GMAT Exam: A Simple Guide To Geometry - Triangles

Descriptions of Triangles

- equilateral triangle has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle has no congruent sides.
All equilateral triangles are isosceles triangles. However, not all isosceles triangles are equilateral triangles.

- acute triangle has three acute angles.

- right triangle has exactly one right angle.

- obtuse triangle has exactly one obtuse angle.

Useful Information About Triangles
- triangle inequality holds that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Basically, for a triangle to exist, the longest side must be shorter than the sum of the other two sides.

For example, 3, 8, and 10 can be the lengths of the sides of a triangle because 3 + 8 = 11, which is greater than 10, the longest side. But 2, 5, and 8 cannot be the lengths of the sides of a triangle, because 2 + 5 = 7, which is less than 8, the longest side.

- If two sides of a triangle are congruent, then the angles opposite those sides are congruent (that is, base angles of an isosceles triangle are congruent); and, conversely, if two angles of a triangle are congruent, then the sides opposite those angles are congruent.

In the figure shown, implies ; and implies .


- If two sides of a triangle are unequal in length, the measures of the angles opposite the two sides are unequal and the angle opposite the greater side is the greater angle.

- If two angles of a triangle are unequal in measure, the lengths of the sides opposite the two angles are unequal and the side opposite the greater angle is the greater side.

- x° = 75° + 35° = 110°, y° = 70° + 35° = 105°, and z° = 70° + 75° = 145°.


- median of a triangle is a line segment from a vertex of the triangle to the midpoint of the opposite side.

- The medians of a triangle meet in a point that is of the way along each median, from the vertex to the opposite side.

- In a right triangle, the median to the hypotenuse is one-half the length of the hypotenuse.

- The line segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.