02.03 Rigid Motion and Congruence

Congruent means a rigid motion exists

Two figures are congruent when one can be stacked on the other by sliding, flipping, or turning. Those three moves keep size and shape, so corresponding sides match and corresponding angles match.

Do not trust a sketch. Pictures lie about length. Prove it with measurements, marks, or a named shortcut.

The letters in a congruence statement are a matching list. ΔABC ≅ ΔDEF means A goes with D, B with E, C with F. Switch the letters and you switch which parts you claimed.

Orientation trap: a reflection reverses naming direction and is still congruent. Congruence is size and shape. Direction of the letters is extra information, not a veto.

Six parts, three is enough

Each triangle has three sides and three angles. You do not need all twelve pieces. Three matching pairs, in a legal pattern, lock the rest.

Hash marks on sides and arcs on angles tell you which pairs already match. Vertical angles, a shared side, a midpoint, and a bisector are how those marks get there without a ruler.

On a coordinate grid, count when a side is horizontal or vertical. Use the distance formula when it slants: square root of (Δx)² + (Δy)².

A shared side is congruent to itself. That reason is the Reflexive Property, not Symmetric.

The five legal patterns

Included means "the piece in between the other two."

SSS: three sides. Angles come along for free after that.
SAS: two sides and the angle they make (the included angle).
ASA: two angles and the side they share (the included side).
AAS: two angles and a side that is not between them.
HL: hypotenuse and one leg, and only after you know both triangles are right.

SAS picture: the marked angle sits between the two marked sides.

Lookalike pair: SAS needs the angle in the middle of the two sides. SSA puts the angle off to the side. SSA is not a congruence shortcut. HL is the one legal cousin of SSA, and only for right triangles, using the hypotenuse plus a leg.

Two patterns that do not work

AAA matches angles only. The triangles are the same shape, so they are similar, not necessarily the same size. You need at least one side to lock the scale.

SSA is the ambiguous case. Two sides and a non-included angle can make zero, one, or two triangles. The course will not let you name SSA as a reason.

If a quiz shows two sides and an angle that is not between them, do not write SAS. Scan for a right angle that would turn it into HL, or for a second angle that would turn it into AAS.

CPCTC is the leftover parts

CPCTC means Corresponding Parts of Congruent Triangles are Congruent. First prove the triangles congruent with SSS, SAS, ASA, AAS, or HL. Then the unmarked leftover pair is congruent too, and the reason on that line is CPCTC.

Typical flow: given marks, plus Reflexive on a shared side, plus a named shortcut, then CPCTC for the piece the question actually asked about.

Example: BF bisects ∠CBA, and ∠CFB ≅ ∠AFB. Shared side BF is Reflexive. That is ASA (angle, included side, angle). Then CF ≅ AF by CPCTC.

Same claim, two reasons: after the triangles are congruent, "definition of congruent segments" is weaker than CPCTC on the leftover pair. HL is wrong if there is no right angle. SSS is wrong if you only have two angles and a side.

A point on a perpendicular bisector

CD is the perpendicular bisector of AB, meeting at E. Prove C is the same distance from A as from B.

AE ≅ EB because a perpendicular bisector cuts AB in half.
∠AEC and ∠BEC are both 90°, so they are congruent.
CE ≅ CE by Reflexive.
ΔAEC ≅ ΔBEC by SAS (leg, right angle, shared side).
CA ≅ CB by CPCTC. Equal lengths means C is equidistant from A and B.

C can slide anywhere on the line except that the same SAS still works. If C sits on AB, it is the midpoint, and the distances still match.

Your turn

Watch for SAS vs SSA, and CPCTC vs a Property of Equality that happens to mention the same length.