Geometry

Triangle Relationships

Big Picture

Triangles are polygons with three sides and three angles. They are classified by their angles as well as by their sides. Like other polygons, triangles  have two  sets  of angles: interior  angles  and exterior angles.  Specifically,  a  triangle’s interior angles always add up to 180°, and the exterior angles add up to 360°.

Key Terms

Triangle: A closed figure made by three line segments intersecting at their endpoints.

Right Triangle: A triangle with a right angle.

Obtuse Triangle: A triangle with an obtuse angle.

Acute Triangle: A triangle with 3 acute angles.

Equiangular Triangle: A triangle whose angles all have the same measure (60°).

Scalene Triangle: A triangle whose sides all have different lengths.

Isosceles Triangle: A triangle with at least two sides of equal length.

Equilateral Triangle: A triangle with three sides of equal length.

Interior Angle: The angle inside of a closed figure with straight sides.

Exterior Angle: The angle formed by one side of a polygon and the extension of the adjacent side.

Corollary: A theorem that follows quickly, easily, and directly from another theorem.

Properties of a Triangle

A triangle is any closed figure formed by three line segments intersecting at their endpoints only.

  1. Three vertices (places where the segments meet)
  2. Three sides (the segments themselves)
  3. Three interior angles (formed at each vertex)
Three vertices

4.  All polygons have two sets of exterior angles - one goes around the triangle clockwise, the other goes around counterclockwise.

exterior angles

5. The interior angle and its adjacent exterior angle form a linear pair by definition.

6. The exterior angles are vertical angles and are therefore congruent.

The interior angle and its adjacent exterior angle

7. If an exterior angle is indicated, the two angles in a triangle that are not adjacent to the exterior angle are called the remote interior angles.

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Geometry

Triangle Relationships Cont.

Properties of a Triangle


Classifying by Angles

Classifying by Angles
  • Right Triangle: A triangle with one right angle.
  • Obtuse Triangle: A triangle with one obtuse angle.
  • Acute Triangle: A triangle with all acute angles.
  • Equiangular Triangle: A triangle with all congruent angles (60°).
  • An equiangular triangle is also an acute triangle.

All THREE angles must be acute in an acute triangle.

Classifying By Side Lengths

Classifying By Side Lengths
  • Scalene Triangle: A triangle where all sides are of different lengths.
  • Isosceles Triangle: A triangle where at least two sides are congruent.
  • Equilateral Triangle: A triangle where all sides are congruent.
  • By definition, an equilateral triangle is also an isos-celes triangle.

Equilateral Triangles Theorem: All equilateral triangles are also equiangular. All equiangular triangles are also equi-lateral.

You can classify a triangle by its angles and side lengths (e.g. an obtuse scalene triangle, an acute isosceles triangle).

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Notation

A triangle is labeled with a \(\triangle\) and its vertices. The order the vertices are listed in does not matter.

Right Triangles

There are some terminology associated with right triangles. The sides adjacent to the right angle are called the legs, while the side opposite to the right angle is called the hypotenuse.

Right Triangles

Isosceles Triangles

Isosceles triangles also have special terminology. The congruent sides of the isosceles triangle are called legs, while the other side is called the base. The angles between the base and the legs are called the base angles. The angle made by the two legs is called the vertex angle.

Isosceles Triangles

Interior and Exterior Angle Theorems

Triangle Sum Theorem: Sum of interior angles in a triangle is always 180°.

  • Applies to any kind of triangle.
  • Can be used to find a missing angle in the triangle.

Corollary to the Triangle Sum Theorem: The acute angles of a right triangle are complementary.

  • m ∠∠A and m ∠∠C are complementary so m ∠A + m ∠C = 90°.∠ A
    Triangle Sum Theorem

    Exterior Angle Sum Theorem: Each set of exterior angles of a polygon add up to 360°.

    • m∠1 + m∠22 + m∠23 = 360° and m∠4 + m∠25 + m∠26 = 360° ∠ A
    • This is true for all polygons, not just triangles.

    Exterior Angle Theorem: Sum of the remote interior angles is equal to the non-adjacent exterior angle.

    • m ∠∠A + m ∠∠B = m∠A CD∠ A
    Triangle Sum Theorem