Triangle Angles

G.CO.10 Prove theorems about triangles... measures of interior angles of a triangle sum to 180.

Goals for this section:

Students should be able to remember that the three angles that make up a triangle will always add up to 180. From this, students should be able to conclude why the triangle exterior angle theorem is true.

Triangle Angle-Sum Theorem

Theorem: The three angles of a triangle add up to 180.

∠a + ∠b + ∠c = 180

Proof of Triangle Angle-Sum Theorem

Consider the picture to the left for the proof. The dotted line (l) is parallel to the bottom segment (BC) of the triangle.

Statements

  1. l is parallel to BC
  2. ∠1 + ∠a + ∠2 = 180
  3. ∠2 ≅ ∠c
  4. ∠1 ≅ ∠b
  5. ∠b + ∠a + ∠c = 180

Reasons

  1. Given
  2. Definition of linear
  3. Alternate Interior Angles
  4. Alternate Interior Angles
  5. Substitution Property

Triangle Exterior Angle Theorem

Theorem: The sum of the remote interior angles is equal to the exterior angle.

∠1 + ∠2 = ∠4

Proof

Statements

  1. ∠3 + ∠4 = 180
  2. ∠1 + ∠2 + ∠3 = 180
  3. ∠1 + ∠2 + ∠3 = ∠3 + ∠4
  4. ∠1 + ∠2 + ∠3 - ∠3 = ∠3 - ∠3 + ∠4
  5. ∠1 + ∠2 = ∠4


Reasons

  1. Definition of Linear Pair (Supplementary)
  2. Triangle Angle-Sum Theorem
  3. Substitution/Transitive Property
  4. Subtraction Property of Equality
  5. Simplify

Keywords: triangle angle sum theorem, triangle exterior angle theorem