Back to Exercise: Prove triangles congruent

Exercises: Prove Triangles Congruent Using Rigid Motions

Work through each section in order. Show your reasoning where indicated.

Grade 9·20 problems·~35 min·Common Core Math - HS Geometry·standard·hsg-co-b-7

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Warm-Up: Review What You Know

These problems review skills from earlier lessons.

$Two congruent triangles in different positions with a transformation arrow between them.$

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

They have the same area.

A sequence of rigid motions maps one figure exactly onto the other.

They have the same perimeter.

One figure is a scaled copy of the other.

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?

$AB$ and $FD$

$AB$ and $DE$

$BC$ and $FD$

$AC$ and $DE$

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

Side lengths only

Angle measures only

Both side lengths and angle measures

Neither side lengths nor angle measures — only the shape is preserved

Fluency Practice

Apply the biconditional and CPCTC directly.

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?

Three — one for each pair of corresponding sides

Three — one for each pair of corresponding angles

Six — three pairs of sides and three pairs of angles

Two — the biconditional has two directions

$Triangle PQR mapped to triangle XYZ by a rigid motion, with corresponding sides PQ and XY marked.$

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?

$PQ = XY$ because the rigid motion maps $P$ to $X$ and $Q$ to $Y$ , and rigid motions preserve distances.

$PQ = XY$ because both segments are sides of triangles.

$\angle P = \angle X$ because they are the first-listed angles in their respective triangles.

$PR = XZ$ because both triangles have the same shape.

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?

$AB = DE$

$\angle B = \angle E$

$\triangle ABC$ and $\triangle DEF$ have the same perimeter

$\angle A + \angle B + \angle C = 180^\circ$ implies $\angle D + \angle E + \angle F = 180^\circ$

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?

The triangles might be congruent, but we need more information.

$\triangle ABC \cong \triangle DEF$ because all six corresponding parts are equal.

The triangles are similar but not necessarily congruent.

We can only conclude congruence after finding an explicit rigid motion.

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?

Vertices $A$ and $D$ must be labeled with the same letter in the figure.

The triangles containing $\angle A$ and $\angle D$ must be congruent.

$\angle A$ and $\angle D$ must be corresponding angles of parallel lines.

The triangles must share a common side.

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