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Triangle Angles
Triangle Angles
G.CO.10 Prove theorems about triangles... measures of interior angles of a triangle sum to 180.
Goals for this section:
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
Triangle Angle-Sum Theorem
Theorem: The three angles of a triangle add up to 180.
Theorem: The three angles of a triangle add up to 180.
∠a + ∠b + ∠c = 180
∠a + ∠b + ∠c = 180
Proof of Triangle Angle-Sum Theorem
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
- l is parallel to BC
- ∠1 + ∠a + ∠2 = 180
- ∠2 ≅ ∠c
- ∠1 ≅ ∠b
- ∠b + ∠a + ∠c = 180
Reasons
- Given
- Definition of linear
- Alternate Interior Angles
- Alternate Interior Angles
- Substitution Property
Triangle Exterior Angle Theorem
Triangle Exterior Angle Theorem
Theorem: The sum of the remote interior angles is equal to the exterior angle.
Theorem: The sum of the remote interior angles is equal to the exterior angle.
∠1 + ∠2 = ∠4
∠1 + ∠2 = ∠4
Proof
Proof
Statements
- ∠3 + ∠4 = 180
- ∠1 + ∠2 + ∠3 = 180
- ∠1 + ∠2 + ∠3 = ∠3 + ∠4
- ∠1 + ∠2 + ∠3 - ∠3 = ∠3 - ∠3 + ∠4
- ∠1 + ∠2 = ∠4
Reasons
- Definition of Linear Pair (Supplementary)
- Triangle Angle-Sum Theorem
- Substitution/Transitive Property
- Subtraction Property of Equality
- Simplify
External Resources
External Resources
Keywords: triangle angle sum theorem, triangle exterior angle theorem
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