All GMAT Math Resources
Example Questions
Example Question #11 : Acute / Obtuse Triangles
In the following triangle:
The angle degrees
The angle degrees
(Figure not drawn on scale)
Find the value of .
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)
Example Question #91 : Triangles
Which of the following is true of a triangle with two angles?
The triangle must be obtuse but it can be either scalene or isosceles.
The triangle must be isosceles and acute.
The triangle must be isosceles but it can be acute, right, or obtuse.
The triangle must be isosceles and obtuse.
The triangle must be scalene and obtuse.
The triangle must be isosceles and obtuse.
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.
Example Question #13 : Calculating An Angle In An Acute / Obtuse Triangle
The measures of the interior angles of a triangle are , , and . Also,
.
Evaluate .
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:
Example Question #14 : Calculating An Angle In An Acute / Obtuse Triangle
The interior angles of a triangle have measures , , and . Also,
.
Which of the following is closest to ?
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.
Example Question #346 : Geometry
A triangle has interior angles whose measures are , , and . A second triangle has interior angles, two of whose measures are and . What is the measure of the third interior angle of the second triangle?
None of the other responses gives the correct answer.
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 .
Example Question #15 : Calculating An Angle In An Acute / Obtuse Triangle
The measures of the interior angles of Triangle 1 are , , and . The measures of two of the interior angles of Triangle 2 are and . Which of the following is the measure of the third interior angle of Triangle 2?
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 .
Example Question #16 : Calculating An Angle In An Acute / Obtuse Triangle
Triangle 1 has three interior angles with measures , , and . Triangle 1 has three interior angles with measures , , and .
Express in terms of .
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:
Example Question #349 : Geometry
Is an acute triangle, a right triangle, or an obtuse triangle?
Statement 1:
Statement 2:
STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide 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.
BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.
EITHER STATEMENT ALONE provides sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.
EITHER STATEMENT ALONE provides sufficient information to answer the question.
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.
Example Question #581 : Problem Solving Questions
is an exterior angle of at .
Is an acute triangle, a right triangle, or an obtuse triangle?
Statement 1: is acute.
Statement 2: and are both acute.
BOTH STATEMENTS TOGETHER do 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.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide 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.
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.
Example Question #11 : Calculating An Angle In An Acute / Obtuse Triangle
, , and are all exterior angles of with vertices , , and , respectively.
Is an acute triangle, a right triangle, or an obtuse triangle?
Statement 1: , , and are all obtuse angles.
Statement 2: .
Note: For purposes of this problem, , , and will refer to the interior angles of the triangle at these vertices.
BOTH STATEMENTS TOGETHER do 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.
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.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
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.