GMAT Math : DSQ: Calculating whether right triangles are congruent

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Example Question #471 : Data Sufficiency Questions

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

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

Statement 1: $\left (DE \right ) ^{2} +(EF) ^{2} = \left ( AC\right )^{2}$

Statement 2: $\left ( AC \right )^{2} - \left ( AB\right )^{2} > \left ( EF\right )^{2}$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question

EITHER 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.

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:

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

Explanation:

Assume Statement 1 only. By the Pythagorean Theorem, $\left (DE \right ) ^{2} +(EF) ^{2} = \left ( DF\right )^{2}$ , 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, $\left ( AC \right )^{2} - \left ( AB\right )^{2} = \left ( BC\right )^{2}$ . Since $\left ( AC \right )^{2} - \left ( AB\right )^{2} > \left ( EF\right )^{2}$ , it follows that $\left ( BC \right )^{2} > \left ( EF\right )^{2}$ , and, subsequently, BC > EF . Since, if $\bigtriangleup ABC \cong \bigtriangleup DEF$ , BC = EF , it follows by contradiction that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is a false statement.

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Example Question #12 : Dsq: Calculating Whether Right Triangles Are Congruent

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

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

Statement 1: A single circle can be constructed that passes through A,B,C,D,E, and .

Statement 2: AB = DE

Possible Answers:

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.

EITHER 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.

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 is true. Since each right triangle is inscribed inside a circle, each of the right angles is inscribed in that circle, and each intercepts a semicircle. That makes the two hypotenuses of the triangles, $\overline{AC}$ and $\overline{DF}$ , diameters; since they are on the same circle, their lengths are the same. However, no information is given about any of the other sides or angles, so no congruence can be proved or disproved.

Statement 2 alone gives us only congruence between one set of corresponding legs, which, along with one angle congruence, is insufficient to prove or to disprove triangler congruence.

Now assume both statements are true. From Statement 1, it follows that congruence of hypotenuses, and Statement 2 gives us congruence of corresponding legs; this sets up the conditions of the Hypotenuse-Leg Theorem, so it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #13 : Dsq: Calculating Whether Right Triangles Are Congruent

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

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

Statement 1: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ have the same area.

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.

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.

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. We show that both statements are insufficient to answer the question.

Suppose AB = DE = 6 and BC = EF= 8 .

AB + BC = 6+8 = 14 and DE+EF = 6 + 8 = 14 , satisfying the conditions of Statement 2.

The area of a right triangle is half the product of the lengths of its legs. Each triangle therefore has as its area $\frac{1}{2} \cdot 6 \cdot 8 = 24$ , making the areas the same; Statement 1 is satisfied.

Since corresponding legs of the triangles are congruent, $\bigtriangleup ABC \cong \bigtriangleup DEF$ by the Side-Angle-Side Postulate.

Now, suppose AB = EF= 6 and BC = DE= 8 .

AB + BC = 6+8 = 14 and DE+EF = 8+6 = 14 , satisfying the conditions of Statement 2.

However, $AB \ne DE$ and $BC \ne EF$ , so it is not true that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #14 : Dsq: Calculating Whether Right Triangles Are Congruent

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 = DE= EF

Statement 2: $m \angle A = m \angle C = m \angle D = m \angle F$

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

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:

Assume Statement 1 alone. The statement gives that all four legs of both triangles are congruent - specifically, AB = DE and BC = EF . Since the right angles are also congruent, then, by the Side-Angle-Side Postulate, $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Assume Statement 2 alone. The statement gives that all four acute angles are congruent - specifically, that $m \angle A = m \angle D$ and $m \angle C = m \angle F$ . However, since we do not have any congruence or noncongruence between corresponding sides, congruence of the triangles cannot be proved or disproved.

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GMAT Math : DSQ: Calculating whether right triangles are congruent

Study concepts, example questions & explanations for GMAT Math

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Example Question #471 : Data Sufficiency Questions

Example Question #12 : Dsq: Calculating Whether Right Triangles Are Congruent

Example Question #13 : Dsq: Calculating Whether Right Triangles Are Congruent

Example Question #14 : Dsq: Calculating Whether Right Triangles Are Congruent

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