Step by step
The question
Question: prove that vertical angles are equal: ∠1 = ∠3.
We must show why ∠1 = ∠3 is always true.
Prove: = ∠1 = ∠3
Careful!
Every row needs a reason: a fact we know is true.
Looking equal in the picture is not enough.
Careful: "it looks equal" is not a reason
A proof is a table.
On the left: what we know.
On the right: why it is true.
1 Use the angle facts
∠1 and ∠2 sit side by side on a straight line.
∠1 + ∠2 = 180°
∠2 and ∠3 are on a straight line too.
∠2 + ∠3 = 180°
2 Reach the goal
Both sums are 180°.
∠1 + ∠2 = ∠2 + ∠3
Take ∠2 away from both sides.
∠1 = ∠3
3 Check
Let's check: every row has a reason.
And the last row is what we had to prove: ∠1 = ∠3.
The proof is done!
Let's look back.
First: use the angle facts.
Then: reach the goal.
Last: check.
Answer
Prove: = ∠1 = ∠3