11.3 Triangle Similarity and Congruence

11.3 Triangle Similarity and Congruence

College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights Geometry concepts for Algebra 2 at 5–10%, typically 1–2 CAT items. Table 11 names triangle similarity and congruency as an Algebra 2 geometry skill. Angle-chasing from Intersecting Lines and Angle Relationships tells you the measures; this section tells you whether two triangles are the same triangle in different positions (congruent) or the same shape at different scales (similar).

Congruent triangles have the same shape and the same size: all three corresponding sides equal, all three corresponding angles equal. Similar triangles have the same shape; size may differ. Corresponding angles are equal, and corresponding sides are in a constant ratio.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

Congruence shortcuts

You do not need all six corresponding parts. These five are enough:

SSA is not a general congruence shortcut. Two sides and a non-included angle can produce zero, one, or two triangles (the ambiguous case). AAF will offer SSA as a distractor named as if it were on the list. It is not.

AAA is not a congruence shortcut. It is a similarity shortcut (and even then, two angles already suffice, so AA is the usual name).

HL is the right-triangle special case of SSA that does work: the right angle is the extra constraint that removes the ambiguity.

Worked: SSS with original triangles MNP and TUV

Triangle MNP has MN = 8, NP = 11, and PM = 14. Triangle TUV has TU = 8, UV = 11, and VT = 14.

Correspondence M ↔ T, N ↔ U, P ↔ V matches three pairs of sides:

• MN = TU = 8
• NP = UV = 11
• PM = VT = 14

By SSS, triangle MNP is congruent to triangle TUV. Corresponding angles are then equal automatically: angle M = angle T, angle N = angle U, angle P = angle V. You do not need to measure them.

If instead VT were 13, SSS would fail. Matching two sides is not enough. This pairing is original teaching material — it is not a restatement of any College Board sample figure.

Worked: SAS, included angle

Triangle WXY has WX = 6, XY = 9, and included angle at X equal to 50°. Triangle ABC has AB = 6, BC = 9, and included angle at B equal to 50°.

The 50° angle is between the 6 and the 9 in both triangles. SAS applies, so triangle WXY ≅ triangle ABC with correspondence W ↔ A, X ↔ B, Y ↔ C.

If the 50° angle in triangle ABC were at A instead of at B, you would have two sides and a non-included angle — SSA, not SAS. Do not call that congruence. Sketch the angle between the sides before you name the shortcut. SAS is a sandwich: side, angle in the middle, side.

Worked: ASA and AAS

Triangle DEF has angle D = 40°, side DE = 10, and angle E = 65°. Triangle RST has angle R = 40°, side RS = 10, and angle S = 65°.

The side of length 10 is between the 40° and 65° angles in both triangles. ASA.

If instead you knew angle D = 40°, angle E = 65°, and side DF (a side not between D and E), that is AAS. It still proves congruence because the third angle is 180 − 40 − 65 = 75° at F and at the corresponding vertex of the other triangle, which converts AAS into ASA once that third angle is filled in. AAF may show two angles and the side opposite one of them; that is AAS, which is legal.

ASA vs AAS is about where the given side sits, not about whether the triangles match. Both prove congruence. SAS vs SSA is different: one proves congruence and one does not.

Worked: HL on a right triangle

Right triangle GHJ has the right angle at H, hypotenuse GJ = 13, and leg GH = 5. Right triangle KLM has the right angle at L, hypotenuse KM = 13, and leg KL = 5.

HL applies: hypotenuses equal, one pair of legs equal, both triangles right. Triangle GHJ ≅ triangle KLM.

The other leg is 12 in both triangles by Pythagoras (5-12-13), but you do not need that for the congruence conclusion. HL is enough. Do not call this SSA even though you have two sides and a non-included right angle; the right-triangle restriction is what makes HL a named theorem.

Similarity shortcuts and corresponding-side ratios

Similar triangles (~) keep angle measures and scale the sides.

SAS~ is not SAS. SAS needs actual equal sides; SAS~ needs a ratio. SSS~ is not SSS for the same reason.

If triangle ABC ~ triangle DEF with correspondence A ↔ D, B ↔ E, C ↔ F, then

AB/DE = BC/EF = CA/FD = k

and angle A = angle D, angle B = angle E, angle C = angle F. Write the letters in corresponding order every time you name the similarity. ABC ~ DEF does not mean the same pairing as ABC ~ EFD.

Worked: AA similarity

Triangle ABC has angle A = 42° and angle B = 73°, so angle C = 180 − 42 − 73 = 65°. Triangle XYZ has angle X = 42° and angle Z = 65°, so angle Y = 73°.

