Triangles - GMAT Math Study Guide

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Pythagorean Theorem

Table of Contents

  1. Properties of All Triangles
    1. Sum of Angles is 180
    2. Triangle Inequality Theorem
    3. Longest Side Opposite Largest Angle
    4. Exterior Angle Theorem
  2. Types of Triangles
  3. Similar Triangles
    1. When Are Triangles Similar?
    2. Properties of Similar Triangles
  4. Congruent Triangles
    1. When Are Triangles Congruent?
    2. Properties of Congruent Triangles
  5. Area of a Triangle
  6. Types of GMAT Problems

Properties of All Triangles

A triangle is a three-sided shape whose three inner angles must sum to 180°. The largest angle will be across from the longest side while the smallest angle will be across from the shortest side of the triangle. If and only if two sides of a triangle are equal, the angles opposite them will be equal as well.

Sum of Angles is 180

The sum of the interior angles in any triangle is 180°

Triangle Inequality Theorem

The sum of any two sides of a triangle must be greater than the third side of a triangle.

For example, the figure below is not physically possible since the sum of two sides is smaller than the third side.

Longest Side Opposite Largest Angle

The longest side of a triangle is opposite the largest angle of a triangle. Conversely, the smallest side of a triangle is opposite the smallest angle of a triangle. Consider the following example:

Rank the size of the angles of triangle ABC from largest to smallest.

Since side AC is the longest side, angle B is the largest angle.
Since side AB is the next longest side, angle C is the next largest angle.
Since side BC is the shortest side, angle A is the smallest angle.

Exterior Angle Theorem

The exterior angle theorem states:
Angle EFG + Angle EGF = Angle DEG

Consider the following example:

In the figure above, if angle EFG is 60° and angle DEG is 120°, what is the measure of angle FGE?

Angle EFG + Angle EGF = Angle DEG
60 + Angle EGF = 120
Angle EGF = 60

Types Triangles

  • Right Triangle
    • One angle is 90°
  • Scalene
    • Each side has a different length
    • Each angle has a different measurement
  • Obtuse
    • One angle of the triangle is greater than 90°
  • Acute
    • Every angle of the triangle is less than 90°
  • Equilateral
    • Every angle of the triangle is equal (i.e., 60°)
    • Every side of the triangle is equal in length
  • Isosceles
    • Two angles of the triangle are equal
    • Two sides of the triangle are equal in length

Similar Triangles

When Are Triangles Similar?

Two triangles are similar if any one of the following three possible scenarios is met:

  1. AAA [Angle Angle Angle] - The corresponding angles of each triangle have the same measurement.
    In other words, the above triangles are similar if:
    Angle L = Angle O; Angle N = Angle Q; Angle M = Angle P
  2. SAS [Side Angle Side] - An angle in one triangle is the same measurement as an angle in the other triangle and the two sides containing these angles have the same ratio.
    In other words, the above triangles are similar if:
    Angle L = Angle O; Side LM/Side OP = Side LN/Side OQ
    Note: Any other combination of side, angle, side also proves similarity.
  3. SSS [Side Side Side] - Each pair of corresponding sides have the same ratio.
    In other words, the above triangles are similar if:
    Side LM/Side OP = Side LN/Side OQ = Side MN/Side PQ

Properties of Similar Triangles

  1. Corresponding angles are the same measurement.
  2. The perimeter of each triangle is in the same ratio as the sides.
  3. Corresponding sides are all in the same proportion.

Congruent Triangles

When Are Triangles Congruent?

Two triangles are congruent if any one of the following three possible scenarios is met:

  1. SAS [Side Angle Side] - Two pairs of corresponding sides are equal and the corresponding angle between the sides is equal.
    In other words, the above triangles are congruent if:
    Side SW = Side UV; Angle W = Angle V; Side WR = Side VT
    Note: Any other combination of side, angle, side also proves congruence.
  2. ASA [Angle Side Angle] - Two pairs of corresponding angles are equal and the corresponding side between them is equal.
    In other words, the above triangles are congruent if:
    Angle R = Angle T; Side RW = Side TV; Angle W = Angle V
    Note: Any other combination of angle, side, angle also proves congruence.
  3. SSS [Side Side Side] - All three pairs of corresponding sides are equal.
    In other words, the above triangles are congruent if:
    Side RS = Side TU; = Side RW = Side TV; Side SW = Side UV

Properties of Congruent Triangles

  1. Corresponding angles have the same measurement.
  2. Corresponding sides have the same measurement.

Area of a Triangle

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Area

The area of a triangle is given by the following formula:

Area = 1/2(base)(height)
The height of the triangle is the length of the line which is perpendicular to the base and goes through the opposite vertex (i.e., line KH in the triangle below).

