To prove the theorem, let's use a two-column proof. This proof lists each statement on the left column and its corresponding justification on the right. Each statement must follow logically from the previous steps. We are given that ∠ 1 and ∠ 3 are vertical angles. This is how we begin our proof!
Statement1)& ∠ 1 and ∠ 3 are vertical
Reason1)& Given
From the diagram, we can see that both ∠ 1 and ∠ 3 form a linear pair with ∠ 2.
Statement2)& ∠ 1 and ∠ 2 form a linear pair
& ∠ 3 and ∠ 2 form a linear pair
Reason2)& Definition of a linear pair,
& as seen in the diagram
Let's show this on the diagram.
Since ∠ 1 and ∠ 2 form a linear pair, by the Linear Pair Postulate, these angles are supplementary. Similarly, since ∠ 3 and ∠ 2 also form a linear pair, we can also say that they are supplementary.
Statement3)& ∠ 1 and ∠ 2 are supplementary
& ∠ 3 and ∠ 2 are supplementary
Reason3)& Linear Pair Postulate
Finally, using the Congruent Supplements Theorem, we can conclude that ∠ 1 is congruent to ∠ 3.
Statement4)& ∠ 1≅ ∠ 3
Reason4)& Congruent Supplements
& Theorem
Let's now write this as a two-column proof.
Statements
Reasons
1.
∠ 1 and ∠ 3 are vertical angles
1.
Given
2.
& ∠ 1 and ∠ 2 are a linear pair &∠ 3 and ∠ 2 are a linear pair
2.
Definition of a linear pair as seen in the diagram
3.
& ∠ 1 and ∠ 2 are supplementary &∠ 3 and ∠ 2 are supplementary
3.
Linear Pair Postulate
4.
∠ 1≅ ∠ 3
4.
Congruent Supplements Theorem
As we can see, in the third row, we need to show that two angles which form a linear pair are supplementary. To do so, we have to use the Linear Pair Postulate. Therefore, our friend is not correct.
