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

Two angles of an isosceles triangle measure $(x + 20) ^{\circ}$ and $\left ( 100-x\right )^{ \circ}$ . What are the possible values of ?

$x = 40 \textrm{ or }x = 80$

$x = 60 \textrm{ or } x = 80$

Explanation

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.

$2x \div 2 =80 \div 2$

The angle measures are $60 ^{\circ}, 60 ^{\circ}, 60 ^{\circ}$ , 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 $(x + 20) ^{\circ}$ .

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

$\left (x + 20 \right ) +\left (x +20 \right )+ \left ( 100-x \right ) = 180$

as before

Case 3: The third angle has measure $\left ( 100-x\right )^{ \circ}$

$\left (x + 20 \right ) +\left (100-x \right )+ \left ( 100-x \right ) = 180$

as before.

Thus, the only possible value of is 40.

$\angle 1$ , $\angle 2$ , and $\angle 3$ are all exterior angles of $\bigtriangleup ABC$ with vertices , , and , respectively.

Is $\bigtriangleup ABC$ an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: $\angle 1$ , $\angle 2$ , and $\angle 3$ are all obtuse angles.

Statement 2: $\angle A \cong \angle B$ .

Note: For purposes of this problem, $\angle A$ , $\angle B$ , and $\angle C$ will refer to the interior angles of the triangle at these vertices.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation

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 $180^{\circ }$ , must be two right angles or one acute angle and one obtuse angle. Since $\angle 1$ , $\angle 2$ , and $\angle 3$ are all obtuse angles, it follows that their respective interior angles - the three angles of $\bigtriangleup ABC$ - are all acute. This makes $\bigtriangleup ABC$ an acute triangle.

Statement 2 alone provides insufficient information to answer the question. For example, if $\angle A$ and $\angle B$ each measure $10^{\circ }$ and $\angle C$ measures $160^{\circ }$ , the sum of the angle measures is $180^{\circ }$ , $\angle A$ and $\angle B$ are congruent, and $\angle C$ is an obtuse angle (measuring more than $90^{\circ }$ ); this makes $\bigtriangleup ABC$ an obtuse triangle. But if $\angle A$ , $\angle B$ , and $\angle C$ each measure $60^{\circ }$ , the sum of the angle measures is again $180^{\circ }$ , $\angle A$ and $\angle B$ are again congruent, and all three angles are acute (measuring less than $90^{\circ }$ ); this makes $\bigtriangleup ABC$ an acute triangle.

Which of the following cannot be the measure of a base angle of an isosceles triangle?

$100 ^{\circ }$

$80 ^{\circ }$

$1 ^{\circ }$

$50^{\circ }$

Each of the other choices can be the measure of a base angle of an isosceles triangle.

Explanation

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 $90^{\circ }$ . The only choice that does not fit this criterion is $100 ^{\circ }$ , making this the correct choice.

Is $\bigtriangleup ABC$ an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: $m \angle A + m \angle B < 90 ^{\circ }$

Statement 2: $(AC)^{2} + (BC ) ^{2}< (AB) ^{2}$

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation

Assume Statement 1 alone. The sum of the measures of interior angles of a triangle is $180 ^{\circ }$ ;

$m \angle A + m \angle B < 90 ^{\circ }$ , or, equivalently, for some positive number ,

$m \angle A + m \angle B =( 90 - N) ^{\circ }$ ,

$m \angle A + m \angle B + m \angle C = 180 ^{\circ }$

$( 90 - N) ^{\circ }+ m \angle C = 180 ^{\circ }$

$m \angle C = 180 ^{\circ } - ( 90 - N) ^{\circ } = ( 90+N) ^{\circ }$

Therefore, $m \angle C > 90 ^{\circ }$ , making $\angle C$ obtuse, and $\bigtriangleup ABC$ 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 $\bigtriangleup ABC$ is an obtuse triangle.

$\Delta ABC \sim \Delta XYZ$

$m \angle A = 20 ^{\circ }; m \angle Y = 120 ^{\circ }$

Which of the following is true of $\Delta ABC$ ?

$\Delta ABC$ is scalene and obtuse.

$\Delta ABC$ is isosceles and obtuse.

$\Delta ABC$ is scalene and acute.

$\Delta ABC$ may be scalene or isosceles, but it is acute,

$\Delta ABC$ may be scalene or isosceles, but it is obtuse.

Explanation

By similarity, $m \angle B = m \angle Y = 120 ^{\circ }$ .

Since measures of the interior angles of a triangle total $180 ^{\circ }$ ,

$m \angle A + m \angle B + m \angle C = 180$

$20 + 120 + m \angle C = 180$

$140 + m \angle C = 180$

$140 + m \angle C -140 = 180-140$

$m \angle C =40^{\circ }$

Since the three angle measures of $\Delta ABC$ are all different, no two sides measure the same; the triangle is scalene. Also, since $m \angle B = 120 ^{\circ } > 90^{\circ }$ , the angle is obtuse, and $\Delta ABC$ is an obtuse triangle.

Which of the following is true of a triangle with three angles whose measures have an arithmetic mean of $90^{\circ }$ ?

The triangle cannot exist.

The triangle must be right and isosceles.

The triangle must be right but may be scalene or isosceles.

The triangle must be obtuse but may be scalene or isosceles.

The triangle may be right or obtuse but must be scalene.

Explanation

The sum of the measures of three angles of any triangle is 180; therefore, their mean is $180 \div 3 = 60$ , making a triangle with angles whose measures have mean 90 impossible.

The measures of the angles of one triangle, in degrees, are .

The measures of the angles of a second triangle, in degrees, are .

What is ?

Explanation

The degree measures of the angles of a triangle add up to a total of 180, so we can set up the following equations:

From the first triangle:

From the second:

$(2X+ 2Y)\div 2 = 160 \div 2$

These equations form a system of equations that can be solved:

$\underline{3X+ Y = 180}$

, so

and .

Two angles of a triangle measure $47 ^{\circ }$ and $23^{\circ }$ . What is the measure of the third angle?

$110^{\circ }$

$20^{\circ }$

$35^{ \circ }$

$70^{\circ }$

$160^{\circ }$

Explanation

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:

$(180-47-23)^{\circ } = 110^{\circ }$

An exterior angle of $\Delta ABC$ with vertex measures $150^{\circ }$ ; an exterior angle of $\Delta ABC$ with vertex measures $110^{\circ }$ . Which is the following is true of $\Delta ABC$ ?

$\Delta ABC$ is acute and scalene

$\Delta ABC$ is acute and isosceles

$\Delta ABC$ is right and scalene

$\Delta ABC$ is obtuse and scalene

$\Delta ABC$ is obtuse and isosceles

Explanation

An interior angle of a triangle measures $180 ^{\circ }$ minus the degree measure of its exterior angle. Therefore:

$m \angle A = 180 ^{\circ } - 150 ^{\circ } = 30 ^{\circ }$

$m \angle B= 180 ^{\circ } - 110 ^{\circ } = 70 ^{\circ }$

The sum of the degree measures of the interior angles of a triangle is $180 ^{\circ }$ , so

$m \angle C = 180 ^{\circ } - (m \angle A + m \angle B) = 180 ^{\circ } - (30 ^{\circ }+ 70 ^{\circ }) = 180 ^{\circ } -100 ^{\circ } = 80 ^{\circ }$ .

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 interior angles of a triangle have measures $x^{\circ }$ , $y^{\circ }$ , and $2x^{\circ }$ . Also,