Two pairs already match (42° and 65°), so AA. Correspondence is A ↔ X, B ↔ Y, C ↔ Z, and the similarity is triangle ABC ~ triangle XYZ.

You did not need the third angle to name AA, but computing it is a cheap check and it names the remaining corresponding pair. AA is why AAA is redundant as a similarity test: the third angle is forced.

Worked: SSS~ and a missing side

Triangle STU has sides 6, 8, and 10. Triangle WXY has sides 9, 12, and 15.

6/9 = 8/12 = 10/15 = 2/3

All three pairs proportional, so SSS~. The similarity ratio of STU to WXY is 2/3 (smaller to larger). The ratio of WXY to STU is 3/2.

If a stem gives sides 6, 8, 10 and 9, 12, and x, and tells you the triangles are similar with that correspondence, then 10/x = 6/9 = 2/3, so x = 15. Set up the proportion from corresponding sides, not from “largest with largest” unless the correspondence says so. In a similarity named STU ~ WXY, side ST corresponds to WX, TU to XY, US to YW.

A 6-8-10 triangle is a scaled 3-4-5, so it is right. That right angle can also feed AA (a right angle plus one more) or HL if the other triangle is congruent rather than merely similar. Use the shortcut the given parts support; do not upgrade similarity to congruence without an equal side.

Worked: SAS~

Triangle MNO has MN = 4, included angle N = 40°, and NO = 6. Triangle FGH has FG = 10, included angle G = 40°, and GH = 15.

MN/FG = 4/10 = 2/5 and NO/GH = 6/15 = 2/5, and the included angles equal. SAS~. Do not use SAS (congruence): the sides are not equal, only proportional. The correspondence is M ↔ F, N ↔ G, O ↔ H, so triangle MNO ~ triangle FGH.

If the 40° angle were not the included angle, SAS~ would fail just as SAS would fail. The “included” requirement survives the change from equal sides to proportional sides.

Scale factor, perimeter, and area

If the similarity ratio of corresponding sides is k, then:

• corresponding perimeters also scale by k
• corresponding areas scale by k^2

AAF Algebra 2 geometry is primarily the side ratio and the congruence-vs-similarity classification. Area scaling is the usual extra; mention it so a “the area is 9 times as large, what is the side ratio” item does not surprise you. If areas are 9:25, the side ratio is 3:5, not 9:5.

The AAA trap and the SSA trap

AAA proves similarity, not congruence. Two triangles with angles 40°, 60°, and 80° are similar. They are congruent only if at least one pair of corresponding sides is also equal (which upgrades AA to ASA or AAS). An option that says “the triangles are congruent by AAA” is false. An option that says “the triangles are similar by AA” is true (two angles suffice).

A concrete size check: triangle ABC with angles 40°-60°-80° and sides 5, about 6.9, and about 7.7 is similar to triangle DEF with the same angles and sides 10, about 13.8, and about 15.4. Same angles, double the sides — similar, not congruent. Equilateral triangles (all angles 60°) are the tempting special case: they are always similar by AAA/AA, and they are congruent only when the side lengths match.

SSA does not prove congruence in general. Suppose triangle ABC has AB = 8, BC = 5, and angle at A = 30°. Swinging side BC from vertex C can intersect ray AC in two places when the 5-length is long enough to hit twice. That is the ambiguous case. Exception: HL, which is SSA in a right triangle with the hypotenuse as one of the sides.

AA similarity is not the SSA ambiguous case. AA uses two angles. SSA uses two sides and a non-included angle. Do not let the shared letters “AA” vs “SSA” blur; one is a similarity theorem and the other is a non-theorem for congruence.

AAF classification checklist

1. Mark tick marks and angle arcs. Write the correspondence in letter order.
2. For congruence, look for SSS, SAS, ASA, AAS, or HL. Reject AAA and SSA.
3. Confirm SAS has the angle between the two sides. Confirm ASA has the side between the two angles.
4. For similarity, look for AA, SAS~, or SSS~. Set up side/side = side/side from the named correspondence.
5. AAA ⇒ similar, not congruent, unless a corresponding side is also given equal.
6. A 5-12-13 triangle is right (HL candidate). A 6-8-10 triangle is a scaled 3-4-5, useful for SSS~.

Original figures, original side lengths. The skill is naming the shortcut and writing the proportion, not recognizing a published sample. Finish the Algebra 2 geometry block with Circles in the Coordinate Plane.