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.
