SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20
演繹幾何中，每一步都要在括號內寫上理由，並使用標準寫法，例如 ∠ABC = 50°（alt. ∠s, AB // CD）。下表按直線、三角形、多邊形及圓分組，列出每個性質應寫的 reason。In deductive geometry every statement needs a reason in brackets, written in the standard form, such as ∠ABC = 50° (alt. ∠s, AB // CD). The table below lists the reason to write for each property, grouped by lines, triangles, polygons and circles.
一條直線截兩條平行線時，同位角相等、內錯角相等，而同旁內角互補（和為 180°）。寫理由時要指明是哪兩條線平行。
When a transversal cuts two parallel lines, corresponding angles are equal, alternate angles are equal, and interior angles on the same side add up to 180°. Name the parallel lines in the reason.
三角形三個內角之和是 180°（∠ sum of △）。外角等於兩個內對角之和（ext. ∠ of △）。等腰三角形的底角相等。
The three angles of a triangle add up to 180° (∠ sum of △). An exterior angle equals the sum of the two interior opposite angles (ext. ∠ of △). In an isosceles triangle the base angles are equal.
兩個三角形通過其中一個判定（SSS、SAS、ASA、AAS 或 RHS）便全等，所有對應邊及對應角都相等，理由寫 corr. sides, ≅ △s 或 corr. ∠s, ≅ △s。
Two triangles are congruent when one passes a test (SSS, SAS, ASA, AAS or RHS). Then every pair of corresponding sides and angles is equal, and you quote corr. sides, ≅ △s or corr. ∠s, ≅ △s.
n 邊形的內角和是 (n − 2) × 180°（∠ sum of polygon）。任何多邊形的外角和都是 360°。
The interior angles of an n-sided polygon add up to (n − 2) × 180° (∠ sum of polygon). The exterior angles of any polygon add up to 360°.
下表列出 DSE 演繹幾何最常用的理由。寫證明時，每個結論後面都要附上其中一個。
| # | 定理 Statement | 寫在括號內的 Reason |
|---|---|---|
| 直線與角 Lines and angles | ||
| 1 | 直線上的相鄰角 Adjacent angles on a straight line add up to 180° | adj. ∠s on st. line |
| 2 | 繞一點的角 Angles at a point add up to 360° | ∠s at a pt. |
| 3 | 對頂角 Vertically opposite angles are equal | vert. opp. ∠s |
| 4 | 同位角（兩線平行） Corresponding angles, parallel lines | corr. ∠s, AB // CD |
| 5 | 內錯角（兩線平行） Alternate angles, parallel lines | alt. ∠s, AB // CD |
| 6 | 同旁內角（兩線平行） Interior angles add up to 180°, parallel lines | int. ∠s, AB // CD |
| 7 | 證明兩線平行 Converse: equal corresponding or alternate angles, or supplementary interior angles | corr. ∠s equal / alt. ∠s equal / int. ∠s supp. |
| 三角形 Triangles | ||
| 8 | 三角形內角和 Angle sum of a triangle is 180° | ∠ sum of △ |
| 9 | 三角形外角 An exterior angle equals the sum of the two interior opposite angles | ext. ∠ of △ |
| 10 | 等腰三角形底角 Base angles of an isosceles triangle are equal | base ∠s, isos. △ |
| 11 | 等邊三角形 All angles of an equilateral triangle are 60° | prop. of equil. △ |
| 12 | 畢氏定理及其逆定理 Pythagoras theorem and its converse | Pyth. theorem / converse of Pyth. theorem |
| 13 | 全等三角形的判定 Tests for congruent triangles | SSS / SAS / ASA / AAS / RHS |
| 14 | 全等三角形的對應部分 Corresponding parts of congruent triangles are equal | corr. sides, ≅ △s / corr. ∠s, ≅ △s |
| 15 | 相似三角形的判定 Tests for similar triangles | AAA / 3 sides proportional / ratio of 2 sides, inc. ∠ |
| 16 | 相似三角形的對應部分 Corresponding parts of similar triangles are proportional or equal | corr. sides, ~ △s / corr. ∠s, ~ △s |
| 17 | 中點定理 The segment joining the mid-points of two sides is parallel to the third side and half as long | mid-pt. theorem |
| 18 | 截線定理 Parallel lines cut transversals in proportion | intercept theorem |
| 多邊形與四邊形 Polygons and quadrilaterals | ||
| 19 | 多邊形內角和 Angle sum of an n-sided polygon is (n − 2) × 180° | ∠ sum of polygon |
| 20 | 多邊形外角和 The exterior angles of a polygon add up to 360° | sum of ext. ∠s of polygon |
| 21 | 平行四邊形的對邊及對角 Opposite sides and opposite angles of a parallelogram are equal | opp. sides of // gram / opp. ∠s of // gram |
| 22 | 平行四邊形的對角線 The diagonals of a parallelogram bisect each other | diags. of // gram |
| 23 | 長方形的對角線 The diagonals of a rectangle are equal | diags. of rectangle |
| 24 | 菱形的對角線 The diagonals of a rhombus are perpendicular and bisect the angles | diags. of rhombus |
| 圓 Circles（詳見圓的基本性質） | ||
| 25 | 圓心角兩倍於圓周角 The angle at the centre is twice the angle at the circumference | ∠ at centre twice ∠ at ⊙ce |
| 26 | 半圓上的圓周角 The angle in a semi-circle is 90° | ∠ in semi-circle |
| 27 | 同弓形內的圓周角 Angles in the same segment are equal | ∠s in the same segment |
| 28 | 圓內接四邊形對角互補 Opposite angles of a cyclic quadrilateral are supplementary | opp. ∠s, cyclic quad. |
| 29 | 切線垂直半徑 A tangent is perpendicular to the radius | tangent ⊥ radius |
在三角形 ABC 中，AB = AC，∠BAC = 40°。求 ∠ABC，並寫出理由。In triangle ABC, AB = AC and ∠BAC = 40°. Find ∠ABC, giving reasons.
∠ABC = ∠ACB (base ∠s, isos. △)∠ABC + ∠ACB + 40° = 180° (∠ sum of △)2∠ABC = 140°答案： Answer: ∠ABC = 70°
AB // CD，直線 EF 與 AB 交於 G，與 CD 交於 H，A 與 D 在 EF 的兩側。若 ∠AGH = 65°，求 ∠GHC。AB // CD. The line EF cuts AB at G and CD at H, with A and D on opposite sides of EF. If ∠AGH = 65°, find ∠GHC.
∠GHD = ∠AGH = 65° (alt. ∠s, AB // CD)∠GHC + ∠GHD = 180° (adj. ∠s on st. line)∠GHC = 180° − 65°答案： Answer: ∠GHC = 115°
證明長方形 ABCD 的兩條對角線相等。Prove that the diagonals of a rectangle ABCD are equal.
AB = DC (opp. sides of rectangle)∠ABC = ∠DCB = 90° (prop. of rectangle)BC = CB (common side)AC = DB (corr. sides, ≅ △s)答案： Answer: 對角線 AC 與 DB 相等。∎The diagonals AC and DB are equal. ∎
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