According to the Triangle Sum Theorem, the sum of the measures of the interior angles of a triangle is 180^(∘).
m∠1+m∠2+m∠3=180^(∘)
The proof will first show that ∠3, ∠4, and ∠5 form a straight angle. After that, we proceed with showing that we have two sets of congruent angles.
∠1≅ ∠5 and ∠2 ≅ ∠4
This will be enough to prove that the sum of the triangles angles equals 180^(∘).
Proving a straight angle at C
Examining the figure, we see that m∠ACD and ∠5 are a linear pair. By the Linear Pair Postulate, we can say the following.
m∠ACD+m∠5=180^(∘)
Additionally, ∠ACD is made up by ∠3 and ∠4 which means we can use the Angle Addition Postulate to write the following equation.
m∠3+m∠4=m∠ACD
Finally, we can prove that the angle at C is a straight angle by using the Substitution Property of Equality and replace m∠ACD with m∠3+m∠4.
m∠3+m∠4+m∠5=180^(∘)
Proving the Triangle Sum Theorem
Since AB∥ CD, we can view AE as a transversal and say that ∠1 and ∠5 are corresponding angles. According to the Corresponding Angles Theorem, if two parallel lines are cut by a transversal, then the pair of corresponding angles are congruent.
∠1≅ ∠5
Additionally, if we view CD as a transversal, ∠2 and ∠4 are alternate interior angles. According to the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, then the pair of alternate interior angles are congruent.
∠2≅ ∠4
Using the definition of congruent angles, we can create the following equations
m∠1=m∠5 and m∠2=m∠4
Finally, we can prove the Triangle Sum Theorem by using the Substitution Property of Equality and replace m∠5 and m∠4 with m∠1 and m∠2.
m∠3+ m∠2 + m∠1=180^(∘)
Thus the Triangle Sum Theorem is true.
