Now that we found m∠ MLP, we can find m∠ JLK. Notice that ∠ JLK and ∠ MLP are vertical angles, which means that they are congruent. Then, m∠ JLK=m∠ MLP.
In this solution, we used the fact that vertical angles are always congruent. This is true due the Vertical Angle Theorem. We will learn a bit more about this theorem by considering the next diagram.
Analyzing the diagram, it can be observed that ∠ 1 and ∠ 2 form a straight angle, so these are supplementary angles. Similarly, ∠ 2 and ∠ 3 are also supplementary angles.
Therefore, by the Angle Addition Postulate, the sum of m∠ 1 and m∠ 2 is 180^(∘), and the sum of m∠ 2 and m∠ 3 is also 180^(∘). These facts can be used to express m∠ 2 in terms of m∠ 1 and also in terms of m∠ 3.
Angle Addition Postulate
Isolate m∠ 2
m∠ 1+m∠ 2 = 180^(∘)
m∠ 2 = 180^(∘)-m∠ 1
m∠ 2+m∠ 3 = 180^(∘)
m∠ 2 = 180^(∘)-m∠ 3
By the Transitive Property of Equality, the expressions representing m∠ 2 can be set equal to each other.
m∠ 2= 180^(∘)-m∠ 1 m∠ 2= 180^(∘)-m∠ 3
⇓
180^(∘)-m∠ 1= 180^(∘)-m∠ 3
Then the obtained equation can be simplified.
By the definition of congruent angles, this means that the vertical angles ∠ 1 and ∠ 3 are congruent angles. Using the same argumentation, ∠ 2 and ∠ 4 can also be proven to be congruent.
