Triangle ABC has angles 40°, 60°, and 80°. Triangle DEF has angles 40° and 60° (the third is unknown). What can you conclude?
AThe triangles are congruent because all three angles will be equal
BThe triangles are similar, and their corresponding sides are equal
CThe triangles are similar, and their corresponding sides are proportional
DYou need to know the side lengths before drawing any conclusion
Two matching angle pairs (40° and 60°) are sufficient for AA similarity — the third angle of DEF must be 80° by the angle sum theorem. Similar triangles have proportional sides, not necessarily equal sides. 'Congruent' would mean identical in size; 'similar' means same shape but possibly different sizes. The side lengths tell you the scale factor but don't determine whether the triangles are similar.
Question 2 Multiple Choice
In the diagram, triangle PQR ~ triangle STU with PQ = 6, QR = 8, and ST = 9. What is TU?
A11 — add the difference of the first pair (9 − 6 = 3) to QR
B12 — the scale factor is 9/6 = 3/2, and 8 × (3/2) = 12
C6 — TU corresponds to PQ, not QR
DCannot determine without knowing the angles
The scale factor from △PQR to △STU is ST/PQ = 9/6 = 3/2. Since QR corresponds to TU (B↔T, R↔U from the similarity statement), TU = QR × (3/2) = 8 × 3/2 = 12. The vertex correspondence in the similarity statement △PQR ~ △STU tells you which sides pair together: P↔S, Q↔T, R↔U. Answer A is the classic error of adding differences rather than multiplying by a scale factor.
Question 3 True / False
To prove two triangles are similar using the AA postulate, you is expected to verify that most three pairs of corresponding angles are congruent.
TTrue
FFalse
Answer: False
Only two pairs of angles are needed. Once two angles of one triangle match two angles of another, the third angles are automatically equal — because the angles of any triangle must sum to 180°. If angle A = angle D and angle B = angle E, then angle C = 180° − A − B = 180° − D − E = angle F. The AA postulate is powerful precisely because it reduces the verification burden from three pairs to two.
Question 4 True / False
If two triangles are similar, then most corresponding sides are equal in length.
TTrue
FFalse
Answer: False
Similar triangles have proportional sides, not equal sides. Equal sides would mean congruent triangles — same shape AND same size. Similarity only requires same shape, which means the ratios of corresponding sides are equal (they share a common scale factor k), but the sides themselves can be any length. A 3-4-5 right triangle and a 6-8-10 right triangle are similar but not congruent.
Question 5 Short Answer
Why does knowing only two pairs of matching angles guarantee that two triangles are similar — why isn't a third angle check required?
Think about your answer, then reveal below.
Model answer: Because the angles in any triangle must sum to 180°. If two angles of triangle A match two angles of triangle B, the third angle of each triangle is completely determined: it equals 180° minus the sum of the other two. Since the first two pairs already match, the third pair must also match. The angle sum theorem makes the third check redundant.
This is the key insight behind AA: the angle sum theorem acts as a 'free' third constraint. Because triangles are closed under the 180° rule, you never need to independently verify the third angle — it's logically forced by the first two. This is also why there is no 'A' (single angle) similarity shortcut: one angle is not enough to determine shape, since many different triangles share a single angle.