Master the three powerful criteria for determining triangle similarity: AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side).
Three different ways to prove triangles are similar
Two pairs of corresponding angles are equal
Two sides proportional and included angle equal
All three pairs of corresponding sides proportional
The most commonly used similarity criterion
AA Similarity Theorem:
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Why it works:
If two angles are equal, the third angle must also be equal (since angles in a triangle sum to 180°). This guarantees all corresponding angles are equal.
Given:
Triangle ABC: ∠A = 60°, ∠B = 80°
Triangle DEF: ∠D = 60°, ∠E = 80°
Are the triangles similar?
Solution:
∠A = ∠D = 60° ✓
∠B = ∠E = 80° ✓
Therefore, ∠C = ∠F = 40° (180° - 60° - 80°)
Yes, the triangles are similar by AA.
When two sides are proportional and the included angle is equal
SAS Similarity Theorem:
If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.
Key Points:
Given:
Triangle ABC: AB = 4, AC = 6, ∠A = 50°
Triangle DEF: DE = 8, DF = 12, ∠D = 50°
Are the triangles similar?
Solution:
Check proportions:
AB/DE = 4/8 = 1/2
AC/DF = 6/12 = 1/2
∠A = ∠D = 50° ✓
Yes, the triangles are similar by SAS.
When all three pairs of corresponding sides are proportional
SSS Similarity Theorem:
If all three pairs of corresponding sides of two triangles are proportional, then the triangles are similar.
Key Points:
Given:
Triangle ABC: sides 3, 4, 5
Triangle DEF: sides 6, 8, 10
Are the triangles similar?
Solution:
Check proportions:
3/6 = 1/2
4/8 = 1/2
5/10 = 1/2
All ratios equal 1/2 ✓
Yes, the triangles are similar by SSS.
Strategy for determining which similarity criterion to use
Explore more sophisticated proof techniques and applications
Two-Column Proofs:
Statement and reason format for logical flow
Flow Proofs:
Visual representation of logical connections
Paragraph Proofs:
Written explanations of logical reasoning
Right Triangles:
HL (Hypotenuse-Leg) similarity criterion
Isosceles Triangles:
Special properties when base angles are equal
Equilateral Triangles:
All equilateral triangles are similar
How triangle similarity is used in practical situations
Height Measurement:
Using shadows and similar triangles
Distance Calculation:
Measuring inaccessible distances
Lens Design:
Calculating focal lengths and magnification
Camera Systems:
Image formation and scaling
Structural Analysis:
Force distribution in truss systems
Scale Models:
Testing designs before construction
Apply the similarity criteria to solve problems
Given: In triangles ABC and DEF, ∠A = ∠D = 45° and ∠B = ∠E = 60°.
Prove that the triangles are similar and find the scale factor if AB = 6 and DE = 9.
Solution:
• ∠A = ∠D = 45° ✓
• ∠B = ∠E = 60° ✓
• Therefore, ∠C = ∠F = 75° (180° - 45° - 60°)
• Triangles are similar by AA
• Scale factor = DE/AB = 9/6 = 1.5
Given: Triangle PQR has sides PQ = 5, PR = 7, and ∠P = 30°. Triangle STU has sides ST = 10, SU = 14, and ∠S = 30°.
Determine if the triangles are similar and state the criterion used.
Solution:
• Check proportions: PQ/ST = 5/10 = 1/2
• PR/SU = 7/14 = 1/2
• ∠P = ∠S = 30° ✓
• Two sides proportional, included angle equal
• Triangles are similar by SAS