Triangle Congruence & Similarity: SSS, SAS, ASA, AA and the Famous Traps
When are two triangles identical, when are they just scaled copies, and why do AAA and SSA never prove congruence? The five valid rules, the AA shortcut for similarity, and the k² / k³ area-and-volume scaling rule.
10 minTEKS 6A,6B,6D,7A,7BGeometry
Same shape vs. same size
Two triangles in front of you. Are they identical? Are they just scaled copies? Or are they unrelated? The Texas Geometry CBE asks this kind of question on roughly one in eight problems — and the answer comes down to two precise vocabulary words.
Big idea
Congruent = same shape and same size (a perfect copy). Similar = same shape, possibly different size (a proportional copy). Every triangle question hinges on knowing which one you're being asked about.
Congruent uses ≅. Similar uses ~. Memorize the symbols — they appear in answer choices.
Congruent (≅)
All corresponding sides equal and all corresponding angles equal. Triangles match perfectly.
Similar (~)
All corresponding angles equal and sides proportional. Same shape, scaled.
Scale factor
The ratio of corresponding sides between similar figures. If 5 → 15, the scale factor is 3.
Congruence: the five rules
You don't have to verify all six parts (3 sides + 3 angles) to prove triangles are congruent. There are exactly five shortcuts:
Five valid rules — and two famous traps. The CBE loves to test the traps.
Trap: AAA & SSA
AAA (three angles match) tells you the triangles are similar — not congruent. They could be different sizes. SSA (two sides and a non-included angle) is the famous “ambiguous case” — it can describe two completely different triangles. Both appear in “which CANNOT prove congruence?” questions.
Pick the right shortcut
Two triangles have all three pairs of sides equal. They are congruent by:
📌 Three pairs of sides → SSS (Side-Side-Side) congruence.
Spot the trap
Which of the following is NOT a valid way to prove two triangles congruent?
📌 **Step 1: List all valid congruence theorems** ✅ SSS (Side-Side-Side) ✅ SAS (Side-Angle-Side) ✅ ASA (Angle-Side-Angle) ✅ AAS (Angle-Angle-Side) ✅ HL (Hypotenuse-Leg, right triangles only) 📌 **Step 2: Why AAA doesn't work** AAA (Angle-Angle-Angle) proves that triangles are **similar** (same shape), but NOT necessarily **congruent** (same size). Two triangles can have identical angles but different side lengths. 📌 **Answer:** **AAA** is NOT valid for proving congruence. 💡 **Example:** A 3-4-5 triangle and a 6-8-10 triangle have the same angles but different sizes.
Similarity: the proportion machine
Two figures are similar when one is a scaled copy of the other. The angles match exactly; the sides are proportional. To find a missing side, set up a proportion of corresponding sides.
If △ ABC ~ △ DEF, then:AB / DE = BC / EF = AC / DFCross-multiply two of these ratios to solve for any missing side.
Worked example
A rectangle measures 12 by 8. A similar rectangle has width 6. What is its length?
12 / L = 8 / 68 · L = 12 · 6L = 72 / 8 = 9The scale factor here is 6/8 = 3/4 — everything in the small rectangle is ¾ of the original.
Set up a proportion
A rectangle is 12×8. A similar rectangle has width 6. What is its length?
📌 Scale = 6/8 = 3/4. Length = 12 × 3/4 = 9
The fastest similarity rule: AA
For triangles only, you don't need three angles — just two. If two angles of one triangle equal two angles of another, the triangles are similar (the third angle is forced because all triangle angles sum to 180°).
AA Similarity
Two pairs of equal angles ⇒ similar triangles. This is the most-used similarity tool on the test — especially in “shadow” word problems where the sun's angle creates two similar right triangles.
Shadow problems — AA in disguise
Classic CBE setup: a tree casts a shadow, a person casts a shadow at the same time, find the height. The two right triangles share the sun's angle of elevation, plus both have a 90° angle — that's two pairs of equal angles, so the triangles are similar.
Both triangles share the sun-angle and a 90° angle → AA similar → sides proportional.
tree height / 5 = 18 / 6tree height = 5 · (18 / 6) = 15 ftSet up tree-side over person-side. Cross multiply. Done.
The shadow problem
A tree casts a shadow 18 feet long. At the same time, a 5-foot-tall fence post casts a shadow 3 feet long. How tall is the tree?
The tree and fence post form similar triangles with their shadows (same sun angle). tree height / tree shadow = fence height / fence shadow h / 18 = 5 / 3 h = 18 × 5/3 = 30 feet.
The square-of-the-scale-factor rule
If two similar figures have a side-length scale factor of k, then their areas are in the ratio k² and their volumes are in the ratio k³. This is the highest-yield rule on similarity word problems.
Side ratio: kArea ratio: k2Volume ratio: k3Example: side ratio 2:3 → area ratio 4:9 → volume ratio 8:27.
3-second recap
Congruent ≅ — same shape & size. Use SSS, SAS, ASA, AAS, HL only.
Similar ~ — same shape, scaled. Set up a proportion of corresponding sides.
AA proves similarity for triangles (because the third angle is forced).
AAA & SSA never prove congruence — classic CBE traps.
Side ratio k → area ratio k² → volume ratio k³.
Check yourself
Quick check #1
Which is NOT a valid triangle congruence postulate?
SSA (side-side-angle, where the angle is NOT between the two given sides) does not guarantee congruence — it can produce two different triangles. The valid postulates are SSS, SAS, ASA, AAS, and HL (right triangles).
Quick check #2
If two triangles have all three corresponding angles equal, they must be:
AA (or AAA) similarity guarantees the triangles are SIMILAR (same shape, proportional sides), but they could be different sizes — so not necessarily congruent.