GMAT Math : DSQ: Calculating an angle in an acute / obtuse triangle

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Example Question #1 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

In \bigtriangleup ABC, if AB=kBC=k+3 and AC=m, which of the three angles of \triangle ABC has the greatest degree measure?

(1) k=3

 (2) m=k+4

Possible Answers:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Correct answer:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Explanation:

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 k+3>k, 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+3\rightarrow SUFFICIENT.

Example Question #2 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

A triangle contains a  angle. What are the other angles in the triangle?

(1) The triangle is isosceles.

(2)The triangle has a perimeter of 12. 

Possible Answers:

Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

EACH statement ALONE is sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Correct answer:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Explanation:

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. 

Example Question #3 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Triangle_b

 

Note: figure NOT drawn to scale.

. What is  ?

Statement 1: 

Statement 2: 

Possible Answers:

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.

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. 

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

Example Question #4 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

In  above, what is the value of ?

(1)

(2)

Possible Answers:

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

 

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

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 .

Example Question #5 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Is triangle  acute, right, or obtuse?

Statement 1: 

Statement 2: 

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

Example Question #6 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Is triangle  acute, right, or obtuse?

Statement 1: 

Statement 2:  

Possible Answers:

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 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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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:

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.

Example Question #7 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Is  an acute, right, or obtuse triangle?

Statement 1:  is complementary to .

Statement 2: The triangle has exactly two acute angles.

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.

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is 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:

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.

Example Question #8 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. Is  an equilateral triangle?

Statement 1: 

Statement 2: 

Possible Answers:

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.

Correct answer:

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

Explanation:

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.

Example Question #9 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. Is  an equilateral triangle?

Statement 1: 

Statement 2: 

Possible Answers:

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.

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is 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:

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.

Example Question #10 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Is  an acute, right, or obtuse triangle?

Statement 1: There are exactly two acute angles.

Statement 2: The exterior angles of the triangle at vertex  are both acute.

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.

EITHER 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.

Correct answer:

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

Explanation:

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.

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