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.
