Back to Explain triangle congruence criteria — Problem 5 · Task Set 8

Exercises: Triangle Congruence Criteria from Rigid Motions

Work through each section in order. For explanation problems, use complete sentences and reference rigid motions where relevant.

Grade 9·21 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-co-b-8

Work through problems with immediate feedback

Fluency Practice

$Two overlapping circles centered at D and E. Their two intersection points, F and F-prime, are symmetric about line DE, illustrating the two possible positions for vertex C-double-prime in the SSS proof.$

In the SSS rigid-motion proof for $\triangle ABC \cong \triangle DEF$ (with $AB = DE$ , $AC = DF$ , $BC = EF$ ), after translating $A$ to $D$ and rotating so $B$ maps to $E$ , vertex $C''$ must satisfy $DC'' = DF$ and $EC'' = EF$ . Why does this guarantee that $C''$ is either $F$ or the reflection of $F$ over line $DE$ ?

Because the angle sum forces $\angle DC''E = \angle DFE$ .

Because $C''$ lies on two circles — one centered at $D$ with radius $DF$ , one centered at $E$ with radius $EF$ — and two distinct circles intersect in at most two points, which are symmetric about the line through their centers.

Because SAS applied to the two sub-triangles forces $C'' = F$ .

Because a translation followed by a rotation always produces a unique image.