Step by step
The question
Question: prove that the angles in a triangle add up to 180°.
We must show why ∠A + ∠B + ∠C = 180° is always true.
Careful!
A new line is fine.
Its row still needs a reason.
A proof is a table.
On the left: what we know.
On the right: why it is true.
1 Write what we know
Draw a line DE through B, parallel to AC.
DE ∥ AC
2 Use the angle facts
∠1 and ∠A make a Z shape with the two parallel lines.
∠1 = ∠A
∠3 and ∠C make a Z shape too.
∠3 = ∠C
∠1, ∠2 and ∠3 fill the straight line DE.
∠1 + ∠2 + ∠3 = 180°
3 Reach the goal
Put ∠A in place of ∠1, and ∠C in place of ∠3.
∠A + ∠2 + ∠C = 180°
∠2 is the angle of the triangle at B.
∠A + ∠B + ∠C = 180°
4 Check
Let's check: every row has a reason.
And the last row is what we had to prove: ∠A + ∠B + ∠C = 180°.
The proof is done!
Let's look back.
First: write what we know.
Then: use the angle facts.
Then: reach the goal.
Last: check.