EXPLAIN THE CONNECTION
Same shape. Clear reasons.
Explain each idea to a classmate, then open a misconception to check your reasoning.識配對、識講原因，再用比例求邊長。
Equal angles → AAA (two pairs imply the third).
Three pairs of lengths → 3 sides proportional.
Two pairs of lengths and their included angle → ratio of 2 sides, included ∠.
Then write the similarity statement in matching order, form a fraction equation, and solve.
For polygons, check both corresponding angles and side ratios.
01 · Same shape, different size
相似三角形的概念
- Similar triangles have the same shape: equal corresponding angles and equal ratios of corresponding sides.形狀相同：對應角相等、對應邊成比例。
- Use one positive scale factor for every length; angles do not scale.所有邊用同一個正數 k；角度唔乘 k。
- Rotation and reflection do not change similarity.轉向或鏡像反轉，仍然可以相似。
Spot the mistake · 搵出錯處
“Similar triangles must be different sizes.”
Read the explanation · 睇解說
A scale factor of 1 is allowed. Congruent triangles are also similar.
02 · Match vertices and sides
配對對應頂點與邊
- Write a vertex map from the order of the similarity statement.先按相似式嘅字母次序配頂點。
- Match both endpoints of each side.一條邊要配對兩個端點。
- Keep every fraction in the same direction.所有分數嘅比較方向保持一致。
Spot the mistake · 搵出錯處
“AB always matches the horizontal side.”
Read the explanation · 睇解說
Rotation changes directions, not correspondence. Use matching vertices in the similarity statement.
03 · Equal angles: AAA
相似條件：三角對應相等
- Match all three angles and write AAA as the similarity reason.三對角相等，理由寫 AAA。
- Two equal angle pairs give the third pair by the angle sum of a triangle.兩對已相等，就可用內角和求第三對。
- Every angle equality in a proof needs a reason.每個角相等都要交代原因。
Spot the mistake · 搵出錯處
“Equal angles tell me the scale factor.”
Read the explanation · 睇解說
They establish similarity. A pair of corresponding lengths is needed for the numerical scale factor.
04 · Three equal side ratios
相似條件：三邊成比例
- Check three corresponding side ratios, all in one direction.三對邊逐一核對，比較方向要一致。
- One common ratio for all three proves similarity.三個比值都相同，先可用呢個條件。
- Two equal side ratios alone do not prove similarity.只驗到兩對邊成比例，未足夠。
Spot the mistake · 搵出錯處
“Two equal side ratios prove similarity.”
Read the explanation · 睇解說
For the three-side test, all three ratios must match. Otherwise an equal included angle is needed.
05 · Two sides and the included angle
相似條件：兩邊比例與夾角
- Identify the two sides and the angle between them.先搵兩條邊中間嘅夾角。
- Show the two corresponding side ratios are equal.核對兩對邊嘅比值相同。
- Use an equal included angle; do not substitute an unrelated angle.要夾角相等，唔係任何一個角相等都得。
Spot the mistake · 搵出錯處
“Any equal angle works with two side ratios.”
Read the explanation · 睇解說
The equal angle must be between the two compared sides for this test.
06 · Parallel lines: prove, then solve
平行線：先證相似，再求邊長
- State the angle equalities and reasons before using similar-triangle side ratios.先寫角相等及原因，再寫相似。
- For parallel lines, identify the specific corresponding or alternate angle pairs.平行線要講清楚用同位角定內錯角。
- Whole side = sum of its segments; match complete triangle sides.完整邊長要加埋各段，唔好用錯剩低嗰段。
Spot the mistake · 搵出錯處
“ABBD = BCDE.”
Read the explanation · 睇解說
For △ABC ∼ △ADE use ABAD = BCDE. AD is the whole side AB + BD.
07 · Crossed & overlapping triangles
交叉與重疊三角形
- Trace the triangles separately and name their vertices in order.先分開描出兩個三角形。
- Use vertically opposite angles and parallel-line angle reasons where appropriate.對頂角、平行線內錯角都要寫原因。
- The same segment can play different roles; match angles before matching sides.先配角再配邊，唔好靠方向估。
Spot the mistake · 搵出錯處
“AO matches CO because both slope the same way.”
Read the explanation · 睇解說
From △AOB ∼ △DOC, AO matches DO and BO matches CO. Match the angles, not the page positions.
08 · A right triangle inside another
直角三角形中的相似證明
- Match the right angles first, then the shared angle.先配直角，再配公共角。
- State the similarity in the order of corresponding vertices.按對應頂點嘅次序寫相似式。
- Form the ratio from that order; do not pair a shared side with itself automatically.共用邊喺兩個三角形可有唔同角色。
Spot the mistake · 搵出錯處
“AC matches AC because the side is shared.”
Read the explanation · 睇解說
From △ACD ∼ △ABC, the small hypotenuse AC matches the large hypotenuse AB.
09 · Similar 2-D figures
相似平面圖形、周界與面積
- For similar polygons, check both corresponding angles and side ratios.多邊形要同時檢查角同邊。
- Corresponding lengths and perimeters use scale factor k.邊、對角線、高、周界都乘 k。
- Areas use k²; use the correct units and check enlargement or reduction.面積乘 k²，單位用平方單位。
Spot the mistake · 搵出錯處
“Doubling every length doubles the area.”
Read the explanation · 睇解說
Area uses both dimensions: its factor is 2² = 4. Perimeter uses factor 2.