f(x) if and only if f(a) = 0.
Proof of Remainder Theorem: By division algorithm: f(x) = (x−a) × q(x) + r When x = a: f(a) = (a−a) × q(a) + r = 0 + r = r ∴ r = f(a) ✓