DSQ: Calculating an angle in a right triangle - GMAT Quantitative

If Statement 1 is assumed, then by the converse of the Pythagorean Theorem, the triangle is a right triangle with right angle , which is explicitly stated in Statement 2. If is a right angle, then the other two angles are acute, since a triangle must have at least two acute angles. A right angle measures and an acute angle measures less, so from either statement, we can deduce that is the angle with greatest measure.

A right triangle must have its two acute angles complementary; if Statement 1 is assumed, then this is false, the triangle is not a right triangle, and is not a right angle.

If Statement 2 is assumed, then we apply the converse of the Pythagorean Theorem to show that the triangle is not right. The sides of a triangle have the relationship

only in a right triangle. If , then the statement to be tested would be

This statement is false, so the triangle is not a right triangle, and is not a right angle.

For , it is necessary that corresponding angles be congruent - specifically, and . We show that the statements together are insufficient by assuming them both to be true and examining two cases:

Case 1: .

Case 2: and .

Both cases fit the main body of the problem and both statements, but in the first case, , and in the second case, . The statement holds only in the first case but not in the second. (Note that in both cases, , but this is a different statement.)

In order to find the angles of right triangle ABC, we would need to find the length of the sides and maybe found that the triangle is isoceles, or is a special triangle with angles 30-60-90.

Statement one tells us that the height is equal to half the hypothenuse of the triangle. From that we can see that the triangle is isoceles. Indeed, an isoceles right triangle will always have its height equal to half the length of the hypothenuse. Therefore we will know that both angles are 45 degrees. Statement 1 alone is sufficient.

Statement 2 alone is insufficient because we don't know anything about the other sides of the triangle. Therefore it doesn't help us.

Since we are already told that triangle ABC is a right triangle, we just need to find information about other angles or other sides.

Statement 1 allows us to calculate , simply by using the sum of the angles of a triangle, since we know AEC is also a right triangle because AE is the height.

Statement 2 is also sufficient because it allows us to know angle . Indeed, in a right triangle, the height divides the triangles in two triangles with similar properties. Therefore angle is the same as .

Therefore, each statement alone is sufficient.

Assume Statement 1 alone. Let be the measure of the angle of second-greatest measure. Since the measures form an arithmetic sequence, the three angles measure for some common difference . Their sum is , so

However, since we do not know that common difference, we cannot determine the other angle measures, or whether the triangle is acute, right, or obtuse. For example, if , the widest of the angles measures ; if , the widest angle measures . In the former case, the triangle is acute, having all of its angles measure less than ; in the latter case, the triangle is obtuse, having an angle that measures greater than .

Statement 2 alone provides insufficient information; a 30-30-120 triangle and a 30-60-90 triangle both fit this condition as well as the conditions of the measures of the angles of a triangle, but the former is obtuse and the latter is right.

Now assume both statements to be true. Then, since one angle measures by Statement 1, and a second measures , the third measures . This angle is a right angle, so is a right triangle.

Corresponding angles of similar triangles - including the same triangle, considered in different configurations - are congruent. It follows from Statement 1 alone that . The measures of the acute angles of a right triangle add up to , so:

Assume Statement 2 alone. By the 45-45-90 Theorem, since the legs of the right triangle are of equal length, the acute angles of the triangle, one of which is , measures .

Assume Statement 1 alone, and let be the measure of . The sum of the measures of the acute angles of a right triangle, and , is , so

The first two terms of the arithmetic sequence given are and . In an arithmetic sequence, each term is obtained by adding the same number, or common difference. This is the difference of the second and first terms, or

,

which is added to the second term to get the third term:

The third term is , so

, the measure of .

Assume Statement 2 alone. is the right angle of the triangle, so is an acute angle; it will have measure less than . Its measure is a whole number less than 90 which is, by Statement 2, a multiple of 5 and 7. However, both 35 and 70 are multiples of 5 and 7, so can be either, and no information is given that eliminates either choice.

An acute angle must have measure less than ; both non-right angles of the triangle, and , must be acute.

Assume Statement 1 alone. The measure of must be a multiple of 6 and 7 that is less than 90; however, there are at least two such numbers, 42 and 84. With no way to eliminate either choice, Statement 1 alone provides insufficient information.

Statement 2 alone also provides insufficient information, for similar reasons. can have measure 24 or 48, for example; the measure of is 90 minus the measure of , so if cannot be determined definitively, neither can .

Now, assume both statements to be true. 6 and 7 have LCM 42, so any multiple of both must also be a multiple of 42. There are only two multiples of 42 less than 90 - 42 and 84.

If , then, since the measures of the angles of the acute angles of a right triangle total ,

48 is a multiple of both 6 and 8, so this scenario is consistent with both statements.

If , then, since the measures of the angles of the acute angles of a right triangle total ,

6 is not a multiple of 8, so this scenario is inconsistent with Statement 2.

Therefore, it can be determined that .

An acute angle must have measure less than ; both non-right angles of the triangle, and , must be acute, and thus have measure less than .

Assume Statement 1 alone. There are four perfect cubes less than 90 - 1, 8, 27, and 64 - so can have any one of these four degree measures. There is no way to eliminate any one of these four.

