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Triangle Congruence
from Rigid Motions

Discover why ASA, SAS, and SSS guarantee congruence: because a rigid motion can always map one triangle exactly onto the other.

Geometry

What Does Congruence Really Mean?

Two figures are congruent if and only if one can be mapped onto the other by a sequence of rigid motions: translations, rotations, and reflections. These transformations preserve distances and angles exactly.

The Three Rigid Motions

Translation
Slides every point the same distance in the same direction. No rotation, no flipping.
Rotation
Turns every point around a fixed center by a given angle. Distances from center stay the same.
Reflection
Flips every point across a line of reflection. Produces a mirror image with all lengths preserved.

Why Do We Need Shortcut Criteria?

A triangle has six measurements: three sides and three angles. Checking all six is tedious. The congruence criteria (SSS, SAS, ASA) tell us that matching only three specific measurements is enough to guarantee a rigid motion exists.

A B C
\(a\)
\(b\)
\(c\)
 Only 3 needed! D E F
\(\cong\)

SSS: Side-Side-Side

If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

\(5\)
\(6\)
\(7\)
 A B C
\(\cong\)
\(5\)
\(6\)
\(7\)
 D E F

SAS: Side-Angle-Side

If two sides and the included angle of one triangle are equal to the corresponding parts of another, the triangles are congruent.

\(50^\circ\)
\(6.5\)
\(7.5\)
 A B C
\(\cong\)
\(50^\circ\)
\(6.5\)
\(7.5\)
 D E F

ASA: Angle-Side-Angle

If two angles and the included side of one triangle equal the corresponding parts of another, the triangles are congruent.

\(55^\circ\)
\(65^\circ\)
\(8\)
 A B C
\(\cong\)
\(55^\circ\)
\(65^\circ\)
\(8\)
 D E F

Included vs. Non-Included Parts

Included Side (ASA)
The side that lies between the two known angles. Both angle rays bound it, locking the third vertex.
Included Angle (SAS)
The angle formed by the two known sides. It fixes the direction of the second side relative to the first.

The Rigid Motion Proof Pattern

1
Translate one triangle so a chosen vertex lands on its corresponding vertex.
2
Rotate around that vertex until one corresponding side aligns perfectly.
3
Use the given equal parts (sides or angles) to show the second vertex maps correctly.
4
Conclude the third vertex must also coincide (possibly after a reflection), completing the rigid motion.

Example 1: SSS in Action

\(5\)
\(6\)
\(7\)
 P Q R
\(\cong\)
\(5\)
\(6\)
\(7\)
 X Y Z

Example 2: SAS in Action

\(40^\circ\)
\(6.5\)
\(7\)
 J K L
\(\cong\)
\(40^\circ\)
\(6.5\)
\(7\)
 M N O

Example 3: ASA in Action

\(50^\circ\)
\(70^\circ\)
\(10\)
 G H I
\(\cong\)
\(50^\circ\)
\(70^\circ\)
\(10\)
 S T U

Why SSA Does Not Work

\(30^\circ\)
\(A_1\)
\(A_2\)
 B C Two possible triangles! The circle from C hits the ray at two distinct points.

Common Mistakes

✕ Using SSA as valid
SSA does not guarantee congruence. The non-included angle allows the ambiguous case with two possible triangles.
✕ Wrong "included" part
For SAS the angle must be between the two sides. For ASA the side must be between the two angles. Mislabeling which part is included voids the criterion.
✕ AAA means congruent
Three equal angles give similarity, not congruence. Without at least one matching side, the triangles can be different sizes.

Comparison of Criteria

SSS SAS ASA SSA
Given Parts 3 sides 2 sides + included angle 2 angles + included side 2 sides + non-included angle
Valid? ✓ Yes ✓ Yes ✓ Yes ✕ No
Rigid Motion Step Circle intersection pins 3rd vertex Angle fixes ray direction Two rays intersect at one point Ambiguous: two intersections
Reflection Needed? Sometimes Sometimes Sometimes N/A

Question 1 of 5

Two figures are defined as congruent when:

Question 2 of 5

In the SAS criterion, the word "included" means the angle must be:

Question 3 of 5

Why does SSA fail to prove triangle congruence?

Question 4 of 5

In the rigid motion proof of ASA, after translating vertex \(E\) to \(B\) and rotating to align \(\overline{EF}\) with \(\overline{BC}\), what guarantees that the third vertex \(D\) maps to \(A\)?

Question 5 of 5

If \(\triangle ABC\) has \(AB = 5\), \(BC = 7\), and \(AC = 5\), and \(\triangle DEF\) has \(DE = 5\), \(EF = 7\), and \(DF = 5\), which criterion proves \(\triangle ABC \cong \triangle DEF\)?