Acute / Obtuse Triangles - GMAT Quantitative

An isosceles triangle has two congruent angles by the Isosceles Triangle Theorem; these are the base angles. Since at least two angles of any triangle must be acute, a base angle must be acute - that is, it must measure under . The only choice that does not fit this criterion is , making this the correct choice.

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

The angle measures are , making the triangle equianglular and, subsequently, equilateral. An equilateral triangle is considered isosceles, so this is a possible scenario.

Case 2: The third angle has measure .

Then, since the sum of the angle measures is 180,

as before

Case 3: The third angle has measure

as before.

Thus, the only possible value of is 40.

The sum of the measures of three angles of any triangle is 180; therefore, their mean is , making a triangle with angles whose measures have mean 90 impossible.

The sum of the measures of three angles of any triangle is 180; therefore, if two angles have measure , the third must have measure . This makes the triangle obtuse. Also, since the triangle has two congruent angles, it is isosceles by the Converse of the Isosceles Triangle Theorem.

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

This is a false statement, indicating that this situation is impossible.

Case 2: The third angle has measure .

Then, since the sum of the angle measures is 180,

This makes the angle measures , a plausible scenario.

Case 3: the third angle has measure

Then, since the sum of the angle measures is 180,

This makes the angle measures , a plausible scenario.

Therefore, either or

By similarity, .

Since measures of the interior angles of a triangle total ,

Since the three angle measures of are all different, no two sides measure the same; the triangle is scalene. Also, since, the angle is obtuse, and is an obtuse triangle.

The sum of the degree measures of the angles of a triangle is 180, so we can subtract the two angle measures from 180 to get the third:

The sum of the measures of the angles of a triangle total , so we can set up and solve for in the following equation:

An interior angle of a triangle measures minus the degree measure of its exterior angle. Therefore:

The sum of the degree measures of the interior angles of a triangle is , so

.

Each angle is acute, so the triangle is acute; each angle is of a different measure, so the triangle has three sides of different measure, making it scalene.

The sum of the exterior angles of a triangle, one per vertex, is . , and are exterior angles at different vertices, so

Since , the following triangles are isoscele: .

If ADC, BDC, and BDA are all isoscele; then:

The angle degrees

The angle degrees, and

The angle degrees

Therefore:

The angle

The angle degrees, and

The angle

Since the sum of angles of a triangle is equal to 180 degrees then:

. So:

.

Now let us solve the equation for x:

(See image below - not drawn on scale)

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 is insufficient. Every triangle has at least two acute angles, and Statement 2 only establishes that and are both acute; the third angle, , can be acute, right, or obtuse, so the question of whether is an acute, right, or obtuse triangle is not settled.

Assume Statement 1 alone. The sum of the measures of interior angles of a triangle is ;

, or, equivalently, for some positive number ,

,

so

Therefore, , making obtuse, and an obtuse triangle.

Assume Statement 2 alone. Since the sum of the squares of the lengths of two sides exceeds the square of the length of the third, it follows that is an obtuse triangle.

Assume Statement 1 alone. An exterior angle of a triangle forms a linear pair with the interior angle of the triangle of the same vertex. The two angles, whose measures total , must be two right angles or one acute angle and one obtuse angle. Since , , and are all obtuse angles, it follows that their respective interior angles - the three angles of - are all acute. This makes an acute triangle.

Statement 2 alone provides insufficient information to answer the question. For example, if and each measure and measures , the sum of the angle measures is , and are congruent, and is an obtuse angle (measuring more than ); this makes an obtuse triangle. But if , , and each measure , the sum of the angle measures is again , and are again congruent, and all three angles are acute (measuring less than ); this makes an acute triangle.

The measures of the interior angles of a triangle have sum , so

Along with , a system of linear equations is formed that can be solved by adding:

The measures of the interior angles of a triangle have sum , so

, or

Along with , a system of linear equations is formed that can be solved by adding:

Of the given choices, 50 comes closest to the correct measure.

The measures of the interior angles of a triangle have sum , so

, or, equivalently,

and are the measures of two interior angles of the second triangle, so if we let be the measure of the third angle, then

By substitution,

and

.

The correct response is .

The measures of the interior angles of a triangle have sum , so

, or, equivalently,

and are the measures of two interior angles of the second triangle, so if we let be the measure of the third angle, then

By substitution,

The correct response is .

The sum of the measures of the interior angles of a triangle is , so it can be determined from Triangle 1 that

From Triangle 2, we can deduce that

By substitution: