Proof: If Three Sides Ratios Equal Then Similar Triangles
Let's prove the following theorem:
if (distance AB) / (distance XY) = (distance BC) / (distance YZ) and (distance BC) / (distance YZ) = (distance CA) / (distance ZX) and distance AC = distance SZ and ST || XY and m∠XSZ = 180 and m∠YTZ = 180, then △ABC ∼ △XYZ
Proof:
Given
1 | (distance AB) / (distance XY) = (distance BC) / (distance YZ) |
---|---|
2 | (distance BC) / (distance YZ) = (distance CA) / (distance ZX) |
3 | distance AC = distance SZ |
4 | ST || XY |
5 | m∠XSZ = 180 |
6 | m∠YTZ = 180 |
# | Claim | Reason |
---|---|---|
1 | XY || ST | if ST || XY, then XY || ST |
2 | m∠TSZ = m∠YXS | if XY || ST and m∠XSZ = 180, then m∠TSZ = m∠YXS |
3 | m∠YXS = m∠YXZ | if m∠XSZ = 180, then m∠YXS = m∠YXZ |
4 | m∠TSZ = m∠YXZ | if m∠TSZ = m∠YXS and m∠YXS = m∠YXZ, then m∠TSZ = m∠YXZ |
5 | m∠YXZ = m∠TSZ | if m∠TSZ = m∠YXZ, then m∠YXZ = m∠TSZ |
6 | m∠XZY = m∠SZT | if m∠XSZ = 180 and m∠YTZ = 180, then m∠XZY = m∠SZT |
7 | △XZY ∼ △SZT | if m∠YXZ = m∠TSZ and m∠XZY = m∠SZT, then △XZY ∼ △SZT |
8 | △XYZ ∼ △STZ | if △XZY ∼ △SZT, then △XYZ ∼ △STZ |
9 | (distance ZS) / (distance ZX) = (distance ST) / (distance XY) | if △XZY ∼ △SZT, then (distance ZS) / (distance ZX) = (distance ST) / (distance XY) |
10 | distance ZS = distance CA | if distance AC = distance SZ, then distance ZS = distance CA |
11 | distance CA = distance ZS | if distance ZS = distance CA, then distance CA = distance ZS |
12 | (distance CA) / (distance ZX) = (distance ST) / (distance XY) | if (distance ZS) / (distance ZX) = (distance ST) / (distance XY) and distance CA = distance ZS, then (distance CA) / (distance ZX) = (distance ST) / (distance XY) |
13 | (distance AB) / (distance XY) = (distance CA) / (distance ZX) | if (distance AB) / (distance XY) = (distance BC) / (distance YZ) and (distance BC) / (distance YZ) = (distance CA) / (distance ZX), then (distance AB) / (distance XY) = (distance CA) / (distance ZX) |
14 | (distance ST) / (distance XY) = (distance AB) / (distance XY) | if (distance AB) / (distance XY) = (distance CA) / (distance ZX) and (distance CA) / (distance ZX) = (distance ST) / (distance XY), then (distance ST) / (distance XY) = (distance AB) / (distance XY) |
15 | distance ST = distance AB | if (distance ST) / (distance XY) = (distance AB) / (distance XY), then distance ST = distance AB |
16 | distance AB = distance ST | if distance ST = distance AB, then distance AB = distance ST |
17 | (distance ST) / (distance XY) = (distance TZ) / (distance YZ) | if △XZY ∼ △SZT, then (distance ST) / (distance XY) = (distance TZ) / (distance YZ) |
18 | (distance ST) / (distance XY) = (distance BC) / (distance YZ) | if (distance ST) / (distance XY) = (distance AB) / (distance XY) and (distance AB) / (distance XY) = (distance BC) / (distance YZ), then (distance ST) / (distance XY) = (distance BC) / (distance YZ) |
19 | (distance TZ) / (distance YZ) = (distance BC) / (distance YZ) | if (distance ST) / (distance XY) = (distance TZ) / (distance YZ) and (distance ST) / (distance XY) = (distance BC) / (distance YZ), then (distance TZ) / (distance YZ) = (distance BC) / (distance YZ) |
20 | distance BC = distance TZ | if (distance TZ) / (distance YZ) = (distance BC) / (distance YZ), then distance BC = distance TZ |
21 | △ABC ≅ △STZ | if distance AB = distance ST and distance BC = distance TZ and distance CA = distance ZS, then △ABC ≅ △STZ |
22 | △ABC ∼ △STZ | if △ABC ≅ △STZ, then △ABC ∼ △STZ |
23 | △ABC ∼ △XYZ | if △ABC ∼ △STZ and △XYZ ∼ △STZ, then △ABC ∼ △XYZ |
Comments
Please log in to add comments