Triangles

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GMAT Quantitative › Triangles

Questions 1 - 10

$\angle 1$ is an exterior angle of $\bigtriangleup ABC$ at $\angle A$ .

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

Statement 1: $\angle 1$ is an acute angle.

Statement 2: $m \angle B + m \angle C = 80 ^{\circ }$

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.

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.

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

Explanation

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

Statement 2 alone also provides sufficient information; the sum of the measures of interior angles of a triangle is $180 ^{\circ }$ ; since the sum of the measures of two of them, $m \angle B$ and $m \angle C$ , is $80 ^{\circ }$ , the other angle, $\angle B AC$ , has measure $180 ^{\circ } - 80 ^{\circ } = 100 ^{\circ } > 90 ^{\circ }$ , making $\angle B AC$ obtuse, and making $\bigtriangleup ABC$ an obtuse triangle.

Is $\Delta ABC$ 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.

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.

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.

EITHER statement ALONE is 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.

Two sides of a triangle are 6 and 6. What is the height of the triangle?

(1) The third side of the triangle is also 6.

(2) One of the angles of the triangle is $60^{\circ}$ .

EACH statement ALONE is 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.

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

Explanation

Statement (1) informs us that the triangle is an equilateral triangle, with all sides equal to 6. Therefore, the height would divide the triangle into two 30-60-90 triangles, which have side lengths in a ratio of $1:\sqrt3:2$ . For this triangle the hypotenuse would be 6 and the base would be 3. The height would therefore be $3\sqrt3$ . SUFFICIENT

Statement (2) also lets you deduce that the triangle is an equilateral triangle. Since the triangle is either isoscles or equilateral (at least 2 sides are equal), that means that two angles are equal. Therefore, if one angle is $60^{\circ}$ , the other two also must be $60^{\circ}$ . See above. SUFFICIENT

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

Statement 1:

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

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.

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

Statement 1 is true for any triangle $\bigtriangleup ABC$ 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 $\bigtriangleup ABC$ is obtuse, since the sum of the squares of the lengths of two sides exceeds the square of the length of the third.

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

Statement 1: $\angle A$ is complementary to $\angle C$ .

Statement 2: The triangle has exactly two acute angles.

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.

EITHER statement ALONE is 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.

Explanation

If we assume Statement 1 alone, that $\angle A$ is complementary to $\angle C$ , then by definition, $m \angle A + m \angle C = 90 ^{\circ }$ . Since $m \angle A+ m \angle B + m \angle C = 180 ^{\circ }$ ,

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

$m \angle B +90 ^{\circ } = 180 ^{\circ }$

$m \angle B +90 ^{\circ } -90 ^{\circ } = 180 ^{\circ }-90 ^{\circ }$

$m \angle B =90 ^{\circ }$

This makes $\angle B$ a right angle and $\Delta ABC$ a right triangle.

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

Is $\Delta ABC$ an isosceles triangle?

Statement 1: $m \angle A + m \angle B = 100 ^{\circ }$

Statement 2: $m \angle A + m \angle C = 100 ^{\circ }$

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.

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

Explanation

From Statement 1 it can be deduced that $m \angle C = 180^{\circ } -\left ( m \angle A + m \angle B \right ) = 180^{\circ } - 100^{\circ } = 80^{\circ }$ . Similarly, from Statement 2 it can be deduced that $m \angle B = 80^{\circ }$ . Neither statement alone gives information about the other two angles. Both statements together, however, prove that $\angle B \cong \angle C$ , making the triangle isosceles by the Isosceles Triangle Theorem.

A triangle contains a $110^{\circ}$ angle. What are the other angles in the triangle?

(1) The triangle is isosceles.

(2)The triangle has a perimeter of 12.

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.

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

Explanation

Statement 1: An isosceles triangle has two equal angles. Since the interior angles of a triangle always sum to $180^{\circ}}$ , the only possible angles the other sides could have are $35^{\circ}$ .

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

Find the perimeter of the obtuse $\Delta PGN$ .

I) .

II) ${}PN=\frac{5}6{}GN$ .

Neither statement is sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Explanation

We are told PGN is obtuse, so it has one angle larger than 90 degrees. However, we don't know what that angle is. To find the perimeter we need all three sides.

I) Relates the two shorter sides.

II) Relates the longest side to one of the short sides.

However, we cannot find any of our side lengths, so we cannot find the perimeter.

Give the perimeter of $\bigtriangleup ABC$ .

Statement 1:

Statement 2:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation

The perimeter of $\bigtriangleup ABC$ is equal to the sum of the lengths of the sides; that is, .

From Statement 1 alone, we get

we can add $2 \cdot BC$ to both sides to get

$AB + AC - BC + 2 \cdot BC = 21 + 2 \cdot BC$

$AB + AC + BC = 21 + 2 \cdot BC$

$p= 21 + 2 \cdot BC$

However, without any further information, we cannot determine the actual perimeter.

A similar argument shows that Statement 2 alone gives insufficient information as well.

However, suppose we were to multiply both sides of the equation in Statement 1 by 2, then add both sides of Statement 2:

$\underline{AB + AC +2 BC = 45}$

Divide both sides by 3:

Since

we can substitute 29 for and find :

While we cannot find or individually, this is not necessary; in the perimeter formula, we can substitute 29 for and 8 for :

True or false: The perimeter of $\bigtriangleup ABC$ is greater than 24.

Statement 1:

Statement 2:

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

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

Statement 1 alone gives insufficient information. By the Triangle Inequality Theorem, the sum of the lengths of the shortest two sides of a triangle must be greater than the length of the longest. Examine these two scenarios:

Case 1: