Geometry Proofs Basics

1Summary 重點摘要

How to write geometry proof: Given, To prove, Proof.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. The three parts of a proofA geometry proof follows a fixed three-part format. Part one, 'Given': copy down every condition the question supplies — angles, lengths, parallel relations. Part two, 'To prove': state the conclusion precisely. Part three, 'Proof': reason step by step from the given facts until the conclusion appears. Every equality and every inference must carry its reason (the theorem or the given condition) on the right. The format is more than ritual: writing given and to-prove gives the reasoning a start and a goal, and reasons on every step let each line be checked, so any error is located at once.
2. 2. Common theorem 1: angle sum of triangleThe three interior angles of any triangle sum to 180° — the most used theorem in school proofs. Application: given two angles of 50° and 60°, let the third be x; then x + 50° + 60° = 180° (angle sum of triangle), giving x = 70°. In an equilateral triangle the three angles are equal, each 180° ÷ 3 = 60°; in an isosceles triangle the base angles are equal, so a vertex angle of 40° leaves each base angle (180° − 40°) ÷ 2 = 70°. Quote the reason as 'angle sum of triangle'; it belongs to triangles only — a quadrilateral's angles sum to 360°, and the two must not be mixed.
3. 3. Common theorem 2: vertically opposite anglesWhen two straight lines cross, the opposite angles (vertically opposite angles) are equal. If two crossing lines form four angles and one of them is 65°, the angle opposite it is also 65° (vertically opposite angles), while adjacent angles on a straight line sum to 180° (angles on a straight line), so the other two are each 115°. The power of the theorem is equality without calculation: wherever an X of crossing lines appears, cite 'vertically opposite angles'. The common slip is confusing adjacent with opposite: adjacent angles are supplementary (adding to 180°), opposite angles are equal; identify which angles share the vertex and face each other before quoting.
4. 4. Common theorem 3: parallel line propertiesWhen a transversal cuts two parallel lines, three special pairs of angles arise. Corresponding angles are equal (an F shape): on the same side of the transversal, in matching positions on the two parallels. Alternate angles are equal (a Z shape): on opposite sides of the transversal, between the parallels. Co-interior angles are supplementary (a C shape, summing to 180°): on the same side of the transversal, between the parallels. Memorise by letter: F for corresponding, Z for alternate, C for co-interior. Quote full reasons such as 'corresponding angles, AB parallel CD', and only after confirming the question really gives the parallel condition.
5. 5. More useful reasonsBeyond the three main theorems, proofs lean on further reasons. A right angle equals 90°: where the square marker appears, quote 'right angle'. A straight angle equals 180°: angles forming a straight line cite 'angles on a straight line'. Common side or common angle: when two triangles share an edge, quote 'common side' — vital in congruence proofs. Base angles of an isosceles triangle are equal: equal sides force equal base angles. Radii of one circle are equal. Write reasons as full statements — '∠BAC = ∠ACB (base angles, isosceles triangle)' rather than just 'base angles' — so the script is clear and no marker misreads it.
6. 6. A demonstration proofA complete demonstration: given two angles of a triangle as 50° and 60°, prove the third is 70°.
   
   Given: ∠A = 50°, ∠B = 60°.
   To prove: ∠C = 70°.
   Proof:
   Let ∠C = x.
   x + 50° + 60° = 180° (angle sum of triangle)
   x = 180° − 110° = 70°
   Hence ∠C = 70°. Q.E.D.
   
   Note three details: introducing the letter x makes the thinking concrete; every line carries its reason; and the closing sentence answers the 'to prove' exactly. Beginners should follow this skeleton word for word; speed follows familiarity. Afterwards, check that each step uses only given or already-proved facts, with no skipped links.
7. 7. Common mistakes summaryFour marking traps in geometry proofs. First, missing reasons: every step needs its justification; a line without a reason is unsupported and loses marks. Second, skipped steps: jumping from 'two angles equal' straight to the conclusion leaves one or two links out; a proof is written for the reader, so every small step appears. Third, the wrong theorem: quoting parallel-line properties when the question never states parallelism, or using 180° for a quadrilateral as if it were a triangle; confirm the condition exists before citing. Fourth, an unclear conclusion: without the final 'hence ∠C = 70°', the marker cannot tell the proof is finished. As you write, ask of each line 'what is the reason here', and close with a sentence answering the goal.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Proof 3 steps: Given, To prove, Proof
• Every step needs reason
• Triangle angle sum = 180°
• Vertically opposite angles equal
• Parallel lines: corresponding, alternate, co-interior