The big idea Similar figures have congruent corresponding angles and one constant ratio between every pair of corresponding sides.
Two tests, both required
Similar polygons have congruent corresponding angles and proportional corresponding sides. Every pair of matching sides uses one constant scale factor.
Write the similarity statement so the letters match: ABCD ~ EFGH means A↔E, B↔F, C↔G, and D↔H. Then AB matches EF, BC matches FG, and ∠D matches ∠H.
A dilation produces similar polygons because it scales every length by the same k. Stretching only one dimension would not.
Check a proportion
Compare a 25-by-5 rectangle with a 15-by-3 rectangle. Test 25/15 and 5/3. Cross products are both 75, so the corresponding side ratios match and the rectangles are similar.
For similar trapezoids, AD = 12 matches EH = x and BC = 4 matches FG = 2. Then 12/x = 4/2. Cross multiply: 24 = 4x, so x = 6.
Flipped fraction: 12/4 = x/2 also works because each fraction keeps one figure together. Mixing 12/2 = 4/x pairs the wrong sides.
Angles come along
Once polygons are similar, every pair of corresponding angles is congruent, including unmarked angles. That is half of the definition of similarity.
A dilation can double a triangle's base from 3 to 6 and height from 4 to 8. Translating the smaller triangle onto a matching corner of the larger one shows that the corresponding angles still line up.
Your turn
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