# GMAT Math : DSQ: Calculating whether right triangles are congruent

## Example Questions

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### Example Question #1 : Right Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram.

True or false:

Statement 1:  is an altitude of

Statement 2:  bisects

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.

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  from reflexivity.

If we only assume Statement 1, then we know that , both angles being right angles. If we only assume Statement 2, then we know that  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 .

### Example Question #2 : Right Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram.

True or false:

Statement 1:  is the perpendicular bisector of .

Statement 2:  is the bisector of  .

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.

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  from reflexivity.

From Statement 1 alone, by definition,  and, since both  and  are right angles, . This sets the conditions to apply the Side-Angle-Side Postulate to prove that .

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

### Example Question #3 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:

Statement 2:

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.

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 , 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 .

### Example Question #4 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:

Statement 2:

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.

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  is proved to be true.

### Example Question #5 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:

Statement 2:

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.

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 , it must hold that both  and . If , then since , then , and since , then . Therefore, . Similarly, if , then , and ; again, . Therefore,  the two statements together prove that  is false.

### Example Question #6 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:  and .

Statement 2:

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true.

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

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.

### Example Question #7 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:

Statement 2:

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.

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, , so ; subsequently, , and one side congruence is proved. However, this, along with one angle congruence - the congruence of right angles  and  - is not enough to prove or disprove triangle congruence.

Assume Statement 2 only. By the Pythagorean Theorem, , so ; 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, .

### Example Question #8 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:

Statement 2:

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

Assume Statement 1 alone. For , it must hold that. However,  and , so . The statement proves that  is false. By a similar argument, Statement 2 proves  is false.

### Example Question #1 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:  and  are complementary angles.

Statement 2:

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.

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,  and  are complementary angles, as are  and  are complementary angles. Also, two angles complementary to the same angle are coongruent, so, since  and  are complementary angles,  and . From the congruences of all three pairs of corresponding angles, it follows that  and  are similar, but without any side comparisons, congruence between the triangles cannot be proved or disproved.

Assume Statement 2 alone. If , then corresponding sides are congruent, so  and . Therefore, . But Statement 2 tells us that this is false. Therefore, we can determine that  is a false statement.

### Example Question #10 : Right Triangles

Given:  and , with right angles

True or false: .

Statement 1:

Statement 2:

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.