Why do all angles in a triangle add up to 180?

Answer 1

Draw two parallel lines (with points marked D and E on one, and F and G on the other). Mark point A on one line between D and E, and mark two points B and C on the other line between F and G. Join ABC to for a triangle:

D_____________A_________________E ← point labels

------->---------+---------------------- ← first of the parallel lines

_______________/\__________________

______________/__\_________________

_____________/____\________________

____________/______\_______________

------->----+---------+---------------- ← second of the parallel lines

F________B__________C__________G ← point labels

(The underscores are there to keep the diagram in place - think of them like lines on the page of an exercise book.)

Angles BAD + BAC + CAE add up to 180° as they are three angles on a straight line.

Now angle ABC is the same as angle BAD as they are alternate (Z) angles

Similarly, angle ACB is the same as angle CAE as they are alternate (Z) angles

So substituting for equal angles, this gives:

BAD + BAC + CAE = 180°

→ ABC + BAC + ACB = 180°

But these three angles are the angles of the triangle ABC;

thus the sum of the angles in a triangle is 180°.