DSQ: Calculating an angle in an acute / obtuse triangle - GMAT Quantitative

There is an implied condition: . Therefore, with each statement, we have 2 unknown numbers and 2 equations. In this case, we can take a guess that we will be able to find the value of by using each statement alone. It’s better to check by actually solving this problem.

For statement (1), we can plug into . Now we have , which means .

For statement (2), we can rewrite the equation to be and then plug into , making it

Then we can solve for and get .

Statement 1: An isosceles triangle has two equal angles. Since the interior angles of a triangle always sum to , the only possible angles the other sides could have are .

Statement 2: This does not provide any information relevant to the question.

Knowing only the triangle is obtuse only tells you that there is one obtuse angle, but along with the fact that there is a angle, this allows no further conclusions.

Knowing only that the triangle is isosceles, you can deduce from the Isosceles Triangle Theorem that there are two angles of equal measure; as the measures of the three angles are , there are two possibilities: the triangle is a triangle, or it is a triangle, but you cannot choose between the two without further information.

Knowing both facts allows you to choose the first of those two options.

The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

If , then by the Isosceles Triangle Theorem, . Since the sum of the measures of a triangle is 180,

After some substitution,

Since and form a linear pair,

, and

If , then by the Triangle Exterior Angle Theorem,

So either statement by itself provides sufficient information.

From Statement 2:

This is enough to prove the triangle is obtuse.

From Statement 2 we can calculate :

We present two cases to demonstrate that this is not enough information to answer the question:

- right triangle.

- acute triangle.

Each statement alone allows us to calculate the measure of one of the angles by subtracting the sum of the other two from 180.

From Statement 1:

From Statement 2:

Neither statement alone is enough to answer the question, since either statement leaves enough angle measurement to allow one of the other triangles to be right or obtuse. But the two statements together allow us to calculate :

:

This allows us to prove acute.

If we assume Statement 1 alone, that is complementary to , then by definition, . Since ,

This makes a right angle and a right triangle.

Statement 2 alone is inufficient, however, since a triangle with exactly two acute angles can be either right or obtuse.

The measure of each of the three angles of the triangle, being angles inscribed in the circle, is one-half the measure of the arc it intercepts. For the triangle to be equilateral, each angle has to measure , and . This is neither proved nor diproved by Statement 1 alone, since one arc can measure without the other two doing so; it is, however, disproved by Statement 2 alone.

The measure of each of the three angles of the triangle, being angles inscribed in the circle, is one-half the measure of the arc it intercepts. For the triangle to be equilateral, each angle has to measure , and .

Each of the arcs mentioned in the statements is a major arc corresponding to one of these minor arcs, so, specifically, and .

From Statement 1 alone, we can calculate:

This does not prove or disprove to be equilateral, since one minor arc can measure without the other two doing so.

From Statement 2 alone, we can calculate

so we know that is not equilateral.

Every triangle has at least two acute angles, so neither statement is sufficient to answer the question. The two statements together, however, are enough to prove to have three acute angles and to therefore be an acute triangle.

Statement 1 tells us that the triangle is either right or obtuse, but nothing more.

Statement 2 tells us that the triangle is obtuse. An exterior angle of a triangle is supplemetary to the interior angle to which it is adjacent. Since the supplement of an acute angle is obtuse, this means the triangle must have an obtuse angle.

From Statement 1 it can be deduced that . Similarly, from Statement 2 it can be deduced that . Neither statement alone gives information about the other two angles. Both statements together, however, prove that , making the triangle isosceles by the Isosceles Triangle Theorem.

An equilateral triangle has three congruent angles, each of which measure . Statement 1 alone establishes the congruence of two angles but not the third; for example, the triangle could be and fit the condition. Statement 2 alone only establishes the measure of one angle.

Assume both statements are true. The degree measures of the angles of a triangle add up to , and, since , we can set up and solve:

, so in .

An equilateral triangle has three congruent angles, each of which measure . Both statements contradict this condition, proving that is not equilateral.

Exterior angle forms a linear pair with its interior angle . Either both are right, or one is acute and one is obtuse. From Statement 1 alone, since is acute, is obtuse, and is an obtuse triangle.

Statement 2 alone also provides sufficient information; the sum of the measures of interior angles of a triangle is ; since the sum of the measures of two of them, and , is , the other angle, , has measure , making obtuse, and making an obtuse triangle.

Statement 1 is true for any triangle by the Triangle Inequality, which states that the sum of the lengths of any two sides is greater than that of the third. Therefore, Statement 1 provides unhelpful information.

Statement 2 alone, however, proves that is obtuse, since the sum of the squares of the lengths of two sides exceeds the square of the length of the third.

The longest side is opposite the largest angle for all triangles.

(1) Substituting 3 for k means that AB=3 and BC=5. But the value of m given for side AC is still unknown rightarrow NOT sufficient.

(2) Since , the longest side must be either k+3 or m. So, knowing whether m>k+3 is sufficient.

If m=k+4, knowing that k+4>k+3 ,

then m>k+3rightarrow SUFFICIENT.

There is an implied condition: . Therefore, with each statement, we have 2 unknown numbers and 2 equations. In this case, we can take a guess that we will be able to find the value of by using each statement alone. It’s better to check by actually solving this problem.

For statement (1), we can plug into . Now we have , which means .

For statement (2), we can rewrite the equation to be and then plug into , making it

Then we can solve for and get .

Statement 1: An isosceles triangle has two equal angles. Since the interior angles of a triangle always sum to , the only possible angles the other sides could have are .

Statement 2: This does not provide any information relevant to the question.