GMAT Math : DSQ: Calculating whether right triangles are congruent

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Example Question #1 : Right Triangles

Altitude

Note: Figure NOT drawn to scale.

Refer to the above diagram.

True or false: $\Delta ABX \cong \Delta ACX$

Statement 1: $\overline{AX}$ is an altitude of $\Delta ABC$

Statement 2: $\overline{AX}$ bisects $\angle CAB$

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

To prove triangle congruence, we need to establish some conditions involving side and angle congruence.

We know $\overline{AX} \cong \overline{AX}$ from reflexivity.

If we only assume Statement 1, then we know that $\angle AXB \cong \angle AXC$ , both angles being right angles. If we only assume Statement 2, then we know that $\angle CAX \cong \angle BAX$ by definition of a bisector. Either way, we only have one angle congruence and one side congruence, not enough to establish congruence between triangles. The two statements together, however, set up the Angle-Side-Angle condition, which does prove that $\Delta ABX \cong \Delta ACX$ .

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Example Question #2 : Right Triangles

Altitude

Note: Figure NOT drawn to scale.

Refer to the above diagram.

True or false: $\Delta ABX \cong \Delta ACX$

Statement 1: $\overline{AX}$ is the perpendicular bisector of $\overline{BC}$ .

Statement 2: $\overline{AX}$ is the bisector of $\angle CAB$ .

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

To prove triangle congruence, we need to establish some conditions involving side and angle congruence.

We know $\overline{AX} \cong \overline{AX}$ from reflexivity.

From Statement 1 alone, by definition, $\overline{XC} \cong \overline{XB }$ and, since both $\angle AXB$ and $\angle AXC$ are right angles, $\angle AXB \cong \angle AXC$ . This sets the conditions to apply the Side-Angle-Side Postulate to prove that $\Delta ABX \cong \Delta ACX$ .

From Statement 2 alone,we know that $\angle CAX \cong \angle BAX$ by definition of a bisector. But we only have one angle congruence and one side congruence, not enough to establish congruence between triangles.

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Example Question #3 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB < EF

Statement 2: BC < DE

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. If $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then AB = DE and BC = EF . That AB < EF does not prove or disprove the congruence statement. Therefore, Statement 1 alone - and by a similar argument, Statement 2 alone - is not sufficient to answer the question.

Now assume both statements to be true.

Suppose BC = EF . Then this, along with the two statements, can be combined to yield the statement

AB <EF = BC < DE .

Similarly, if AB = DE ,

BC < DE = AB <EF ,

and BC < EF .

AB = DE and BC = EF cannot both be true, so it is impossible for $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #4 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AC = DF

Statement 2: AB = DE

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 1 alone, we are only give one angle congruency and one side congruency, which, without further information, is not enough to prove or to disprove triangle congruence. By a similar argument, Statement 2 alone is not sufficient either.

Assume both statements are true. By Statement 1, the hypotenuses are congruent, and by Statement 2, one pair of corresponding legs are congruent. These are the conditions of the Hypotenuse Leg Theorem, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ is proved to be true.

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Example Question #5 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB = BC

Statement 2: DE < EF

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Each statement alone give information about only one of the triangles; without information about the other triangle, it is impossible to prove or to disprove triangle congruence.

Assume both statements are true. For $\bigtriangleup ABC \cong \bigtriangleup DEF$ , it must hold that both AB = DE and BC = EF . If AB = DE , then since AB = BC , then BC= DE , and since DE < EF , then BC< EF . Therefore, $\bigtriangleup ABC \ncong \bigtriangleup DEF$ . Similarly, if BC = EF , then AB = EF , and AB > DE ; again, $\bigtriangleup ABC \ncong \bigtriangleup DEF$ . Therefore, the two statements together prove that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is false.

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Example Question #6 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB = 60 and BC = 80 .

Statement 2: DF = 100

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true.

