SAS Similarity
If two sides of one triangle are proportional to two sides of another, and the included angles are congruent, the triangles are similar.
Introduction
SAS Similarity is the “middle ground” between AA (angles only) and SSS (sides only). You need one angle and two sides — but the angle must be between the two sides (included). This is analogous to SAS Congruence, but with proportionality instead of equality.
Past Knowledge
AA Similarity (7.2.1). SSS Similarity (7.2.2). SAS Congruence (5.2.3).
Today's Goal
Prove similarity using two proportional sides with a congruent included angle.
Future Success
Triangle Proportionality Theorem (7.2.4), geometric mean (7.3.2).
Key Concepts
SAS Similarity Theorem
If two sides of one triangle are proportional to two sides of another, and their included angles are congruent, the triangles are similar.
If and , then .
“Included” Is Critical
The angle must be between the two sides. If it's not the included angle, SAS Similarity does not apply — just like SAS Congruence requires the included angle.
Theorem & Proof
Two-Column Proof: SAS Similarity Theorem
Given: and
Prove:
Strategy: Dilate so two sides match those of , then use SAS Congruence.
| # | Statement | Reason |
|---|---|---|
| 1 | and | Given |
| 2 | Apply a dilation centered at with scale factor to get | A dilation exists for every center and scale factor |
| 3 | and | Dilation scales all lengths by |
| 4 | Dilations preserve angle measures; given | |
| 5 | SAS Congruence (steps 3–4: two sides equal, included angle congruent) | |
| 6 | A dilation produces a similar figure | |
| 7 | Steps 5–6: |
∎ Two proportional sides with a congruent included angle guarantee similarity.
Worked Examples
Verifying SAS Similarity
and . Are the triangles similar?
and ✓
(included between the proportional sides) ✓
by SAS Similarity
Finding a Missing Side
by SAS with . Find the scale factor and if .
Scale factor:
Check: ✓
Choosing the Right Theorem
and . Which similarity theorem applies?
— one angle pair.
Check sides around the angle: , ✓
The congruent angle is included between the proportional sides.
by SAS Similarity (scale factor 2.5)
Common Pitfalls
Non-Included Angle
If the congruent angle is NOT between the two proportional sides, SAS Similarity does not apply. There is no “SSA Similarity” theorem.
Confusing SAS Similarity with SAS Congruence
SAS Congruence needs sides to be equal. SAS Similarity only needs them to be proportional. Don't reject similarity just because the sides aren't equal lengths.
Real-Life Applications
Surveying
Surveyors measure two sides and the included angle of a triangle created by landmarks. By creating a similar triangle on paper with the same angle and proportional sides, they can find distances they can't directly measure.
Engineering Scale Models
When building a bridge model, engineers maintain the same angles at each joint (included angles) and scale all connecting member lengths by the same factor — this is exactly SAS Similarity in 3D.
Practice Quiz
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