Assume Statement 2 alone. The least common multiple of 7 and 9 is 63, so, since is a multiple of both 7 and 9, it must be a multiple of 63. The only multiple of 63 less than 90 is 63 itself, so . The acute angles of a right angle have degree measures that total , so

If Statement 1 is assumed, then by the converse of the Pythagorean Theorem, the triangle is a right triangle with right angle , which is explicitly stated in Statement 2. If is a right angle, then the other two angles are acute, since a triangle must have at least two acute angles. A right angle measures and an acute angle measures less, so from either statement, we can deduce that is the angle with greatest measure.

A right triangle must have its two acute angles complementary; if Statement 1 is assumed, then this is false, the triangle is not a right triangle, and is not a right angle.

If Statement 2 is assumed, then we apply the converse of the Pythagorean Theorem to show that the triangle is not right. The sides of a triangle have the relationship

only in a right triangle. If , then the statement to be tested would be

This statement is false, so the triangle is not a right triangle, and is not a right angle.

For , it is necessary that corresponding angles be congruent - specifically, and . We show that the statements together are insufficient by assuming them both to be true and examining two cases:

Case 1: .

Case 2: and .

Both cases fit the main body of the problem and both statements, but in the first case, , and in the second case, . The statement holds only in the first case but not in the second. (Note that in both cases, , but this is a different statement.)

In order to find the angles of right triangle ABC, we would need to find the length of the sides and maybe found that the triangle is isoceles, or is a special triangle with angles 30-60-90.

Statement one tells us that the height is equal to half the hypothenuse of the triangle. From that we can see that the triangle is isoceles. Indeed, an isoceles right triangle will always have its height equal to half the length of the hypothenuse. Therefore we will know that both angles are 45 degrees. Statement 1 alone is sufficient.

Statement 2 alone is insufficient because we don't know anything about the other sides of the triangle. Therefore it doesn't help us.

Since we are already told that triangle ABC is a right triangle, we just need to find information about other angles or other sides.

Statement 1 allows us to calculate , simply by using the sum of the angles of a triangle, since we know AEC is also a right triangle because AE is the height.

Statement 2 is also sufficient because it allows us to know angle . Indeed, in a right triangle, the height divides the triangles in two triangles with similar properties. Therefore angle is the same as .

Therefore, each statement alone is sufficient.

Assume Statement 1 alone. Let be the measure of the angle of second-greatest measure. Since the measures form an arithmetic sequence, the three angles measure for some common difference . Their sum is , so

However, since we do not know that common difference, we cannot determine the other angle measures, or whether the triangle is acute, right, or obtuse. For example, if , the widest of the angles measures ; if , the widest angle measures . In the former case, the triangle is acute, having all of its angles measure less than ; in the latter case, the triangle is obtuse, having an angle that measures greater than .

Statement 2 alone provides insufficient information; a 30-30-120 triangle and a 30-60-90 triangle both fit this condition as well as the conditions of the measures of the angles of a triangle, but the former is obtuse and the latter is right.

Now assume both statements to be true. Then, since one angle measures by Statement 1, and a second measures , the third measures . This angle is a right angle, so is a right triangle.

Corresponding angles of similar triangles - including the same triangle, considered in different configurations - are congruent. It follows from Statement 1 alone that . The measures of the acute angles of a right triangle add up to , so:

Assume Statement 2 alone. By the 45-45-90 Theorem, since the legs of the right triangle are of equal length, the acute angles of the triangle, one of which is , measures .

Assume Statement 1 alone, and let be the measure of . The sum of the measures of the acute angles of a right triangle, and , is , so

The first two terms of the arithmetic sequence given are and . In an arithmetic sequence, each term is obtained by adding the same number, or common difference. This is the difference of the second and first terms, or

,

which is added to the second term to get the third term:

The third term is , so

, the measure of .

Assume Statement 2 alone. is the right angle of the triangle, so is an acute angle; it will have measure less than . Its measure is a whole number less than 90 which is, by Statement 2, a multiple of 5 and 7. However, both 35 and 70 are multiples of 5 and 7, so can be either, and no information is given that eliminates either choice.

An acute angle must have measure less than ; both non-right angles of the triangle, and , must be acute.

Assume Statement 1 alone. The measure of must be a multiple of 6 and 7 that is less than 90; however, there are at least two such numbers, 42 and 84. With no way to eliminate either choice, Statement 1 alone provides insufficient information.

Statement 2 alone also provides insufficient information, for similar reasons. can have measure 24 or 48, for example; the measure of is 90 minus the measure of , so if cannot be determined definitively, neither can .

Now, assume both statements to be true. 6 and 7 have LCM 42, so any multiple of both must also be a multiple of 42. There are only two multiples of 42 less than 90 - 42 and 84.

If , then, since the measures of the angles of the acute angles of a right triangle total ,

48 is a multiple of both 6 and 8, so this scenario is consistent with both statements.

If , then, since the measures of the angles of the acute angles of a right triangle total ,

6 is not a multiple of 8, so this scenario is inconsistent with Statement 2.

Therefore, it can be determined that .

An acute angle must have measure less than ; both non-right angles of the triangle, and , must be acute, and thus have measure less than .

Assume Statement 1 alone. There are four perfect cubes less than 90 - 1, 8, 27, and 64 - so can have any one of these four degree measures. There is no way to eliminate any one of these four.

Assume Statement 2 alone. The least common multiple of 7 and 9 is 63, so, since is a multiple of both 7 and 9, it must be a multiple of 63. The only multiple of 63 less than 90 is 63 itself, so . The acute angles of a right angle have degree measures that total , so