AB = 60 and BC = 80 , so hypotenuse can be calculated using the Pythagorean Theorem:

$AC = \sqrt{\left ( AB\right )^{2}+\left ( BC\right )^{2} }$

$= \sqrt{60^{2}+80^{2} }$

$= \sqrt{3,600+6,400 }$

$= \sqrt{10,000 }$

= 100

This establishes that AC = DF - that is, that the hypotenuses of triangles are congruent. This gives us one side congruence and one angle congruence, the right angles, between the triangles; however, we are not given any other side or angle congruences, so we cannot determine whether or not the triangles are congruent.

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Example Question #7 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: $\left (AB \right ) ^{2} +(BC) ^{2} = \left ( DF\right )^{2}$

Statement 2: $\left ( DF\right )^{2} - \left ( DE\right )^{2} = \left ( BC\right )^{2}$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 only. By the Pythagorean Theorem, $\left (AB \right ) ^{2} +(BC) ^{2} = \left ( AC\right )^{2}$ , so $\left ( AC\right )^{2}= \left ( DF\right )^{2}$ ; subsequently, AC= DF , and one side congruence is proved. However, this, along with one angle congruence - the congruence of right angles $\angle B$ and $\angle E$ - is not enough to prove or disprove triangle congruence.

Assume Statement 2 only. By the Pythagorean Theorem, $\left ( DF\right )^{2} - \left ( DE\right )^{2} = \left ( EF\right )^{2}$ , so $\left ( BC\right )^{2} = \left ( EF\right )^{2}$ ; subsequently, BC = EF , and one side congruence is proved. For the same reason as with Statement 1, this provides insufficient information.

The two statements together, however, are sufficient. Statements 1 and 2 estabish congruence between the hypotenuses and corresponding legs, respectively, setting up the conditions of the Hypotenuse-Leg Theorem. As a consequence, $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #8 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB = DF

Statement 2: BC = DF

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

In any right triangle, the hypotenuse must have length greater than either leg. Therefore, DF > DE and DF > EF .

Assume Statement 1 alone. For $\bigtriangleup ABC \cong \bigtriangleup DEF$ , it must hold that AB = DE . However, AB = DF and DF > DE , so AB > DE . The statement proves that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is false. By a similar argument, Statement 2 proves $\bigtriangleup ABC \cong \bigtriangleup DEF$ is false.

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Example Question #1 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: $\angle A$ and $\angle F$ are complementary angles.

Statement 2: AB + BC < DE + EF

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. In any right triangle, the two acute angles are complementary. Therefore, $\angle A$ and $\angle C$ are complementary angles, as are $\angle D$ and $\angle F$ are complementary angles. Also, two angles complementary to the same angle are coongruent, so, since $\angle A$ and $\angle F$ are complementary angles, $\angle A \cong \angle D$ and $\angle C \cong \angle F$ . From the congruences of all three pairs of corresponding angles, it follows that $\bigtriangleup ABC$ and $\bigtriangleup DEF$ are similar, but without any side comparisons, congruence between the triangles cannot be proved or disproved.

Assume Statement 2 alone. If $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then corresponding sides are congruent, so AB = DE and BC = EF . Therefore, AB + BC = DE + EF . But Statement 2 tells us that this is false. Therefore, we can determine that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is a false statement.

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Example Question #10 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB = BC

Statement 2: $DF = DE \sqrt {2}$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true. Statement 1 establishes that $\bigtriangleup ABC$ has two congruent legs, making it a 45-45-90 triangle. Statement 2 establishes that $\bigtriangleup DEF$ has a hypotenuse that has length $\sqrt{ 2}$ times that of a leg, making it also a 45-45-90 triangle. The triangles have the same angle measures, so they are similar by the Angle-Angle Postulate. However, we are not given any actual lengths or any relationship between the lengths of the corresponding sides of different triangles, so we cannot determine whether the triangles are congruent or not.

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GMAT Math : DSQ: Calculating whether right triangles are congruent

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Example Question #1 : Right Triangles

Example Question #2 : Right Triangles

Example Question #3 : Right Triangles

Example Question #4 : Right Triangles

Example Question #5 : Right Triangles

Example Question #6 : Right Triangles

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