Practice Similarity Criteria in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Three sets of conditions that guarantee two triangles are similar: AA (two pairs of equal angles), SAS~ (two pairs of proportional sides with equal included angle), and SSS~ (all three pairs of sides in the same ratio).
Think of a photo and its enlargement. They look the same but are different sizes. For triangles, you only need to check that two angles match (AA)—if the angles are the same, the shape is the same, even if the size differs. It's like verifying two buildings have the same blueprint, even if one is a scale model.
Example 1
easyTriangle ABC has angles 40°, 60°, 80°. Triangle DEF has angles 40°, 60°, 80°. Are the triangles similar? Which criterion applies?
Example 2
mediumIn \triangle ABC: AB = 6, BC = 9, AC = 12. In \triangle DEF: DE = 4, EF = 6, DF = 8. Are the triangles similar? State the criterion.
Example 3
mediumTriangle ABC has sides 5, 12, 13. Triangle DEF has sides 10, 24, 26. Prove the triangles are similar and state the criterion used.
Example 4
easyA tree casts a 15 m shadow, and at the same time a 2 m stick casts a 3 m shadow. How tall is the tree? Which similarity criterion justifies this method?
Example 5
hardIn \triangle PQR and \triangle STU: PQ = 8, QR = 12, \angle Q = 50°; ST = 6, TU = 9, \angle T = 50°. Show the triangles are similar and find the scale factor.