DSQ: Calculating whether right triangles are congruent - GMAT Quantitative Reasoning

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Question

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 $\angle CAB$

Answer

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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Question

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 $\angle CAB$ .

Answer

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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Question

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

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

Statement 1:

Statement 2:

Answer

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 and . If , then since , then , and since , then . Therefore, $\bigtriangleup ABC cong \bigtriangleup DEF$ . Similarly, if , then , and ; again, $\bigtriangleup ABC cong \bigtriangleup DEF$ . Therefore, the two statements together prove that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is false.

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Question

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

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

Statement 1:

Statement 2:

Answer

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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Question

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

Answer

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, , 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, , 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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Question

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:

Answer

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 and . Therefore, . 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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Question

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

Answer

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, , 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, . Since, if $\bigtriangleup ABC \cong \bigtriangleup DEF$ , , it follows by contradiction that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is a false statement.

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Question

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:

Answer

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

Suppose and .

and , 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 and .

and , 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.

However, and , so it is not true that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Question

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

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

Statement 1:

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

Answer

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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Question

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 and .

Statement 2:

Answer

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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Question

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

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

Statement 1:

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

Answer

Assume Statement 1 alone. The statement gives that all four legs of both triangles are congruent - specifically, and . 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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Question

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

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

Statement 1:

Statement 2:

Answer

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

Assume Statement 1 alone. For $\bigtriangleup ABC \cong \bigtriangleup DEF$ , it must hold that. However, and , so . 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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Question

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

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

Statement 1: and .

Statement 2:

Answer

Assume both statements to be true.

and , so hypotenuse can be calculated using the Pythagorean Theorem:

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

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

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

$= \sqrt{10,000 }$

This establishes that - 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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Question

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

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

Statement 1:

Statement 2:

Answer

Assume Statement 1 alone. If $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then and . That 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 . Then this, along with the two statements, can be combined to yield the statement

.

Similarly, if ,

,

and .

and cannot both be true, so it is impossible for $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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