Exercises: Prove Triangles Congruent Using Rigid Motions

Two figures are congruent (under the rigid-motion definition) if and only if which condition holds?

The notation "$\triangle ABC \cong \triangle FDE$" tells us that vertex $A$ corresponds to vertex $F$, vertex $B$ corresponds to vertex $D$, and vertex $C$ corresponds to vertex $E$. Under this correspondence, which pair of sides must be congruent?

Rigid motions preserve which of the following properties of a figure? Select the best answer.

The biconditional for triangle congruence states: $\triangle ABC \cong \triangle DEF$ if and only if all corresponding sides and angles are congruent. How many separate congruence conditions does this require?

A rigid motion maps $\triangle PQR$ onto $\triangle XYZ$ with $P \mapsto X$, $Q \mapsto Y$, $R \mapsto Z$. Which statement is justified by the forward direction of the biconditional?

A rigid motion maps $\triangle ABC$ onto $\triangle DEF$ with $A \mapsto D$, $B \mapsto E$, $C \mapsto F$. Which of the following is NOT directly justified by the preservation properties of rigid motions?

Two triangles have the following measurements. Triangle $ABC$: $AB = 5$, $BC = 7$, $AC = 6$, $\angle A = 82^\circ$, $\angle B = 55^\circ$, $\angle C = 43^\circ$. Triangle $DEF$: $DE = 5$, $EF = 7$, $DF = 6$, $\angle D = 82^\circ$, $\angle E = 55^\circ$, $\angle F = 43^\circ$. What does the biconditional guarantee?

In a geometric proof, a student writes: "By CPCTC, $\angle A = \angle D$." Which condition must have been established in a prior step for this conclusion to be valid?

Triangle $RST$ is congruent to triangle $UVW$. A student claims that $RS \cong UV$ because "$R$ and $U$ are both the first vertices in their triangles." What is wrong with this reasoning?

The reverse direction of the triangle congruence biconditional states: if all six corresponding parts of two triangles are congruent, then a sequence of rigid motions exists that maps one triangle onto the other. Describe the three steps of this construction in order, naming the type of rigid motion used in each step.

Triangle $ABC$ has vertices $A(0, 0)$, $B(4, 0)$, $C(2, 3)$. Triangle $DEF$ has vertices $D(1, 2)$, $E(5, 2)$, $F(3, 5)$. A student claims the two triangles are congruent with $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$. Which rigid motion maps $\triangle ABC$ onto $\triangle DEF$?

Complete the logical order of a CPCTC argument. First,   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . Second,   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . Third, conclude that corresponding parts are congruent by CPCTC.

It is established that $\triangle MNP \cong \triangle QRS$. A student lists only the side congruences as the conclusion: $MN = QR$, $NP = RS$, $MP = QS$. What is missing from the complete CPCTC conclusion?

What is the fundamental error in Jordan's proof?

What is wrong with Alex's conclusion?

Triangle $ABC$ has vertices $A(0, 0)$, $B(6, 0)$, $C(3, 4)$. Triangle $DEF$ has vertices $D(2, 1)$, $E(2, 7)$, $F(-2, 4)$.

Verify that all six corresponding parts are congruent under the correspondence $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$. Then describe the sequence of rigid motions (translate, rotate, and check for reflection) that maps $\triangle ABC$ onto $\triangle DEF$.

A classmate argues: "The reverse direction of the triangle congruence biconditional is obvious. If two triangles have the same measurements, of course you can move one onto the other. Why do we need a formal proof?" Write a response that (a) acknowledges the intuition, (b) explains what the formal construction adds that the intuition does not, and (c) explains why this argument would fail for quadrilaterals — even if all four sides and all four angles match.