To reiterate, the area of a triangle can be found using the equation: A = 1/2bh. In this case, b stands for the base of the triangle and h stands for the height. Any side can be chosen to be the base, but the height is the line that is perpendicular to the base and goes through the opposing vertex. The perimeter of a triangle is the sum of the three sides.

Consider the following example:

Referring to the triangle immediately above, what is the area of triangle KIJ if KJ = 13, KH = 5, IH = 3, and JI = 9?

Area = 1/2(JI)(HK)
Area = 1/2(9)(5) = 22.5

Types of GMAT Problems

  1. Using Triangles to find Diagonals

    For rectangles and rectangular solids, triangles can be used to determine the length of the diagonal. For two-dimensional objects, the Pythagorean theorem can be used to find the length of the diagonal given the lengths of two adjacent sides. To find the length of a diagonal in a rectangular solid, the Pythagorean theorem must first be used to find the diagonal of one side. Then, a new triangle can be formed with the diagonal of the side, the diagonal of the solid and one other edge. Using these three lines and the Pythagorean theorem, the length of the diagonal can be calculated.

    If a cube has a volume of 1000, what is the length of a diagonal that goes through the center of the cube?
    Correct Answer: E
    1. Remember that x(1/2) = the square root of x.
    2. Since volume = V = 1000 = a3, where a is the length of an edge of the cube, a must be 10 since 10001/3 = 10.
    3. The following is a diagram of the cube:
      Figure 1
    4. To find the length of a diagonal of a rectangular solid (i.e., the length from the bottom right to the upper left--the length from D2 to D5), a diagonal for one side must first be calculated.
      A triangle can be formed with two edges of one side and the corresponding diagonal (e.g., the triangle formed by D1, D2, and D7).
    5. Use the Pythagorean theorem to calculate the length of the diagonal of the face touching the ground (e.g., D2 to D7) (the length of this diagonal is equivalent to the length of any other diagonal of a face of the cube since all faces are the same size).
      a2 + b2 = c2; a and b are the lengths of the edges of the cube and c is the diagonal of the side. For a cube, all edges are the same. Thus a = 10, b = 10.
      102 + 102 = c2
      100 + 100 = c2
      c = 200(1/2) = D2 to D7 (do not simplify)
    6. Form a triangle with the diagonal of the side of the cube (D2 to D7, which you just calculated), the diagonal of the solid (i.e., the distance from the bottom right of the cube to the top left of the cube--D2 to D5), and the edge that connects them (i.e., D7 to D5).
    7. Use the Pythagorean theorem to solve for the diagonal of the solid.
      d2 + c2 = e2; where c is the length of the diagonal found earlier, d is the length of the second edge (10), and e is the length of the diagonal for the solid (i.e., the distance from the bottom left to the upper right, or the distance from the bottom right to the upper left since these two distances will be the same in a cube).
      102 + ((200)(1/2))2 = d2
      100 + 200 = d2
      d2 = 300
      d2(1/2) = d = 300(1/2)
      d = 300(1/2) = (100*3)(1/2)
      d = 100(1/2) * 3(1/2) = 10 * (3)(1/2) = D2 to D5
  2. Similar Triangles

    Two triangles are similar if: (a) 3 angles of one triangle are the same as 3 angles of the other triangle (b) 3 pairs of corresponding sides are in the same ratio (c) an angle of 1 triangle is the same as the angle of the other triangle and the sides containing these angles are in the same ratio. For example:

    What is the value of x?
    Figure 1
    Correct Answer: A
    1. The two triangles are similar triangles since they share two sets of common angles. Particularly, both triangles have a 90 degree angle and a 20 degree angle.
    2. The ratio of the corresponding sides of the similar triangles is 14/7 = 2.
    3. To determine x, set up a ratio of the sides' length since one property of common triangles is that their corresponding sides have the same ratio.
      x/3 = 14/7 = 2
      x=6