Geometry

Congruence: SSS, SAS, ASA

Two triangles are congruent if they have exactly the same shape and size. But you don’t need to check all six measurements (3 sides + 3 angles). There are shortcut criteria — SSS, SAS, and ASA — that let you prove congruence with just three measurements.

SSS (Side-Side-Side)

If all three sides of one triangle equal all three sides of another, the triangles are congruent. Let’s see this: adjust the sides of Triangle 2 to match Triangle 1.

Triangle 1 has fixed sides: a = 4, b = 5, c = 6.

Side a' (Triangle 2)3
18
Side b' (Triangle 2)3
18
Side c' (Triangle 2)3
18
Triangle 1: a=4,  b=5,  c=6∣Triangle 2: a′=3,  b′=3,  c′=3\text{Triangle 1: } a=4,\; b=5,\; c=6 \quad|\quad \text{Triangle 2: } a'=3,\; b'=3,\; c'=3

The graph below shows the side lengths as horizontal bars. When all three pairs match, the triangles are congruent.

-112345678910-5-4-3-2-112345T1 side a = 4T1 side b = 5T1 side c = 6T2 side a'T2 side b'T2 side c'Divider
Try This

Try this: Set a’ = 4, b’ = 5, c’ = 6. The bar lengths match perfectly above and below the divider line — that’s SSS congruence!

SAS (Side-Angle-Side)

If two sides and the included angle (the angle between them) match, the triangles are congruent. The included angle determines how “open” the triangle is.

Below, both triangles share sides of length 4 and 5. Adjust the included angle of Triangle 2 to match Triangle 1’s angle of 60 degrees.

Included angle of Triangle 2 (degrees)90
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T1: sides 4, 5, angle 60°∣T2: sides 4, 5, angle 90°\text{T1: sides 4, 5, angle } 60° \quad|\quad \text{T2: sides 4, 5, angle } 90°

We can visualize two sides of each triangle as lines from the origin. The angle between them determines the third side via the Law of Cosines.

2468-224681012T1 side along x-axis (len 4)T1 second side (60 deg)T2 second side (adjustable)
Try This

Try this: Set the angle to 60 degrees. The red line overlaps the blue line — the triangles are congruent by SAS! Move the angle away and see how the triangle shape changes.

ASA (Angle-Side-Angle)

If two angles and the included side (the side between those angles) match, the triangles are congruent. Since the angles of a triangle sum to 180 degrees, knowing two angles automatically tells you the third.

Angle A of Triangle 2 (degrees)50
1080
Angle B of Triangle 2 (degrees)60
1080
T1: A=50°,  B=60°,  C=70°∣T2: A=50°,  B=60°,  C=\text{T1: } A=50°,\; B=60°,\; C=70° \quad|\quad \text{T2: } A=50°,\; B=60°,\; C=
180°−50°−60°=Angle C of T2180° - 50° - 60° = \text{Angle C of T2}

Below, lines from the origin show the two base angles of each triangle. The shared side is along the x-axis.

-11234567-112345678910T1 angle A = 50 degT1 angle B = 60 degT2 angle AT2 angle BShared side (len 5)
Try This

Try this: Set A = 50 and B = 60. The colored lines overlap — that’s ASA congruence! The intersection point (the triangle’s apex) lands in exactly the same spot.

Why SSA Doesn’t Always Work

You might wonder: what about SSA (Side-Side-Angle)? It turns out SSA is ambiguous — there can be zero, one, or two possible triangles. This is the famous “ambiguous case.” That’s why SSA is NOT a valid congruence criterion.

Connection

Connection: Congruence criteria are the foundation of geometric proofs. Whenever you need to show two triangles are identical, you look for SSS, SAS, or ASA. These shortcuts save you from measuring all six parts!
Challenge

Challenge: Two triangles have angles 40 degrees, 60 degrees, and 80 degrees. Are they necessarily congruent? Why or why not?
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