Calculating whether acute / obtuse triangles are congruent - GMAT Quantitative Reasoning

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Question

Is it true that $\Delta ABC \cong \Delta DEF$ ?

Suppose you want to answer this question, and you know that $\overline{AB} \cong \overline{DE}$ and $\overline{AC} \cong \overline{DF}$ . Which of the following additional facts would help you to answer this question one way or the other?

Answer

If you know either that $\angle B\cong \angle E$ or $\angle C\cong \angle F$ , you have three congruencies - two sides and a nonincluded angle. This is not enough to establish triangle congruence,

If you know that $\overline{AB} \cong \overline{DF}$ , this, along with the other two statements, establishes that $\overline{AB} \cong \overline{AC} \cong \overline{DE} \cong \overline{DF}$ ; all this proves is that both triangles are isosceles.

If you also know that $\overline{BC} \cong \overline{EF}$ , however, the three statements together make three side congruencies, setting up the Side-Side-Side criterion for triangle congruence.

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Question

Note: Figure NOT drawn to scale

Refer to the above diagram. Which of the following statements is NOT a consequence of the fact that $\Delta ABX \cong \Delta ACX$ ?

Answer

We use the congruence of corresponding sides and angles of congruent triangles to prove four of these statements:

$\angle CAX \cong \angle BAX$ , so $\overline{AX}$ bisects $\angle BAC$ .

$\overline{BX} \cong \overline{CX}$ , so is the midpoint of $\overline{BC}$ .

$\angle AXC \cong \angle AXB$ , and, since they form a linear pair, they are supplementary - therefore, they are right angles. Also, by definition, this makes $\overline{AX}$ an altitude.

However, nothing in this congruence proves that $\overline{BC}$ is congruent to the other two sides of $\Delta ABC$ (which are congruent). The correct statement to exclude is that $\Delta ABC$ is equilateral.

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