QuestionWhich statement proves $\triangle D E F \cong \triangle A B C$ ? a. $A B=D E$ , $B C=E F$ b. $\angle D \cong \angle A$ , $\angle B \cong \angle E$ , $\angle C \cong \angle F$ c. Rigid motions map $A$ to $D$ , $A B$ to $D E$ , $\angle B$ to $\angle E$ d. Rigid motions map $A B$ to $D E$ , $B C$ to $E F$ , $A C$ to $D F$ .

Studdy Solution

STEP 1

Assumptions1. We are given four statements and we need to determine which one provides sufficient evidence that $\triangle DEF$ is congruent to $\triangle ABC$ . . Congruence of triangles is determined by certain postulates such as Side-Side-Side (SS), Side-Angle-Side (AS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) for right triangles.

STEP 2

Let's analyze each statement one by one.
a. $AB=DE$ and $BC=EF$
This statement is saying that two sides of $\triangle ABC$ are equal to two sides of $\triangle DEF$ . However, for theSS postulate, we need all three sides of one triangle to be equal to the three sides of the other triangle. So, this statement is not sufficient.

STEP 3

b. $\angle D \cong \angle A, \angle B \cong \angle, \angle C \cong \angle F$
This statement is saying that all three angles of $\triangle ABC$ are equal to the three angles of $\triangle DEF$ . However, for the ASA or AAS postulates, we need two angles and the included side or the side between them to be equal in both triangles. So, this statement is not sufficient.

STEP 4

c. There is a sequence of rigid motions that maps point $A$ onto $$, $\overline{AB}$ onto $\overline{DE}$, and $\angle B$ onto $\angle$.
This statement is saying that there is a sequence of rigid motions (transformations that do not change the shape or size of the figure) that maps $\triangle ABC$ onto $\triangle DEF$ . However, it only mentions one side and one angle. For the SAS or ASA postulates, we need two sides and the included angle or two angles and the included side to be equal in both triangles. So, this statement is not sufficient.

STEP 5

d. There is a sequence of rigid motions that maps $\overline{AB}$ onto $\overline{DE}$ , $\overline{BC}$ onto $\overline{EF}$ , and $\overline{AC}$ onto $\overline{DF}$ .
This statement is saying that there is a sequence of rigid motions that maps all sides of $\triangle ABC$ onto $\triangle DEF$ . This satisfies theSS postulate, which states that if all three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.
So, statement d is sufficient evidence that $\triangle DEF$ is congruent to $\triangle ABC$ .

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