GMAT Math : Triangles

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Example Question #1 : Calculating Whether Acute / Obtuse Triangles Are Congruent

Is it true that $\Delta ABC \cong \Delta DEF$ ?

Suppose you want to answer this question, and you know that $\overline{AB} \cong \overline{DE}$ and $\overline{AC} \cong \overline{DF}$ . Which of the following additional facts would help you to answer this question one way or the other?

Possible Answers:

$\overline{BC} \cong \overline{EF}$

$\overline{AB} \cong \overline{DF}$

None of these statements would be sufficient to answer the question.

$\angle B\cong \angle E$

$\angle C\cong \angle F$

Correct answer:

$\overline{BC} \cong \overline{EF}$

Explanation:

If you know either that $\angle B\cong \angle E$ or $\angle C\cong \angle F$ , you have three congruencies - two sides and a nonincluded angle. This is not enough to establish triangle congruence,

If you know that $\overline{AB} \cong \overline{DF}$ , this, along with the other two statements, establishes that $\overline{AB} \cong \overline{AC} \cong \overline{DE} \cong \overline{DF}$ ; all this proves is that both triangles are isosceles.

If you also know that $\overline{BC} \cong \overline{EF}$ , however, the three statements together make three side congruencies, setting up the Side-Side-Side criterion for triangle congruence.

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Example Question #1 : Acute / Obtuse Triangles

Altitude

Note: Figure NOT drawn to scale

Refer to the above diagram. Which of the following statements is NOT a consequence of the fact that $\Delta ABX \cong \Delta ACX$ ?

Possible Answers:

$\angle AXC$ is a right angle

is the midpoint of $\overline{BC}$

$\overline{AX}$ bisects $\angle BAC$

$\overline{AX}$ is an altitude of $\Delta ABC$

$\Delta ABC$ is an equilateral triangle

Correct answer:

$\Delta ABC$ is an equilateral triangle

Explanation:

We use the congruence of corresponding sides and angles of congruent triangles to prove four of these statements:

$\angle CAX \cong \angle BAX$ , so $\overline{AX}$ bisects $\angle BAC$ .

$\overline{BX} \cong \overline{CX}$ , so is the midpoint of $\overline{BC}$ .

$\angle AXC \cong \angle AXB$ , and, since they form a linear pair, they are supplementary - therefore, they are right angles. Also, by definition, this makes $\overline{AX}$ an altitude.

However, nothing in this congruence proves that $\overline{BC}$ is congruent to the other two sides of $\Delta ABC$ (which are congruent). The correct statement to exclude is that $\Delta ABC$ is equilateral.

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

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

Possible Answers:

$1 ^{\circ }$

$80 ^{\circ }$

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

$100 ^{\circ }$

$50^{\circ }$

Correct answer:

$100 ^{\circ }$

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.

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Example Question #1 : Acute / Obtuse Triangles

Let the three interior angles of a triangle measure x, x+30 , and 2x - 10 . Which of the following statements is true about the triangle?

Possible Answers:

The triangle is scalene and acute.

The triangle is isosceles and acute.

The triangle is isosceles and obtuse.

The triangle is scalene and obtuse.

The triangle is scalene and right.

Correct answer:

The triangle is isosceles and acute.

Explanation:

If these are the measures of the interior angles of a triangle, then they total $180^{\circ }$ . Add the expressions, and solve for .

$x +\left ( x + 30 \right ) +\left ( 2x - 10 \right ) = 180$

4x +20 = 180

4x = 160

x = 40

One angle measures $40^{\circ }$ . The others measure:

$x + 30 = 40 + 30 = 70^{\circ }$

$2x-10 = 2 \cdot 40 - 10 = 80 - 10 = 70 ^{\circ }$

All three angles measure less than $90 ^{\circ }$ , so the triangle is acute. Also, there are two congruent angles, so by the converse of the Isosceles Triangle Theorem, two sides are congruent, and the triangle is isosceles.

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Example Question #81 : Triangles

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

Possible Answers:

x = 40

x = 80

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

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

x = 60

Correct answer:

x = 40

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.

x + 20 = 100 - x

x+ 20 + x = 100 - x + x

2x+ 20 = 100

2x + 20 - 20= 100 -20

2x = 80

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

x = 40

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,

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

x + 140= 180

x + 140 - 140 = 180- 140

x = 40

as before

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

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

220 -x = 180

220 -x -220 = 180-220

-x = -40

x = 40

as before.

Thus, the only possible value of is 40.

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Example Question #82 : Triangles

Two angles of an isosceles triangle measure $\left (x + 30 \right )^{\circ }$ and $\left (x + 36 \right )^{\circ }$ . What are the possible value(s) of ?

Possible Answers:

x= 12

x=6

$x= 6, \; x=26 \textrm{, or }x=28$

$x=26 \textrm{ or }x=28$

$x= 12, \; x=26 \textrm{, or }x=28$

Correct answer:

$x=26 \textrm{ or }x=28$

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.

x + 30 = x + 36

x + 30-x = x + 36-x

30=36

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

Case 2: The third angle has measure $\left (x + 30 \right )^{\circ }$ .

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

$\left (x + 30 \right ) +\left (x + 30 \right )+ \left ( x + 36 \right ) = 180$

3x + 96 = 180

3x =84

x=28

This makes the angle measures $58^{\circ },58^{\circ },64^{\circ }$ , a plausible scenario.

Case 3: the third angle has measure $\left (x + 36 \right )^{\circ }$

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

$\left (x + 30 \right ) +\left (x + 36 \right )+ \left ( x + 36 \right ) = 180$

3x +102 = 180

3x =78

x=26

This makes the angle measures $56^{\circ }, 62^{\circ }, 62^{\circ }$ , a plausible scenario.

Therefore, either x=26 or x=28

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Example Question #83 : Triangles

$\Delta ABC \sim \Delta XYZ$

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

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

Possible Answers:

$\Delta ABC$ is isosceles and obtuse.

$\Delta ABC$ is scalene and obtuse.

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

$\Delta ABC$ is scalene and acute.

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

Correct answer:

$\Delta ABC$ is scalene and 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.

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

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

Possible Answers:

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

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

The triangle cannot exist.

The triangle must be right and isosceles.

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

Correct answer:

The triangle cannot exist.

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.

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Example Question #81 : Triangles

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

Possible Answers:

$35^{ \circ }$

$70^{\circ }$

$110^{\circ }$

$20^{\circ }$

$160^{\circ }$

Correct answer:

$110^{\circ }$

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:

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

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

The angles of a triangle measure $x^{\circ }, x^{\circ },\left ( x -54 \right ) ^{\circ }$ . Evaluate .

Possible Answers:

x = 72

x = 78

x = 112

x = 84

x = 96

Correct answer:

x = 78

Explanation:

The sum of the measures of the angles of a triangle total $180^{\circ }$ , so we can set up and solve for in the following equation:

x + x + (x-54) = 180

3x-54 = 180

3x-54 + 54 = 180+ 54

3x =234

$3x\div 3 =234 \div 3$

x = 78

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GMAT Math : Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating Whether Acute / Obtuse Triangles Are Congruent

Example Question #1 : Acute / Obtuse Triangles

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

Example Question #1 : Acute / Obtuse Triangles

Example Question #81 : Triangles

Example Question #82 : Triangles

Example Question #83 : Triangles

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

Example Question #81 : Triangles

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

All GMAT Math Resources

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