Introduction to Geometry

1Summary 重點摘要

Learn basic geometry concepts, know how to write geometry proofs.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Basic geometry conceptsGeometry studies points, lines, planes and solids: a point has position but no size, two points determine a straight line, and two lines meeting form an angle. Angles are measured in degrees with the symbol °; a straight angle formed along a line is 180°, and a full turn is 360°. Acute angles are below 90°, right angles equal 90°, and obtuse angles lie between 90° and 180°. These names recur throughout proofs and calculations, forming the vocabulary of the whole topic.
2. Step 2: The angle sum of a triangleThe three interior angles of a triangle always sum to 180°, one of the most important theorems in geometry and the starting point of countless problems. For example, with two angles of 60° and 70°, the third is 180 - 60 - 70 = 50°, and in the worked example 40° and 80° leave 60°. Two corollaries follow: the acute angles of a right triangle sum to 90°, and each angle of an equilateral triangle is 60°. Memorising these together with the main theorem speeds up problem solving considerably.
3. Step 3: How to write a proofA geometry proof has a standard format in three parts: first write 'Given', listing the conditions supplied by the question; then 'To prove', stating the target conclusion; finally 'Proof', reasoning step by step with a reason attached to every step. Skipping steps is not allowed, nor is stating a conclusion without justification; each reason must be an axiom, a proven theorem, or a given condition. The habit of 'one inference, one reason' keeps longer proofs orderly and easy for the examiner to follow.
4. Step 4: A complete worked demonstrationHere is a complete model. Given: in triangle ABC, ∠A = 50° and ∠B = 60°. To prove: ∠C = 70°. Proof: since the angle sum of a triangle is 180° (angle sum of triangle), ∠C = 180° - 50° - 60° = 70°. Proven. Observe the structure — three parts present, one inference with its reason attached. Beginners can imitate this skeleton, filling the question's data into Given, the goal into To prove, and the calculation with its theorem into the body of the proof.
5. Step 5: A list of common reasonsCommon reasons at the beginner stage include: the angle sum of a triangle is 180°; vertically opposite angles are equal; if two lines are parallel, corresponding angles are equal; if two lines are parallel, alternate angles are equal; and a straight angle is 180°. Write reasons in full, including the parallel-line condition, such as 'corresponding angles, AB∥CD', rather than the bare name. These theorems will be studied in detail later; at this stage recognising them inside proofs is enough.
6. Step 6: Points to watch in examinationsIn examinations remember: every step needs a reason, and answers without reasons lose marks even when numerically correct. Draw figures neatly with a ruler, marking angles and sides unambiguously; double-check subtractions such as 180 - 40 - 80 before moving on. Finally confirm the format is complete — Given, To prove and Proof all present, with a reason on every line — so that every method mark is secured.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Triangle angle sum = 180°
• Proof has given, prove, reasoning, reason
• Each step must write reason
• Can't skip steps
• Remember to write reason in exam